CS-Notes/notes/算法.md

<!-- GFM-TOC -->
* [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
    * [ջ](#ջ)
        * [1. <20><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>](#1-<2D><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>)
        * [2. <20><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>](#2-<2D><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>)
    * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
    * [<EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23>㷨<EFBFBD><E3B7A8><EFBFBD><EFBFBD>)
        * [1. <20><><EFBFBD><EFBFBD>ת<EFBFBD><D7AA>](#1-<2D><><EFBFBD><EFBFBD>ת<EFBFBD><D7AA>)
        * [2. <20><>ѧģ<D1A7><C4A3>](#2-<2D><>ѧģ<D1A7><C4A3>)
        * [3. ThreeSum](#3-threesum)
        * [4. <20><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>](#4-<2D><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>)
        * [5. ע<><D7A2><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#5-ע<><D7A2><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
    * [union-find](#union-find)
        * [1. quick-find <20>㷨](#1-quick-find-<2D>㷨)
        * [2. quick-union <20>㷨](#2-quick-union-<2D>㷨)
        * [3. <20><>Ȩ quick-union <20>㷨](#3-<2D><>Ȩ-quick-union-<2D>㷨)
        * [4. ·<><C2B7>ѹ<EFBFBD><D1B9><EFBFBD>ļ<EFBFBD>Ȩ quick-union <20>㷨](#4-·<><C2B7>ѹ<EFBFBD><D1B9><EFBFBD>ļ<EFBFBD>Ȩ-quick-union-<2D>㷨)
        * [5. <20><><EFBFBD><EFBFBD> union-find <20>㷨<EFBFBD>ıȽ<C4B1>](#5-<2D><><EFBFBD><EFBFBD>-union-find-<2D>㷨<EFBFBD>ıȽ<C4B1>)
* [<EFBFBD>ڶ<EFBFBD><EFBFBD><EFBFBD> <20><><EFBFBD><EFBFBD>](#<23>ڶ<EFBFBD><DAB6><EFBFBD>-<2D><><EFBFBD><EFBFBD>)
    * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨)
        * [Լ<EFBFBD><EFBFBD>](#Լ<><D4BC>)
        * [ѡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#ѡ<><D1A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
        * [ѡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ͳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ıȽ<EFBFBD>](#ѡ<><D1A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ͳ<EFBFBD><CDB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ıȽ<C4B1>)
        * [ϣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#ϣ<><CFA3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
    * [<EFBFBD>鲢<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23>鲢<EFBFBD><E9B2A2><EFBFBD><EFBFBD>)
        * [<EFBFBD>鲢<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23>鲢<EFBFBD><E9B2A2><EFBFBD><EFBFBD>)
        * [<EFBFBD>Զ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>¹鲢<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23>Զ<EFBFBD><D4B6><EFBFBD><EFBFBD>¹鲢<C2B9><E9B2A2><EFBFBD><EFBFBD>)
        * [<EFBFBD>Ե<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϲ鲢<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23>Ե<EFBFBD><D4B5><EFBFBD><EFBFBD>Ϲ鲢<CFB9><E9B2A2><EFBFBD><EFBFBD>)
    * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨](#<23><><EFBFBD><EFBFBD><EFBFBD>㷨)
        * [<EFBFBD>з<EFBFBD>](#<23>з<EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD>ܷ<EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD>ܷ<EFBFBD><DCB7><EFBFBD>)
        * [<EFBFBD>㷨<EFBFBD>Ľ<EFBFBD>](#<23>㷨<EFBFBD>Ľ<EFBFBD>)
            * [<EFBFBD>л<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23>л<EFBFBD><D0BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
            * [<EFBFBD><EFBFBD>ȡ<EFBFBD><EFBFBD>](#<23><>ȡ<EFBFBD><C8A1>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>)
    * [<EFBFBD><EFBFBD><EFBFBD>ȶ<EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD>ȶ<EFBFBD><C8B6><EFBFBD>)
        * [<EFBFBD><EFBFBD>](#<23><>)
        * [<EFBFBD>ϸ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>³<EFBFBD>](#<23>ϸ<EFBFBD><CFB8><EFBFBD><EFBFBD>³<EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>Ԫ<EFBFBD><D4AA>)
        * [ɾ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD>](#ɾ<><C9BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><D4AA>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
    * [Ӧ<EFBFBD><EFBFBD>](#Ӧ<><D3A6>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD>ıȽ<EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD>ıȽ<C4B1>)
        * [Java <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨ʵ<E3B7A8><CAB5>](#java-<2D><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨ʵ<E3B7A8><CAB5>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>зֵĿ<EFBFBD><EFBFBD><EFBFBD>ѡ<EFBFBD><EFBFBD><EFBFBD>㷨](#<23><><EFBFBD><EFBFBD><EFBFBD>зֵĿ<D6B5><C4BF><EFBFBD>ѡ<EFBFBD><D1A1><EFBFBD>㷨)
* [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> <20><><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>-<2D><><EFBFBD><EFBFBD>)
    * [<EFBFBD><EFBFBD><EFBFBD>ű<EFBFBD>](#<23><><EFBFBD>ű<EFBFBD>)
        * [API](#api)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ű<EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ű<EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ķ<EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ķ<EFBFBD><C4B6>ֲ<EFBFBD><D6B2><EFBFBD>)
        * [<EFBFBD>Զ<EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD>ҵķ<EFBFBD><EFBFBD><EFBFBD>](#<23>Զ<EFBFBD><D4B6>ֲ<EFBFBD><D6B2>ҵķ<D2B5><C4B7><EFBFBD>)
    * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
        * [get()](#get)
        * [put()](#put)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
        * [floor()](#floor)
        * [rank()](#rank)
        * [min()](#min)
        * [deleteMin()](#deletemin)
        * [delete()](#delete)
        * [keys()](#keys)
        * [<EFBFBD><EFBFBD><EFBFBD>ܷ<EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD>ܷ<EFBFBD><DCB7><EFBFBD>)
    * [ƽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#ƽ<><C6BD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
        * [2-3 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#2-3-<2D><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD>ڶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD>ڶ<EFBFBD><DAB6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת](#<23><><EFBFBD><EFBFBD>ת)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת](#<23><><EFBFBD><EFBFBD>ת)
            * [<EFBFBD><EFBFBD>ɫת<EFBFBD><EFBFBD>](#<23><>ɫת<C9AB><D7AA>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
            * [ɾ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD>](#ɾ<><C9BE><EFBFBD><EFBFBD>С<EFBFBD><D0A1>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
    * [ɢ<EFBFBD>б<EFBFBD>](#ɢ<>б<EFBFBD>)
        * [ɢ<EFBFBD>к<EFBFBD><EFBFBD><EFBFBD>](#ɢ<>к<EFBFBD><D0BA><EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>̽<EFBFBD>ⷨ<EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>̽<EFBFBD>ⷨ<EFBFBD><E2B7A8>ɢ<EFBFBD>б<EFBFBD>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD>)
            * [ɾ<EFBFBD><EFBFBD>](#ɾ<><C9BE>)
            * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С)
    * [Ӧ<EFBFBD><EFBFBD>](#Ӧ<><D3A6>)
        * [<EFBFBD><EFBFBD><EFBFBD>ַ<EFBFBD><EFBFBD>ű<EFBFBD>ʵ<EFBFBD>ֵıȽ<EFBFBD>](#<23><><EFBFBD>ַ<EFBFBD><D6B7>ű<EFBFBD>ʵ<EFBFBD>ֵıȽ<C4B1>)
        * [Java <20>ķ<EFBFBD><C4B7>ű<EFBFBD>ʵ<EFBFBD><CAB5>](#java-<2D>ķ<EFBFBD><C4B7>ű<EFBFBD>ʵ<EFBFBD><CAB5>)
        * [<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>](#<23><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>)
        * [ϡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˷<EFBFBD>](#ϡ<><CFA1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˷<EFBFBD>)
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# <20><><EFBFBD><EFBFBD>

## ջ

### 1. <20><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>

```java
public class ResizeArrayStack<Item> implements Iterable<Item> {
    
    // <20><>Ҫ<EFBFBD><D2AA> Object <20><><EFBFBD><EFBFBD>Ȼ<EFBFBD><C8BB>ת<EFBFBD>ͣ<EFBFBD><CDA3><EFBFBD><EFBFBD><EFBFBD>ֱ<EFBFBD><D6B1>ʹ<EFBFBD><CAB9> new Item[1];
    private Item[] a = (Item[]) new Object[1];
    // ջ<>е<EFBFBD>Ԫ<EFBFBD>ظ<EFBFBD><D8B8><EFBFBD>
    private int N = 0;

    public void push(Item item) {
        if (N >= a.length) {
            resize(2 * a.length);
        }
        a[N++] = item;
    }

    public Item pop() {
        Item item = a[--N];
        if (N <= a.length / 4) {
            resize(a.length / 2);
        }
        return item;
    }

    // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><D0A1>ʹ<EFBFBD><CAB9>ջ<EFBFBD><D5BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
    private void resize(int size) {
        Item[] tmp = (Item[]) new Object[size];
        for (int i = 0; i < N; i++) {
            tmp[i] = a[i];
        }
        a = tmp;
    }

    public boolean isEmpty() {
        return N == 0;
    }

    public int size() {
        return N;
    }

    @Override
    public Iterator<Item> iterator() {
        // <20><>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĵ<EFBFBD><C4B5><EFBFBD><EFBFBD><EFBFBD>
        return new ReverseArrayIterator();
    }

    private class ReverseArrayIterator implements Iterator<Item> {
        private int i = N;

        @Override
        public boolean hasNext() {
            return i > 0;
        }

        @Override
        public Item next() {
            return a[--i];
        }
    }
}
```

### 2. <20><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>

<EFBFBD><EFBFBD>Ҫʹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͷ<EFBFBD>巨<EFBFBD><EFBFBD>ʵ<EFBFBD>֣<EFBFBD><EFBFBD><EFBFBD>Ϊͷ<EFBFBD>巨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ѹ<EFBFBD><EFBFBD>ջ<EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ<EFBFBD>ͷ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> next ָ<><D6B8>ָ<EFBFBD><D6B8>ǰһ<C7B0><D2BB>ѹ<EFBFBD><D1B9>ջ<EFBFBD><D5BB>Ԫ<EFBFBD>أ<EFBFBD><D8A3>ڵ<EFBFBD><DAB5><EFBFBD>Ԫ<EFBFBD><D4AA>ʹ<EFBFBD>Ϳ<EFBFBD><CDBF><EFBFBD><EFBFBD><EFBFBD>ǰһ<C7B0><D2BB>ѹ<EFBFBD><D1B9>ջ<EFBFBD><D5BB>Ԫ<EFBFBD>س<EFBFBD>Ϊջ<CEAA><D5BB>Ԫ<EFBFBD>ء<EFBFBD>

```java
public class Stack<Item> {
    
    private Node top = null;
    private int N = 0;

    private class Node {
        Item item;
        Node next;
    }

    public boolean isEmpty() {
        return N == 0;
    }

    public int size() {
        return N;
    }

    public void push(Item item) {
        Node newTop = new Node();
        newTop.item = item;
        newTop.next = top;
        top = newTop;
        N++;
    }

    public Item pop() {
        Item item = top.item;
        top = top.next;
        N--;
        return item;
    }
}
```

## <20><><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƕ<EFBFBD><EFBFBD>е<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD>֣<EFBFBD><EFBFBD><EFBFBD>Ҫά<EFBFBD><EFBFBD> first <20><> last <20>ڵ<EFBFBD>ָ<EFBFBD>룬<EFBFBD>ֱ<EFBFBD>ָ<EFBFBD><D6B8><EFBFBD><EFBFBD><EFBFBD>׺Ͷ<D7BA>β<EFBFBD><CEB2>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĸ<EFBFBD>ָ<EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͷ<EFBFBD><EFBFBD><EFBFBD>ڵ㣬<EFBFBD>ĸ<EFBFBD>ָ<EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>β<EFBFBD><EFBFBD><EFBFBD>ڵ㡣<EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>в<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>ö<EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>ص<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD>Ԫ<EFBFBD>س<EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD>ף<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD>׻<EFBFBD>ȡ<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͷ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD> next ָ<><D6B8>ָ<EFBFBD><D6B8><EFBFBD><EFBFBD>һ<EFBFBD><D2BB>Ԫ<EFBFBD>أ<EFBFBD><D8A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ö<EFBFBD><C3B6><EFBFBD>ָ<EFBFBD><D6B8> first ָ<><D6B8><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ<EFBFBD>ͷ<EFBFBD><CDB7>

```java
public class Queue<Item> {
    private Node first;
    private Node last;
    int N = 0;
    private class Node{
        Item item;
        Node next;
    }

    public boolean isEmpty(){
        return N == 0;
    }

    public int size(){
        return N;
    }

    // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
    public void enqueue(Item item){
        Node newNode = new Node();
        newNode.item = item;
        newNode.next = null;
        if(isEmpty()){
            last = newNode;
            first = newNode;
        } else{
            last.next = newNode;
            last = newNode;
        }
        N++;
    }

    // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
    public Item dequeue(){
        Node node = first;
        first = first.next;
        N--;
        return node.item;
    }
}
```

## <20>㷨<EFBFBD><E3B7A8><EFBFBD><EFBFBD>

### 1. <20><><EFBFBD><EFBFBD>ת<EFBFBD><D7AA>

ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD>Ժ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ӷ<EFBFBD><EFBFBD>ں<EFBFBD><EFBFBD><EFBFBD>ͼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʾ<EFBFBD>ĸ<EFBFBD>ֱ<EFBFBD>ۡ<EFBFBD>

T(N)=aN<sup>3</sup> ת<><D7AA>Ϊ lg(T(N))=3lgN+lga

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/5510045a-8f32-487f-a756-463e51a6dab0.png)

### 2. <20><>ѧģ<D1A7><C4A3>

**<2A><><EFBFBD><EFBFBD>**

ʹ<EFBFBD><EFBFBD> \~f(N) <20><><EFBFBD><EFBFBD>ʾ<EFBFBD><CABE><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> N <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> f(N) <20>Ľ<EFBFBD><C4BD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> 1 <20>ĺ<EFBFBD><C4BA><EFBFBD> , <20><><EFBFBD><EFBFBD> N<sup>3</sup>/6-N<sup>2</sup>/2+N/3 \~ N<sup>3</sup>/6<><36>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/ca3a793e-06e5-4ff3-b28e-a9c20540d164.png)

**<2A><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>**

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD>ָ<EFBFBD><EFBFBD>뿪<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ N<sup>3</sup> <20><><EFBFBD><EFBFBD><EFBFBD>Ƿ<EFBFBD><C7B7><EFBFBD> Java ʵ<>֣<EFBFBD><D6A3>Ƿ<EFBFBD><C7B7><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ض<EFBFBD><D8B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>޹ء<DEB9>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/1ea4dc9a-c4dd-46b5-bb11-49f98d57ded1.png)

**<2A><>ѭ<EFBFBD><D1AD>**

ִ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˳<EFBFBD><EFBFBD><EFBFBD>ִ<EFBFBD>е<EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD>䣬<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Щָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ѭ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>

**<2A>ɱ<EFBFBD>ģ<EFBFBD><C4A3>**

ʹ<EFBFBD>óɱ<EFBFBD>ģ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ķ<EFBFBD><EFBFBD>ʴ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD>ֳɱ<EFBFBD>ģ<EFBFBD>͡<EFBFBD>

### 3. ThreeSum

ThreeSum <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͳ<EFBFBD><CDB3>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><D4AA><EFBFBD>ĺ<EFBFBD>Ϊ 0 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

```java
public class ThreeSum {
    public static int count(int[] a) {
        int N = a.length;
        int cnt = 0;
        for (int i = 0; i < N; i++) {
            for (int j = i + 1; j < N; j++) {
                for (int k = j + 1; k < N; k++) {
                    if (a[i] + a[j] + a[k] == 0) {
                        cnt++;
                    }
                }
            }
        }
        return cnt;
    }
}
```

<EFBFBD>ó<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ѭ<EFBFBD><EFBFBD>Ϊ if (a[i] + a[j] + a[k] == 0) <20><><EFBFBD>䣬<EFBFBD>ܹ<EFBFBD>ִ<EFBFBD>еĴ<D0B5><C4B4><EFBFBD>Ϊ N<sup>3</sup>/6-N<sup>2</sup>/2+N/3<><33><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ľ<EFBFBD><C4BD><EFBFBD>ִ<EFBFBD>д<EFBFBD><D0B4><EFBFBD>Ϊ \~N<sup>3</sup>/6<><36><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ N<sup>3</sup><EFBFBD><EFBFBD>

**<2A>Ľ<EFBFBD>**

ͨ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>򣬶<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ö<EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD>ҷ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڸú͵<EFBFBD><EFBFBD>෴<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڣ<EFBFBD><EFBFBD><EFBFBD>˵<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD>ĺ<EFBFBD>Ϊ 0<><30>

<EFBFBD>÷<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Խ<EFBFBD> ThreeSum <20>㷨<EFBFBD><E3B7A8><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ N<sup>2</sup>logN<EFBFBD><EFBFBD>

```java
public class ThreeSumFast {
    public static int count(int[] a) {
        Arrays.sort(a);
        int N = a.length;
        int cnt = 0;
        for (int i = 0; i < N; i++) {
            for (int j = i + 1; j < N; j++) {
                for (int k = j + 1; k < N; k++) {
                    // rank() <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><D4AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>е<EFBFBD><D0B5>±꣬<C2B1><EAA3AC><EFBFBD><EFBFBD>Ԫ<EFBFBD>ز<EFBFBD><D8B2><EFBFBD><EFBFBD>ڣ<EFBFBD><DAA3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>᷵<EFBFBD><E1B7B5> -1<><31>Ӧ<EFBFBD><D3A6>ע<EFBFBD><D7A2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>±<EFBFBD><C2B1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> j<><6A><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ͳ<EFBFBD><CDB2><EFBFBD><EFBFBD>ظ<EFBFBD>ͳ<EFBFBD><CDB3><EFBFBD>ˡ<EFBFBD>
                    if (BinarySearch.rank(-a[i] - a[j], a) > j) {
                        cnt++;
                    }
                }
            }
        }
        return cnt;
    }
}
```

### 4. <20><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD> T(N) \~ aN<sup>b</sup>lgN<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ô T(2N)/T(N) \~ 2<sup>b</sup><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> ThreeSum <20>㷨<EFBFBD><E3B7A8><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1>Ϊ \~N<sup>3</sup>/6<><36><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>б<EFBFBD><D0B1><EFBFBD>ʵ<EFBFBD><CAB5><EFBFBD>õ<EFBFBD><C3B5><EFBFBD><EFBFBD>½<EFBFBD><C2BD><EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/6f5ed46f-86d7-4852-a34f-c1cf1b6343a0.png)

<EFBFBD>ɼ<EFBFBD> T(2N)/T(N)\~2<sup>3</sup><EFBFBD><EFBFBD>Ҳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD> b Ϊ 3<><33>

### 5. ע<><D7A2><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

**<2A><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>**

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ij<EFBFBD><EFBFBD><EFBFBD>ϵ<EFBFBD><EFBFBD><EFBFBD>ܴ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD><EFBFBD><EFBFBD>ƵĽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ǵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ġ<EFBFBD>

**<2A><><EFBFBD><EFBFBD>**

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϵͳ<EFBFBD><EFBFBD>ʹ<EFBFBD>û<EFBFBD><EFBFBD>漼<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֯<EFBFBD>ڴ棬<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>Ԫ<EFBFBD>ػ<EFBFBD><EFBFBD>ȷ<EFBFBD><EFBFBD>ʲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>Ԫ<EFBFBD>ؿ<EFBFBD><EFBFBD>ܶࡣ

**<2A><><EFBFBD><EFBFBD><EEBBB5><EFBFBD><EFBFBD><EFBFBD>µ<EFBFBD><C2B5><EFBFBD><EFBFBD>ܵı<DCB5>֤**

<EFBFBD>ں˷<EFBFBD>Ӧ<EFBFBD>ѡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>е<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>µ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʮ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>ġ<EFBFBD>

**<2A><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨**

ͨ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>룬ȥ<EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

**<2A><>̯<EFBFBD><CCAF><EFBFBD><EFBFBD>**

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>в<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܳɱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Բ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɱ<EFBFBD><EFBFBD><EFBFBD>̯<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ջ<EFBFBD><EFBFBD><EFBFBD><EFBFBD> N <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> push() <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><D4AA>Ϊ N+4+8+16+...+2N=5N-4<><34>N <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>д<EFBFBD><D0B4>Ԫ<EFBFBD>أ<EFBFBD><D8A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ķ<EFBFBD><C4B6>ǵ<EFBFBD><C7B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Сʱ<D0A1><CAB1><EFBFBD>и<EFBFBD><D0B8><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>ķ<EFBFBD><C4B7><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>̯<EFBFBD><CCAF>ÿ<EFBFBD>β<EFBFBD><CEB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ƽ<EFBFBD><C6BD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><CEAA><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

## union-find

**<2A><><EFBFBD><EFBFBD>**

<EFBFBD><EFBFBD><EFBFBD>ڽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>̬<EFBFBD><EFBFBD>ͨ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>⣬<EFBFBD>ܶ<EFBFBD>̬<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㣬<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ж<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӡ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/5d387d02-6f96-44d6-b5d0-4538349f868e.png)

**API**

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/a9b91b7d-65d7-4aa3-8ef6-21876b05ad16.png)

**<2A><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ݽṹ**

```java
public class UF {
    // ʹ<><CAB9> id <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͨ<EFBFBD><CDA8>Ϣ
    private int[] id;

    public UF(int N) {
        id = new int[N];
        for (int i = 0; i < N; i++) {
            id[i] = i;
        }
    }

    public boolean connected(int p, int q) {
        return find(p) == find(q);
    }
}
```

### 1. quick-find <20>㷨

<EFBFBD><EFBFBD>֤<EFBFBD><EFBFBD>ͬһ<EFBFBD><EFBFBD>ͨ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>д<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> id ֵ<><D6B5><EFBFBD>ȡ<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>ַ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Կ<EFBFBD><EFBFBD><EFBFBD>ȡ<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> id ֵ<><D6B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ж<EFBFBD><D0B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƿ<EFBFBD><C7B7><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD><EFBFBD><EFBFBD><EFBFBD> union <20>IJ<EFBFBD><C4B2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȴ<EFBFBD>ܸߣ<DCB8><DFA3><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD><EFBFBD><EFBFBD>е<EFBFBD><D0B5><EFBFBD><EFBFBD>нڵ<D0BD> id ֵ<><D6B5><EFBFBD>޸<EFBFBD>Ϊ<EFBFBD><CEAA>һ<EFBFBD><D2BB><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD> id ֵ<><D6B5>

```java
    public int find(int p) {
        return id[p];
    }
    public void union(int p, int q) {
        int pID = find(p);
        int qID = find(q);

        if (pID == qID) return;
        for (int i = 0; i < id.length; i++) {
            if (id[i] == pID) id[i] = qID;
        }
    }
```

### 2. quick-union <20>㷨

<EFBFBD><EFBFBD> union ʱֻ<CAB1><D6BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> id ֵָ<D6B5><D6B8><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD> id ֵ<><D6B5><EFBFBD><EFBFBD>ֱ<EFBFBD><D6B1><EFBFBD><EFBFBD> id <20><><EFBFBD>洢<EFBFBD><E6B4A2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>͹<EFBFBD><CDB9><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD>õ<EFBFBD><C3B5><EFBFBD><EFBFBD>νṹ<CEBD><E1B9B9><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD>Ҫָ<D2AA><D6B8><EFBFBD>Լ<EFBFBD><D4BC><EFBFBD><EFBFBD>ڽ<EFBFBD><DABD>в<EFBFBD><D0B2><EFBFBD>һ<EFBFBD><D2BB><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1>Ҫһֱ<D2BB><D6B1><EFBFBD>ϲ<EFBFBD><CFB2><EFBFBD>ֱ<EFBFBD><D6B1><EFBFBD><EFBFBD><EFBFBD>ڵ㣬<DAB5><E3A3AC>ʹ<EFBFBD>ø<EFBFBD><C3B8>ڵ<EFBFBD><DAB5><EFBFBD> id ֵ<><D6B5>Ϊ<EFBFBD><CEAA><EFBFBD><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD><EFBFBD><EFBFBD><EFBFBD> idֵ<64><D6B5>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/9192dc0a-a7cd-4030-8df6-e388600644cf.jpg)

```java
    public int find(int p) {
        while (p != id[p]) p = id[p];
        return p;
    }

    public void union(int p, int q) {
        int pRoot = find(p);
        int qRoot = find(q);
        if (pRoot == qRoot) return;
        id[pRoot] = qRoot;
    }
```

<EFBFBD><EFBFBD><EFBFBD>ַ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Կ<EFBFBD><EFBFBD>ٽ<EFBFBD><EFBFBD><EFBFBD> union <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> find <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>߳<EFBFBD><DFB3><EFBFBD><EFBFBD>ȣ<EFBFBD><C8A3><EFBFBD><EEBBB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĸ߶<C4B8>Ϊ<EFBFBD><CEAA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ<EFBFBD><C4BF>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/d206d090-d911-4263-a1fe-d6f63f5d1776.png)

### 3. <20><>Ȩ quick-union <20>㷨

Ϊ<EFBFBD>˽<EFBFBD><EFBFBD><EFBFBD> quick-union <20><><EFBFBD><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD><EFBFBD><EFBFBD>ܸߵ<DCB8><DFB5><EFBFBD><EFBFBD>⣬<EFBFBD><E2A3AC>Ȩ quick-union <20><> union <20><><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1><EFBFBD>ý<EFBFBD>С<EFBFBD><D0A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽϴ<D3BD><CFB4><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>档

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>о<EFBFBD>֤<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ȩ quick-union <20>㷨<EFBFBD><E3B7A8><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>಻<EFBFBD><E0B2BB><EFBFBD><EFBFBD> lgN<67><4E>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/8d6af5ac-74eb-4e07-99aa-654b9f21f1d3.jpg)

```java
public class WeightedQuickUnionUF {
    private int[] id;
    // <20><><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϣ
    private int[] sz;

    public WeightedQuickUnionUF(int N) {
        id = new int[N];
        sz = new int[N];
        for (int i = 0; i < N; i++) {
            id[i] = i;
            sz[i] = 1;
        }
    }

    public boolean connected(int p, int q) {
        return find(p) == find(q);
    }

    public int find(int p) {
        while (p != id[p]) p = id[p];
        return p;
    }

    public void union(int p, int q) {
        int i = find(p);
        int j = find(q);
        if (i == j) return;
        if (sz[i] < sz[j]) {
            id[i] = j;
            sz[j] += sz[i];
        } else {
            id[j] = i;
            sz[i] += sz[j];
        }
    }
}
```

### 4. ·<><C2B7>ѹ<EFBFBD><D1B9><EFBFBD>ļ<EFBFBD>Ȩ quick-union <20>㷨

<EFBFBD>ڼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD>ͬʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ㣬ֻ<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD> find <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB>ѭ<EFBFBD><D1AD><EFBFBD><EFBFBD><EFBFBD>ɡ<EFBFBD>

### 5. <20><><EFBFBD><EFBFBD> union-find <20>㷨<EFBFBD>ıȽ<C4B1>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/e5baeb38-0ec9-4ad7-8374-1cdb0dba74a6.jpg)

# <20>ڶ<EFBFBD><DAB6><EFBFBD> <20><><EFBFBD><EFBFBD>

## <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨

### Լ<><D4BC>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫʵ<EFBFBD><EFBFBD> Java <20><> Comparable <20>ӿڣ<D3BF><DAA3>ýӿ<C3BD><D3BF><EFBFBD> compareTo() <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD>о<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD>ijɱ<EFBFBD>ģ<EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>DZȽϺͽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

ʹ<EFBFBD>ø<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> less() <20><> exch() <20><><EFBFBD><EFBFBD><EFBFBD>бȽϺͽ<CFBA><CDBD><EFBFBD><EFBFBD>IJ<EFBFBD><C4B2><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>ô<EFBFBD><C3B4><EFBFBD><EFBFBD>Ŀɶ<C4BF><C9B6>ԺͿ<D4BA><CDBF><EFBFBD>ֲ<EFBFBD>Ը<EFBFBD><D4B8>á<EFBFBD>

```java
private static boolean less(Comparable v, Comparable w){
    return v.compareTo(w) < 0;
}

private void exch(Comparable[] a, int i, int j){
    Comparable t = a[i];
    a[i] = a[j];
    a[j] = t;
}
```

### ѡ<><D1A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD>ҵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>е<EFBFBD><EFBFBD><EFBFBD>СԪ<EFBFBD>أ<EFBFBD>Ȼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĵ<EFBFBD>һ<EFBFBD><EFBFBD>Ԫ<EFBFBD>ؽ<EFBFBD><EFBFBD><EFBFBD>λ<EFBFBD>á<EFBFBD>Ȼ<EFBFBD><EFBFBD><EFBFBD>ٴ<EFBFBD>ʣ<EFBFBD>µ<EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ҵ<EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĵڶ<EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>ؽ<EFBFBD><EFBFBD><EFBFBD>λ<EFBFBD>á<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ͻ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>IJ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/222768a7-914f-4d64-b874-d98f3b926fb6.jpg)

```java
public class Selection {
    public static void sort(Comparable[] a) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            int min = i;
            for (int j = i + 1; j < N; j++) {
                if (less(a[j], a[min])) min = j;
            }
            exch(a, i, min);
        }
    }
}
```

ѡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ \~N<sup>2</sup>/2 <20>αȽϺ<C8BD> \~N <20>ν<EFBFBD><CEBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>޹أ<DEB9><D8A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ص<EFBFBD>ʹ<EFBFBD><CAB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD>Ѿ<EFBFBD><D1BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD><D2B2>Ҫ<EFBFBD><D2AA>ô<EFBFBD><C3B4><EFBFBD>ıȽϺͽ<CFBA><CDBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD>Ԫ<EFBFBD>ز<EFBFBD><EFBFBD>뵽<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>У<EFBFBD>ʹ<EFBFBD>ò<EFBFBD><EFBFBD><EFBFBD>֮<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ġ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD>ÿ<EFBFBD>β<EFBFBD><EFBFBD><EFBFBD>֮<EFBFBD><EFBFBD><EFBFBD>󲿵<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ġ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/065c3bbb-3ea0-4dbf-8f26-01d0e0ba7db7.png)

```java
public class Insertion {
    public static void sort(Comparable[] a) {
        int N = a.length;
        for (int i = 1; i < N; i++) {
            for (int j = i; j > 0 && less(a[j], a[j - 1]); j--) {
                exch(a, j, j - 1);
            }
        }
    }
}
```

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĸ<EFBFBD><EFBFBD>Ӷ<EFBFBD>ȡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ij<EFBFBD>ʼ˳<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ѿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ˣ<EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܿ졣ƽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>²<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ \~N<sup>2</sup>/4 <20>Ƚ<EFBFBD><C8BD>Լ<EFBFBD> \~N<sup>2</sup>/4 <20>ν<EFBFBD><CEBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EEBBB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ \~N<sup>2</sup>/2 <20>Ƚ<EFBFBD><C8BD>Լ<EFBFBD> \~N<sup>2</sup>/2 <20>ν<EFBFBD><CEBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EEBBB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ģ<EFBFBD><C4A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>õ<EFBFBD><C3B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ N-1 <20>αȽϺ<C8BD> 0 <20>ν<EFBFBD><CEBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>õ<EFBFBD><C3B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ѿ<EFBFBD><D1BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ˡ<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD>ģ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ر<EFBFBD><EFBFBD><EFBFBD>Ч<EFBFBD><EFBFBD>

### ѡ<><D1A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ͳ<EFBFBD><CDB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ıȽ<C4B1>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ظ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ѡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ƽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ģ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֮<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>ij<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### ϣ<><CFA3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>ڴ<EFBFBD><EFBFBD><EFBFBD>ģ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD>ֻ<EFBFBD>ܽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD>Ԫ<EFBFBD>ش<EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>Ƶ<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD>ˣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>ܶ<EFBFBD><EFBFBD>β<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

ϣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ij<EFBFBD><EFBFBD>־<EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD>˸Ľ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>־<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ԣ<EFBFBD><EFBFBD><EFBFBD>ͨ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD>ʹ<EFBFBD><EFBFBD>Ԫ<EFBFBD>ظ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƶ<EFBFBD><EFBFBD><EFBFBD>ȷ<EFBFBD><EFBFBD>λ<EFBFBD><EFBFBD><EFBFBD>ϡ<EFBFBD>

ϣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>ò<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Լ<EFBFBD><EFBFBD><EFBFBD> h <20><><EFBFBD><EFBFBD><EFBFBD>н<EFBFBD><D0BD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> h <20>ܴ<EFBFBD><DCB4><EFBFBD><EFBFBD><EFBFBD>ôԪ<C3B4>ؾ<EFBFBD><D8BE>ܺܿ<DCBA><DCBF><EFBFBD><EFBFBD>Ƶ<EFBFBD><C6B5><EFBFBD>Զ<EFBFBD>ĵط<C4B5><D8B7><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD><EFBFBD><EFBFBD>ϼ<EFBFBD>С h<><68><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> h=1<><31><EFBFBD>Ϳ<EFBFBD><CDBF><EFBFBD>ʹ<EFBFBD><CAB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ġ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/8320bad6-3f91-4a15-8e3d-68e8f39649b5.png)

```java
public class Shell {
    public static void sort(Comparable[] a) {
        int N = a.length;
        int h = 1;
        while (h < N / 3) {
            h = 3 * h + 1;// 1, 4, 13, 40, ...
        }
        while (h >= 1) {
            for (int i = h; i < N; i++) {
                for (int j = i; j >= h && less(a[j], a[j - h]); j -= h) {
                    exch(a, j, j - h);
                }
            }
            h = h / 3;
        }
    }
}
```

ϣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD>ﲻ<EFBFBD><EFBFBD>ƽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>õ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> 1, 4, 13, 40, ... <20><>ϣ<EFBFBD><CFA3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>ıȽϴ<C8BD><CFB4><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ᳬ<EFBFBD><E1B3AC> N <20><><EFBFBD><EFBFBD><EFBFBD>ɱ<EFBFBD><C9B1><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>еij<D0B5><C4B3>ȡ<EFBFBD><C8A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܵĸ߼<C4B8><DFBC><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨ֻ<E3B7A8><D6BB><EFBFBD><EFBFBD>ϣ<EFBFBD><CFA3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ҡ<EFBFBD>

## <20>鲢<EFBFBD><E9B2A2><EFBFBD><EFBFBD>

<EFBFBD>鲢<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˼<EFBFBD><EFBFBD><EFBFBD>ǽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֣<EFBFBD><EFBFBD>ֱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ȼ<EFBFBD><EFBFBD><EFBFBD>鲢<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/dcf265ad-fe35-424d-b4b7-d149cdf239f4.png)

### <20>鲢<EFBFBD><E9B2A2><EFBFBD><EFBFBD>

```java
public class MergeSort {
    private static Comparable[] aux;

    private static void merge(Comparable[] a, int lo, int mid, int hi) {
        int i = lo, j = mid + 1;

        for (int k = lo; k <= hi; k++) {
            aux[k] = a[k]; // <20><><EFBFBD><EFBFBD><EFBFBD>ݸ<EFBFBD><DDB8>Ƶ<EFBFBD><C6B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
        }

        for (int k = lo; k <= hi; k++) {
            if (i > mid) a[k] = aux[j++];
            else if (j > hi) a[k] = aux[i++];
            else if (aux[i].compareTo(a[j]) < 0) a[k] = aux[i++]; // <EFBFBD>Ƚ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֤<EFBFBD>ȶ<EFBFBD><EFBFBD><EFBFBD>
            else a[k] = aux[j++];
        }
    }
}
```

### <20>Զ<EFBFBD><D4B6><EFBFBD><EFBFBD>¹鲢<C2B9><E9B2A2><EFBFBD><EFBFBD>

```java
    public static void sort(Comparable[] a) {
        aux = new Comparable[a.length];
        sort(a, 0, a.length - 1);
    }

    private static void sort(Comparable[] a, int lo, int hi) {
        if (hi <= lo) return;
        int mid = lo + (hi - lo) / 2;
        sort(a, lo, mid);
        sort(a, mid + 1, hi);
        merge(a, lo, mid, hi);
    }
```

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/6468a541-3a9a-4008-82b6-03a0fe941d2a.png)

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/c7665f73-c52f-4ce4-aed3-592bbd76265b.png)

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>׿<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD>ʱ<EFBFBD>临<EFBFBD>Ӷ<EFBFBD>Ϊ O(Nlg<sub>N</sub>)<29><>

<EFBFBD><EFBFBD>ΪС<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĵݹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>ò<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD>齫<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ø<EFBFBD><EFBFBD>ߵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܡ<EFBFBD>

### <20>Ե<EFBFBD><D4B5><EFBFBD><EFBFBD>Ϲ鲢<CFB9><E9B2A2><EFBFBD><EFBFBD>

<EFBFBD>ȹ鲢<EFBFBD><EFBFBD>Щ΢<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬Ȼ<EFBFBD><EFBFBD><EFBFBD>ɶԹ鲢<EFBFBD>õ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>顣

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/c7b9b4c8-83d1-4eb0-8408-ea6576a9ed90.png)

```java
    public static void busort(Comparable[] a) {
        int N = a.length;
        aux = new Comparable[N];
        for (int sz = 1; sz < N; sz += sz) {
            for (int lo = 0; lo < N - sz; lo += sz + sz) {
                merge(a, lo, lo + sz - 1, Math.min(lo + sz + sz - 1, N - 1));
            }
        }
    }
```

## <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### <20><><EFBFBD><EFBFBD><EFBFBD>㷨

<EFBFBD>鲢<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>򣬲<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>鲢ʹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>򣻿<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͨ<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>Ԫ<EFBFBD>ؽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD>ͽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ˡ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/61b4832d-71f3-413c-84b6-237e219b9fdc.png)

```java
public class QuickSort {
    public static void sort(Comparable[] a) {
        shuffle(a);
        sort(a, 0, a.length - 1);
    }

    private static void sort(Comparable[] a, int lo, int hi) {
        if (hi <= lo) return;
        int j = partition(a, lo, hi);
        sort(a, lo, j - 1);
        sort(a, j + 1, hi);
    }
}
```

### <20>з<EFBFBD>

ȡ a[lo] <20><>Ϊ<EFBFBD>з<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD>Ȼ<EFBFBD><C8BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɨ<EFBFBD><C9A8>ֱ<EFBFBD><D6B1><EFBFBD>ҵ<EFBFBD><D2B5><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><D8A3>ٴ<EFBFBD><D9B4><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҷ<EFBFBD><D2B6><EFBFBD><EFBFBD><EFBFBD>ɨ<EFBFBD><C9A8><EFBFBD>ҵ<EFBFBD><D2B5><EFBFBD>һ<EFBFBD><D2BB>С<EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><D8A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><D8A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϼ<EFBFBD><CFBC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>̣<EFBFBD><CCA3>Ϳ<EFBFBD><CDBF>Ա<EFBFBD>֤<EFBFBD><D6A4>ָ<EFBFBD><D6B8><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>ض<EFBFBD><D8B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><D8A3><EFBFBD>ָ<EFBFBD><D6B8> j <20><><EFBFBD>Ҳ<EFBFBD>Ԫ<EFBFBD>ض<EFBFBD><D8B6><EFBFBD>С<EFBFBD><D0A1><EFBFBD>з<EFBFBD>Ԫ<EFBFBD>ء<EFBFBD><D8A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ָ<EFBFBD><D6B8><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>Ԫ<EFBFBD><D4AA> a[lo] <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD><D2B2><EFBFBD>Ԫ<EFBFBD><D4AA> a[j] <20><><EFBFBD><EFBFBD>Ȼ<EFBFBD>󷵻<EFBFBD> j <20><><EFBFBD>ɡ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/e198c201-f386-4491-8ad6-f7e433bf992d.png)

```java
    private static int partition(Comparable[] a, int lo, int hi) {
        int i = lo, j = hi + 1;
        Comparable v = a[lo];
        while (true) {
            while (less(a[++i], v)) if (i == hi) break;
            while (less(v, a[--j])) if (j == lo) break;
            if (i >= j) break;
            exch(a, i, j);
        }
        exch(a, lo, j);
        return j;
    }
```

### <20><><EFBFBD>ܷ<EFBFBD><DCB7><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ԭ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>򣬲<EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬<EFBFBD><EFBFBD><EFBFBD>ǵݹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ջ<EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>õ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD>ζ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>԰<EFBFBD><EFBFBD>֣<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ݹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ٵġ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>±Ƚϴ<EFBFBD><EFBFBD><EFBFBD>Ϊ C<sub>N</sub>=2C<sub>N/2</sub>+N<><4E>Ҳ<EFBFBD><D2B2><EFBFBD>Ǹ<EFBFBD><C7B8>Ӷ<EFBFBD>Ϊ O(Nlg<sub>N</sub>)<29><>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>£<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD>δ<EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD>з֣<EFBFBD><EFBFBD>ڶ<EFBFBD><EFBFBD>δӵڶ<EFBFBD>С<EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD>з֣<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㡣<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>Ƚ<EFBFBD> N<sup>2</sup>/2<><32>Ϊ<EFBFBD>˷<EFBFBD>ֹ<EFBFBD><D6B9><EFBFBD><EFBFBD><EFBFBD>ʼ<EEBFAA><CABC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ģ<EFBFBD><C4A3>ڽ<EFBFBD><DABD>п<EFBFBD><D0BF><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>顣

### <20>㷨<EFBFBD>Ľ<EFBFBD>

#### <20>л<EFBFBD><D0BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Լ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD>飬<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܸ<EFBFBD><EFBFBD>ã<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>п<EFBFBD><EFBFBD><EFBFBD><EFBFBD>л<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

#### <20><>ȡ<EFBFBD><C8A1>

<EFBFBD><EFBFBD><EFBFBD>õ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD>ζ<EFBFBD><EFBFBD><EFBFBD>ȡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>λ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD>з<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ǽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>λ<EFBFBD><EFBFBD><EFBFBD>Ĵ<EFBFBD><EFBFBD>ۺܸߡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƿ<EFBFBD><EFBFBD><EFBFBD>ȡ 3 <20><>Ԫ<EFBFBD>ز<EFBFBD><D8B2><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><D0A1><EFBFBD>е<EFBFBD>Ԫ<EFBFBD><D4AA><EFBFBD><EFBFBD>Ϊ<EFBFBD>з<EFBFBD>Ԫ<EFBFBD>ص<EFBFBD>Ч<EFBFBD><D0A7><EFBFBD><EFBFBD><EFBFBD>á<EFBFBD>

#### <20><><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>д<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ظ<EFBFBD>Ԫ<EFBFBD>ص<EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬<EFBFBD><EFBFBD><EFBFBD>Խ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֣<EFBFBD><EFBFBD>ֱ<EFBFBD><EFBFBD><EFBFBD>ӦС<EFBFBD>ڡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ںʹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>Ԫ<EFBFBD>ء<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>зֿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֻ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɲ<EFBFBD>ͬ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/9d2226dc-c4a3-40ec-9b3e-a46bf86af499.png)

```java
public class Quick3Way {
    public static void sort(Comparable[] a, int lo, int hi) {
        if (hi <= lo) return;
        int lt = lo, i = lo + 1, gt = hi;
        Comparable v = a[lo];
        while (i <= gt) {
            int cmp = a[i].compareTo(v);
            if (cmp < 0) exch(a, lt++, i++);
            else if (cmp > 0) exch(a, i, gt--);
            else i++;
        }
        sort(a, lo, lt - 1);
        sort(a, gt + 1, hi);
    }
}
```

## <20><><EFBFBD>ȶ<EFBFBD><C8B6><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>ȶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD>ڴ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>ء<EFBFBD>

### <20><>

<EFBFBD><EFBFBD><EFBFBD>壺һ<EFBFBD>Ŷ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD><EFBFBD><EFBFBD>ڵ㶼<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ㡣

<EFBFBD>ѿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʾ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>׾ʹ洢<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>λ<EFBFBD><EFBFBD> k <20>Ľڵ<C4BD><DAB5>ĸ<EFBFBD><C4B8>ڵ<EFBFBD>λ<EFBFBD><CEBB>Ϊ k/2<><32><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<D3BD><DAB5><EFBFBD>λ<EFBFBD>÷ֱ<C3B7>Ϊ 2k <20><> 2k+1<><31><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Dz<EFBFBD>ʹ<EFBFBD><CAB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ 0 <20><>λ<EFBFBD>ã<EFBFBD><C3A3><EFBFBD>Ϊ<EFBFBD>˸<EFBFBD><CBB8><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5>Ĺ<EFBFBD>ϵ<EFBFBD><CFB5>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/a9b6c1db-0f4a-4e91-8ac8-6b19bd106b51.png)

```java
public class MaxPQ<Key extends Comparable<Key> {
    private Key[] pq;
    private int N = 0;

    public MaxPQ(int maxN) {
        pq = (Key[]) new Comparable[maxN + 1];
    }

    public boolean isEmpty() {
        return N == 0;
    }

    public int size() {
        return N;
    }

    private boolean less(int i, int j) {
        return pq[i].compareTo(pq[j]) < 0;
    }

    private void exch(int i, int j) {
        Key t = pq[i];
        pq[i] = pq[j];
        pq[j] = t;
    }
}
```

### <20>ϸ<EFBFBD><CFB8><EFBFBD><EFBFBD>³<EFBFBD>

<EFBFBD>ڶ<EFBFBD><EFBFBD>У<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD>ȸ<EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ㡣<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>󻹿<EFBFBD><EFBFBD>ܱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>µĸ<EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD>ϵؽ<EFBFBD><EFBFBD>бȽϺͽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD>ϸ<EFBFBD><EFBFBD><EFBFBD>

```java
private void swim(int k) {
    while (k > 1 && less(k / 2, k)) {
        exch(k / 2, k);
        k = k / 2;
    }
}
```

<EFBFBD><EFBFBD><EFBFBD>Ƶأ<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD>Ҳ<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD>ϵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>±ȽϺͽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD>³<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ㣬Ӧ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>н<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

```java
private void sink(int k) {
    while (2 * k <= N) {
        int j = 2 * k;
        if (j < N && less(j, j + 1)) j++;
        if (!less(k, j)) break;
        exch(k, j);
        k = j;
    }
}
```

### <20><><EFBFBD><EFBFBD>Ԫ<EFBFBD><D4AA>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>طŵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĩβ<EFBFBD><EFBFBD>Ȼ<EFBFBD><EFBFBD><EFBFBD>ϸ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD>λ<EFBFBD>á<EFBFBD>

```java
public void insert(Key v) {
    pq[++N] = v;
    swim(N);
}
```

### ɾ<><C9BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><D4AA>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>鶥<EFBFBD><EFBFBD>ɾ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD>Ԫ<EFBFBD>طŵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ˣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD><EFBFBD>³<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD>λ<EFBFBD>á<EFBFBD>

```java
public Key delMax() {
    Key max = pq[1];
    exch(1, N--);
    pq[N + 1] = null;
    sink(1);
    return max;
}
```

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>ڶѿ<EFBFBD><EFBFBD>Ժ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>׵õ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>ز<EFBFBD>ɾ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϵؽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Եõ<EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>ݼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>غ͵<EFBFBD>ǰ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD>Ԫ<EFBFBD>ؽ<EFBFBD><EFBFBD><EFBFBD>λ<EFBFBD>ã<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD>ɾ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD>Ϳ<EFBFBD><EFBFBD>Եõ<EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>β<EFBFBD><EFBFBD>ͷ<EFBFBD>ĵݼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>У<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD>˺<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>ö<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>򣬲<EFBFBD><EFBFBD>Ҷ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ԭ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>򣬲<EFBFBD>ռ<EFBFBD>ö<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ռ䡣 

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>׶Σ<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>׶<EFBFBD><EFBFBD>ǰ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>齨<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>ѣ<EFBFBD><EFBFBD>ڶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>׶<EFBFBD><EFBFBD>ǽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>غ͵<EFBFBD>ǰ<EFBFBD>ѵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ҽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>³<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ά<EFBFBD>ֶѵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>״̬<EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>齨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֱ<EFBFBD>ӵķ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ǵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ұ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬Ȼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϸ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ч<EFBFBD>ķ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ǵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>³<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ㶼<EFBFBD>Ѿ<EFBFBD><EFBFBD>Ƕ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>³<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD>Ķ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҷ<EFBFBD>ӽڵ㲻<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>³<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˿<EFBFBD><EFBFBD>Ժ<EFBFBD><EFBFBD><EFBFBD>Ҷ<EFBFBD>ӽڵ<EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֻ<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>ؼ<EFBFBD><EFBFBD>ɡ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/a2670745-a7b1-497b-90a4-dbddc4e2006d.jpg)

```java
public static void sort(Comparable[] a){
    int N = a.length;
    for(int k = N/2; k >= 1; k--){
        sink(a, k, N);
    }
    while(N > 1){
        exch(a, 1, N--);
        sink(a, 1, N);
    }
}
```

### <20><><EFBFBD><EFBFBD>

һ<EFBFBD><EFBFBD><EFBFBD>ѵĸ߶<EFBFBD>Ϊ lgN<67><4E><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڶ<EFBFBD><DAB6>в<EFBFBD><D0B2><EFBFBD>Ԫ<EFBFBD>غ<EFBFBD>ɾ<EFBFBD><C9BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>صĸ<D8B5><C4B8>Ӷȶ<D3B6>Ϊ lgN<67><4E>

<EFBFBD><EFBFBD><EFBFBD>ڶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD> N <20><><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>³<EFBFBD><C2B3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˸<EFBFBD><CBB8>Ӷ<EFBFBD>Ϊ NlgN<67><4E>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱһ<EFBFBD><EFBFBD>ԭ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>û<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ö<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀռ䡣

<EFBFBD>ִ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϵͳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>ö<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD>޷<EFBFBD><EFBFBD><EFBFBD><EFBFBD>û<EFBFBD><EFBFBD>棬Ҳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>غ<EFBFBD><EFBFBD>ٺ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>Ԫ<EFBFBD>ؽ<EFBFBD><EFBFBD>бȽϡ<EFBFBD>

## Ӧ<><D3A6>

### <20><><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD>ıȽ<C4B1>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/be53c00b-2534-4dc6-ad03-c55995c47db9.jpg)

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͨ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ѭ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>٣<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>û<EFBFBD><EFBFBD>棬<EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˳<EFBFBD><EFBFBD><EFBFBD>ط<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ݡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ $\~cNlgN$<24><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> c <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Զ<EFBFBD><D4B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><E3B7A8>ҪС<D2AA><D0A1>ʹ<EFBFBD><CAB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>з<EFBFBD>֮<EFBFBD><D6AE><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>Ӧ<EFBFBD><D3A6><EFBFBD>п<EFBFBD><D0BF>ܳ<EFBFBD><DCB3>ֵ<EFBFBD>ijЩ<C4B3>ֲ<EFBFBD><D6B2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܹ<EFBFBD><DCB9>ﵽ <20><><EFBFBD>Լ<EFBFBD><D4BC>𣬶<EFBFBD><F0A3ACB6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><E3B7A8>Ȼ<EFBFBD><C8BB>Ҫ<EFBFBD><D2AA><EFBFBD>Զ<EFBFBD><D4B6><EFBFBD>ʱ<EFBFBD>䡣

### Java <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨ʵ<E3B7A8><CAB5>

Java ϵͳ<CFB5><CDB3><EFBFBD>е<EFBFBD><D0B5><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD>򷽷<EFBFBD>Ϊ java.util.Arrays.sort()<29><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ԭʼ<D4AD><CABC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD><CAB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>зֵĿ<D6B5><C4BF><EFBFBD><EFBFBD><EFBFBD><EFBFBD>򣬶<EFBFBD><F2A3ACB6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>ù鲢<C3B9><E9B2A2><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### <20><><EFBFBD><EFBFBD><EFBFBD>зֵĿ<D6B5><C4BF><EFBFBD>ѡ<EFBFBD><D1A1><EFBFBD>㷨

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> partition() <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ὣ<EFBFBD><E1BDAB><EFBFBD><EFBFBD><EFBFBD><EFBFBD> a[lo] <20><> a[hi] <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>򲢷<EFBFBD><F2B2A2B7><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD> j ʹ<><CAB9> a[lo..j-1] С<>ڵ<EFBFBD><DAB5><EFBFBD> a[j]<5D><><EFBFBD><EFBFBD> a[j+1..hi] <20><><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD> a[j]<5D><><EFBFBD><EFBFBD>ô<EFBFBD><C3B4><EFBFBD><EFBFBD> j=k<><6B>a[j] <20><><EFBFBD>ǵ<EFBFBD> k <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Լ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ģ<EFBFBD><EFBFBD><EFBFBD>Ϊÿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ý<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֣<EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD>Ƚϵ<EFBFBD><EFBFBD>ܴ<EFBFBD><EFBFBD><EFBFBD>Ϊ (N+N/2+N/4+..)<29><>ֱ<EFBFBD><D6B1><EFBFBD>ҵ<EFBFBD><D2B5><EFBFBD> k <20><>Ԫ<EFBFBD>أ<EFBFBD><D8A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȻС<C8BB><D0A1> 2N<32><4E>

```java
    public static Comparable select(Comparable[] a, int k) {
        int lo = 0, hi = a.length - 1;
        while (hi > lo) {
            int j = partion(a, lo, hi);
            if (j == k) return a[k];
            else if (j > k) hi = j - 1;
            else lo = j + 1;
        }
        return a[k];
    }
```


# <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD> <20><><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>־<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD>ָ<EFBFBD>Ч<EFBFBD>ķ<EFBFBD><EFBFBD>ű<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD><EFBFBD><EFBFBD>

## <20><><EFBFBD>ű<EFBFBD>

### API

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/b69d7184-ab62-4957-ba29-fb4fa25f9b65.jpg)

<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵΪ null ʱ<><CAB1><EFBFBD><EFBFBD>ʾ<EFBFBD><CABE><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˿<EFBFBD><CBBF><EFBFBD>ʹ<EFBFBD><CAB9> put(key, null) <20><>Ϊ delete(key) <20><>һ<EFBFBD><D2BB><EFBFBD>ӳ<EFBFBD>ʵ<EFBFBD>֡<EFBFBD>

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ű<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/ba6ae411-82da-4d86-a434-6776d1731e8e.jpg)

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ű<EFBFBD><EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD>Ҫʵ<EFBFBD><EFBFBD> Comparable <20>ӿڡ<D3BF>

<EFBFBD><EFBFBD><EFBFBD>ҵijɱ<EFBFBD>ģ<EFBFBD>ͣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ıȽϴ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>бȽ<EFBFBD>ʱʹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ķ<EFBFBD><EFBFBD>ʴ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5>

<EFBFBD><EFBFBD><EFBFBD>Ӷȣ<EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>ձ<EFBFBD><EFBFBD>в<EFBFBD><EFBFBD><EFBFBD> N <20><><EFBFBD><EFBFBD>ͬ<EFBFBD>ļ<EFBFBD><C4BC><EFBFBD>Ҫ \~N<sup>2</sup>/2 <20>αȽϡ<C8BD>

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ķ<EFBFBD><C4B6>ֲ<EFBFBD><D6B2><EFBFBD>

ʹ<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD>ƽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬һ<EFBFBD><EFBFBD><EFBFBD>洢<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>洢ֵ<EFBFBD><EFBFBD>

<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD> Key <20><><EFBFBD>͵<EFBFBD> Comparable <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB> Value <20><><EFBFBD>͵<EFBFBD> Object <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>顣

rank() <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڱ<EFBFBD><DAB1><EFBFBD>ʱ<EFBFBD><CAB1><EFBFBD><EFBFBD><EFBFBD>ܹ<EFBFBD>֪<EFBFBD><D6AA><EFBFBD>ü<EFBFBD><C3BC><EFBFBD>λ<EFBFBD>ã<EFBFBD><C3A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڱ<EFBFBD><DAB1><EFBFBD>ʱ<EFBFBD><CAB1><EFBFBD><EFBFBD>Ҳ<EFBFBD><D2B2>֪<EFBFBD><D6AA><EFBFBD>ںδ<DABA><CEB4><EFBFBD><EFBFBD><EFBFBD><EFBFBD>¼<EFBFBD><C2BC><EFBFBD>

```java
public class BinarySearchST<Key extends Comparable<Key>, Value> {
    private Key[] keys;
    private Value[] values;
    private int N;

    public BinarySearchST(int capacity) {
        keys = (Key[]) new Comparable[capacity];
        values = (Value[]) new Object[capacity];
    }

    public int size() {
        return N;
    }

    public Value get(Key key) {
        int i = rank(key);
        if (i < N && keys[i].compareTo(key) == 0) {
            return values[i];
        }
        return null;
    }

    public int rank(Key key) {
        int lo = 0, hi = N - 1;
        while (lo <= hi) {
            int mid = lo + (hi - lo) / 2;
            int cmp = key.compareTo(keys[mid]);
            if (cmp == 0) return mid;
            else if (cmp < 0) hi = mid - 1;
            else lo = mid + 1;
        }
        return lo;
    }

    public void put(Key key, Value value) {
        int i = rank(key);
        if (i < N && keys[i].compareTo(key) == 0) {
            values[i] = value;
            return;
        }
        for (int j = N; j > i; j--) {
            keys[j] = keys[j - 1];
            values[j] = values[j - 1];
        }
        keys[i] = key;
        values[i] = value;
        N++;
    }

    public Key ceiling(Key key){
        int i = rank(key);
        return keys[i];
    }
}
```

### <20>Զ<EFBFBD><D4B6>ֲ<EFBFBD><D6B2>ҵķ<D2B5><C4B7><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>Ӷȣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ lgN+1 <20>αȽϣ<C8BD>ʹ<EFBFBD>ö<EFBFBD><C3B6>ֲ<EFBFBD><D6B2><EFBFBD>ʵ<EFBFBD>ֵķ<D6B5><C4B7>ű<EFBFBD><C5B1>IJ<EFBFBD><C4B2>Ҳ<EFBFBD><D2B2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA>ʱ<EFBFBD><CAB1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ƕ<EFBFBD><C7B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ġ<EFBFBD><C4A1><EFBFBD><EFBFBD>Dz<EFBFBD><C7B2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>ƶ<EFBFBD><C6B6><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD><D8A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Լ<EFBFBD><D4BC><EFBFBD><EFBFBD>ġ<EFBFBD>

## <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>壺<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊһ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӵĽڵ㣬ÿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ӷ<EFBFBD>ָ<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>Ӷ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>BST<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD>Ŷ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>е<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/25226bb2-92cc-40cb-9e7f-c44e79fbb64a.jpg)

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>IJ<EFBFBD><EFBFBD>Ҳ<EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD>ε<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD>룬<EFBFBD>Ͷ<EFBFBD><EFBFBD>ֲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ơ<EFBFBD>

```java
public class BST<Key extends Comparable<Key>, Value> {
    private Node root;

    private class Node {
        private Key key;
        private Value val;
        private Node left, right;
        private int N; // <20>Ըýڵ<C3BD>Ϊ<EFBFBD><CEAA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>нڵ<D0BD><DAB5><EFBFBD><EFBFBD><EFBFBD>

        public Node(Key key, Value val, int N) {
            this.key = key;
            this.val = val;
            this.N = N;
        }
    }

    public int size() {
        return size(root);
    }

    private int size(Node x) {
        if (x == null) return 0;
        return x.N;
    }
}
```

### get()

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ǿյģ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>δ<EFBFBD><EFBFBD><EFBFBD>У<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ҵļ<EFBFBD><EFBFBD>͸<EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>У<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ݹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>в<EFBFBD><EFBFBD>ң<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ҵļ<EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>в<EFBFBD><EFBFBD>ң<EFBFBD><EFBFBD>ϴ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>в<EFBFBD><EFBFBD>ҡ<EFBFBD>

```java
public Value get(Key key) {
    return get(root, key);
}
private Value get(Node x, Key key) {
    if (x == null) return null;
    int cmp = key.compareTo(x.key);
    if (cmp == 0) return x.val;
    else if (cmp < 0) return get(x.left, key);
    else return get(x.right, key);
}
```

### put()

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>У<EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD>½ڵ㣬<EFBFBD><EFBFBD><EFBFBD>Ҹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϲ<EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>øýڵ<EFBFBD><EFBFBD><EFBFBD>ȷ<EFBFBD><EFBFBD><EFBFBD>ӵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>

```java
public void put(Key key, Value val) {
    root = put(root, key, val);
}
private Node put(Node x, Key key, Value val) {
    if (x == null) return new Node(key, val, 1);
    int cmp = key.compareTo(x.key);
    if (cmp == 0) x.val = val;
    else if (cmp < 0) x.left = put(x.left, key, val);
    else x.right = put(x.right, key, val);
    x.N = size(x.left) + size(x.right) + 1;
    return x;
}
```

### <20><><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD>ȡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>״<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>״<EFBFBD><EFBFBD>ȡ<EFBFBD><EFBFBD><EFBFBD>ڼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ⱥ<EFBFBD>˳<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>õ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȫƽ<EFBFBD><EFBFBD><EFBFBD>ģ<EFBFBD>ÿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ӻ͸<EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD>ľ<EFBFBD><EFBFBD>붼Ϊ lgN<67><4E><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EEBBB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>£<EFBFBD><C2A3><EFBFBD><EFBFBD>ĸ߶<C4B8>Ϊ N<><4E>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/73a3983d-dd18-4373-897e-64b706a7e370.jpg)

<EFBFBD><EFBFBD><EFBFBD>Ӷȣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ҺͲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### floor()

<EFBFBD><EFBFBD><EFBFBD><EFBFBD> key С<>ڸ<EFBFBD><DAB8>ڵ<EFBFBD><DAB5><EFBFBD> key<65><79><EFBFBD><EFBFBD>ôС<C3B4>ڵ<EFBFBD><DAB5><EFBFBD> key <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>У<EFBFBD><D0A3><EFBFBD><EFBFBD><EFBFBD> key <20><><EFBFBD>ڸ<EFBFBD><DAB8>ڵ<EFBFBD><DAB5><EFBFBD> key<65><79>ֻ<EFBFBD>е<EFBFBD><D0B5><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>д<EFBFBD><D0B4><EFBFBD>С<EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD> key <20>Ľڵ㣬С<E3A3AC>ڵ<EFBFBD><DAB5><EFBFBD> key <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>У<EFBFBD><D0A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD> key <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ㡣

```java
public Key floor(Key key) {
    Node x = floor(root, key);
    if (x == null) return null;
    return x.key;
}
private Node floor(Node x, Key key) {
    if (x == null) return null;
    int cmp = key.compareTo(x.key);
    if (cmp == 0) return x;
    if (cmp < 0) return floor(x.left, key);
    Node t = floor(x.right, key);
    if (t != null) {
        return t;
    } else {
        return x;
    }
}
```

### rank()

```java
public int rank(Key key) {
    return rank(key, root);
}
private int rank(Key key, Node x) {
    if (x == null) return 0;
    int cmp = key.compareTo(x.key);
    if (cmp == 0) return size(x.left);
    else if (cmp < 0) return rank(key, x.left);
    else return 1 + size(x.left) + rank(key, x.right);
}
```

### min()

```java
private Node min(Node x) {
    if (x.left == null) return x;
    return min(x.left);
}
```

### deleteMin()

<EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/6e2cb20a-8d2a-46fe-9ac7-68a2126b7bd5.jpg)

```java
public void deleteMin() {
    root = deleteMin(root);
}
public Node deleteMin(Node x) {
    if (x.left == null) return x.right;
    x.left = deleteMin(x.left);
    x.N = size(x.left) + size(x.right) + 1;
    return x;
}
```

### delete()

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɾ<EFBFBD><EFBFBD><EFBFBD>Ľڵ<EFBFBD>ֻ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ôֻ<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD>Ψһ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD>ڵ<EFBFBD><EFBFBD>滻<EFBFBD>ýڵ㡣

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/b488282d-bfe0-464f-9e91-1f5b83a975bd.jpg)

```java
public void delete(Key key) {
    root = delete(root, key);
}
private Node delete(Node x, Key key) {
    if (x == null) return null;
    int cmp = key.compareTo(x.key);
    if (cmp < 0) x.left = delete(x.left, key);
    else if (cmp > 0) x.right = delete(x.right, key);
    else {
        if (x.right == null) return x.left;
        if (x.left == null) return x.right;
        Node t = x;
        x = min(t.right);
        x.right = deleteMin(t.right);
        x.left = t.left;
    }
    x.N = size(x.left) + size(x.right) + 1;
    return x;
}
```

### keys()

<EFBFBD><EFBFBD><EFBFBD>ö<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ľ<EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>е<EFBFBD><EFBFBD>ص㡣

```java
public Iterable<Key> keys(Key lo, Key hi) {
    Queue<Key> queue = new LinkedList<>();
    keys(root, queue, lo, hi);
    return queue;
}
private void keys(Node x, Queue<Key> queue, Key lo, Key hi) {
    if (x == null) return;
    int cmpLo = lo.compareTo(x.key);
    int cmpHi = hi.compareTo(x.key);
    if (cmpLo < 0) keys(x.left, queue, lo, hi);
    if (cmpLo <= 0 && cmpHi >= 0) queue.add(x.key);
    if (cmpHi > 0) keys(x.right, queue, lo, hi);
}
```

### <20><><EFBFBD>ܷ<EFBFBD><DCB7><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>Ӷȣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>в<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD>ʱ<EFBFBD>䶼<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĸ߶ȳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȡ<EFBFBD>

## ƽ<><C6BD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### 2-3 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/2548f2ec-7b00-4ec7-b286-20fc3022e084.jpg)

һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ƽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD> 2-3 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>п<EFBFBD><D0BF><EFBFBD><EFBFBD>ӵ<EFBFBD><D3B5><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5>ľ<EFBFBD><C4BE><EFBFBD>Ӧ<EFBFBD><D3A6><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͬ<EFBFBD>ġ<EFBFBD>

#### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֮<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ 4- <20>ڵ<EFBFBD>ʱ<EFBFBD><CAB1><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA> 4- <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD>ѳ<EFBFBD> 3 <20><> 2- <20>ڵ㣬<DAB5><E3A3AC><EFBFBD><EFBFBD><EFBFBD>м<EFBFBD><D0BC><EFBFBD> 2- <20>ڵ<EFBFBD><DAB5>Ƶ<EFBFBD><C6B5>ϲ<EFBFBD><CFB2>ڵ<EFBFBD><DAB5>У<EFBFBD><D0A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ʋ<EFBFBD><C6B2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ 4- <20>ڵ<EFBFBD><DAB5><EFBFBD>һֱ<D2BB><D6B1><EFBFBD>з<EFBFBD><D0B7><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ƣ<EFBFBD>ֱ<EFBFBD><D6B1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ 4- <20>ڵ㡣

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/912174d8-0786-4222-b7ef-a611d36e5db9.jpg)

#### <20><><EFBFBD><EFBFBD>

2-3 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ı任<C4B1><E4BBBB><EFBFBD>Ǿֲ<C7BE><D6B2>ģ<EFBFBD><C4A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>صĽڵ<C4BD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֮<EFBFBD>ⲻ<EFBFBD><E2B2BB><EFBFBD>޸Ļ<DEB8><C4BB>߼<EFBFBD><DFBC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֣<EFBFBD><D6A3><EFBFBD><EFBFBD><EFBFBD>Щ<EFBFBD>ֲ<EFBFBD><D6B2>任<EFBFBD><E4BBBB><EFBFBD><EFBFBD>Ӱ<EFBFBD><D3B0><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȫ<EFBFBD><C8AB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ժ<EFBFBD>ƽ<EFBFBD><C6BD><EFBFBD>ԡ<EFBFBD>

2-3 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>IJ<EFBFBD><C4B2>ҺͲ<D2BA><CDB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӶȺͲ<C8BA><CDB2><EFBFBD>˳<EFBFBD><CBB3><EFBFBD>޹أ<DEB9><D8A3><EFBFBD><EFBFBD><EFBFBD><EEBBB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>²<EFBFBD><C2B2>ҺͲ<D2BA><CDB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵĽڵ<C4BD><DAB5><EFBFBD>Ȼ<EFBFBD><C8BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD> logN <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> 10 <20>ڸ<EFBFBD><DAB8>ڵ<EFBFBD><DAB5><EFBFBD> 2-3 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֻ<EFBFBD><D6BB>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD> 30 <20><><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD>ܽ<EFBFBD><DCBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>IJ<EFBFBD><C4B2>ҺͲ<D2BA><CDB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### <20><><EFBFBD>ڶ<EFBFBD><DAB6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

2-3 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>õ<EFBFBD> 2- <20>ڵ<EFBFBD><DAB5><EFBFBD> 3- <20>ڵ㣬<DAB5><E3A3AC><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>ú<EFBFBD><C3BA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5> 3- <20>ڵ㡣ָ<E3A1A3><D6B8>һ<EFBFBD><D2BB><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɫ<EFBFBD><C9AB><EFBFBD><EFBFBD>Ϊ<EFBFBD><CEAA>ɫ<EFBFBD><C9AB><EFBFBD><EFBFBD>ô<EFBFBD><C3B4><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD>ϲ<EFBFBD><CFB2>ڵ<EFBFBD><DAB5><EFBFBD>ʾ<EFBFBD><CABE><EFBFBD><EFBFBD>һ<EFBFBD><D2BB> 3- <20>ڵ㣬<DAB5><E3A3AC><EFBFBD><EFBFBD>ɫ<EFBFBD><C9AB><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͨ<EFBFBD><CDA8><EFBFBD>ӡ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/7080a928-06ba-4e10-9792-b8dd190dc8e2.jpg)

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʣ<EFBFBD>

1. <20><><EFBFBD><EFBFBD><EFBFBD>Ӷ<EFBFBD>Ϊ<EFBFBD><CEAA><EFBFBD><EFBFBD><EFBFBD>ӣ<EFBFBD>
2. <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɫƽ<C9AB>⣬<EFBFBD><E2A3AC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӵ<EFBFBD><D3B5><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD>·<EFBFBD><C2B7><EFBFBD>ϵĺ<CFB5><C4BA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͬ<EFBFBD><CDAC>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD>Խ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӻ<EFBFBD>ƽ<EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/62077f5d-a06d-4129-9b43-78715b82cb03.png)

```java
public class RedBlackBST<Key extends Comparable<Key>, Value> {
    private Node root;
    private static final boolean RED = true;
    private static final boolean BLACK = false;

    private class Node {
        Key key;
        Value val;
        Node left, right;
        int N;
        boolean color;

        Node(Key key, Value val, int n, boolean color) {
            this.key = key;
            this.val = val;
            N = n;
            this.color = color;
        }
    }

    private boolean isRed(Node x) {
        if (x == null) return false;
        return x.color == RED;
    }
}
```

#### <20><><EFBFBD><EFBFBD>ת

<EFBFBD><EFBFBD>Ϊ<EFBFBD>Ϸ<EFBFBD><EFBFBD>ĺ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ӷ<EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӣ<EFBFBD><EFBFBD><EFBFBD>ô<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/33a4e822-2dd0-481e-ac89-7f6161034402.jpg)

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/5e0cef33-4087-4f21-a428-16d5fddda671.jpg)

```java
public Node rotateLeft(Node h) {
    Node x = h.right;
    h.right = x.left;
    x.left = h;
    x.color = h.color;
    h.color = RED;
    x.N = h.N;
    h.N = 1 + size(h.left) + size(h.right);
    return x;
}
```

#### <20><><EFBFBD><EFBFBD>ת

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD>ת<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֮<EFBFBD><EFBFBD><EFBFBD>IJ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>̽<EFBFBD>֡<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/dfd078b2-aa4f-4c50-8319-232922d822b8.jpg)

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/3f8d8c9d-a9a9-4d7a-813c-2de05ee5a97e.jpg)

```java
public Node rotateRight(Node h) {
    Node x = h.left;
    h.left = x.right;
    x.color = h.color;
    h.color = RED;
    x.N = h.N;
    h.N = 1 + size(h.left) + size(h.right);
    return x;
}
```

#### <20><>ɫת<C9AB><D7AA>

һ<EFBFBD><EFBFBD> 4- <20>ڵ<EFBFBD><DAB5>ں<EFBFBD><DABA><EFBFBD><EFBFBD><EFBFBD><EFBFBD>б<EFBFBD><D0B1><EFBFBD>Ϊһ<CEAA><D2BB><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ㶼<DAB5>Ǻ<EFBFBD>ɫ<EFBFBD>ġ<EFBFBD><C4A1><EFBFBD><EFBFBD><EFBFBD> 4- <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD>ӽڵ<D3BD><DAB5><EFBFBD><EFBFBD><EFBFBD>ɫ<EFBFBD>ɺ<EFBFBD><C9BA><EFBFBD><EFBFBD><EFBFBD>֮<EFBFBD>⣬ͬʱ<CDAC><CAB1>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD>ɫ<EFBFBD>ɺڱ<C9BA><DAB1>죬<EFBFBD><ECA3AC> 2-3 <20><><EFBFBD>ĽǶȿ<C7B6><C8BF><EFBFBD><EFBFBD>ǽ<EFBFBD><C7BD>м<EFBFBD><D0BC>ڵ<EFBFBD><DAB5>Ƶ<EFBFBD><C6B5>ϲ<EFBFBD><CFB2>ڵ㡣

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/de7c5a31-55f5-4e9d-92ec-4ed5b2ec3828.jpg)

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/e5ad625e-729d-4a8d-923a-7c3df5773e1c.jpg)

```java
void flipColors(Node h){
    h.color = RED;
    h.left.color = BLACK;
    h.right.color = BLACK;
}
```

#### <20><><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>㷨<EFBFBD><EFBFBD>

- <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<D3BD><DAB5>Ǻ<EFBFBD>ɫ<EFBFBD>Ķ<EFBFBD><C4B6><EFBFBD><EFBFBD>ӽڵ<D3BD><DAB5>Ǻ<EFBFBD>ɫ<EFBFBD>ģ<EFBFBD><C4A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><D7AA>
- <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<D3BD><DAB5>Ǻ<EFBFBD>ɫ<EFBFBD><C9AB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<D3BD>Ҳ<EFBFBD>Ǻ<EFBFBD>ɫ<EFBFBD>ģ<EFBFBD><C4A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><D7AA>
- <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<D3BD><DAB5><EFBFBD>Ϊ<EFBFBD><CEAA>ɫ<EFBFBD>ģ<EFBFBD><C4A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɫת<C9AB><D7AA><EFBFBD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/40639782-5df2-4e96-a4f3-f9dd664d0ca1.jpg)

```java
public void put(Key key, Value val) {
    root = put(root, key, val);
    root.color = BLACK;
}

private Node put(Node x, Key key, Value val) {
    if (x == null) return new Node(key, val, 1, RED);
    int cmp = key.compareTo(x.key);
    if (cmp == 0) x.val = val;
    else if (cmp < 0) x.left = put(x.left, key, val);
    else x.right = put(x.right, key, val);

    if (isRed(x.right) && !isRed(x.left)) x = rotateLeft(x);
    if (isRed(x.left) && isRed(x.left.left)) x = rotateRight(x);
    if (isRed(x.left) && isRed(x.right)) flipColors(x);

    x.N = size(x.left) + size(x.right) + 1;
    return x;
}
```

<EFBFBD><EFBFBD><EFBFBD>Կ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ò<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> BST <20>IJ<EFBFBD><C4B2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ƣ<EFBFBD>ֻ<EFBFBD><D6BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><D7AA><EFBFBD><EFBFBD>ɫ<EFBFBD>任<EFBFBD><E4BBBB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɡ<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>һ<EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD>ɫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>û<EFBFBD><EFBFBD><EFBFBD>ϲ<EFBFBD><EFBFBD>ڵ㣬Ҳ<EFBFBD><EFBFBD>û<EFBFBD><EFBFBD><EFBFBD>ϲ<EFBFBD><EFBFBD>ڵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ָ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ㡣flipColors() <20>п<EFBFBD><D0BF>ܻ<EFBFBD>ʹ<EFBFBD>ø<EFBFBD><C3B8>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD>ɫ<EFBFBD><C9AB>Ϊ<EFBFBD><CEAA>ɫ<EFBFBD><C9AB>ÿ<EFBFBD><C3BF><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5>ɺ<EFBFBD>ɫ<EFBFBD><C9AB><EFBFBD>ɺ<EFBFBD>ɫʱ<C9AB><CAB1><EFBFBD>ĺ<EFBFBD><C4BA><EFBFBD><EFBFBD>Ӹ߶ȼ<DFB6> 1.

#### ɾ<><C9BE><EFBFBD><EFBFBD>С<EFBFBD><D0A1>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD> 2- <20>ڵ<EFBFBD><DAB5>У<EFBFBD><D0A3><EFBFBD>ôɾ<C3B4><C9BE><EFBFBD>ü<EFBFBD><C3BC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӣ<EFBFBD><D3A3><EFBFBD><EFBFBD>ƻ<EFBFBD><C6BB><EFBFBD>ƽ<EFBFBD><C6BD><EFBFBD>ԣ<EFBFBD><D4A3><EFBFBD><EFBFBD><EFBFBD>Ҫȷ<D2AA><C8B7><EFBFBD><EFBFBD>С<EFBFBD><D0A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD> 2- <20>ڵ<EFBFBD><DAB5>С<EFBFBD><D0A1><EFBFBD> 2- <20>ڵ<EFBFBD>ת<EFBFBD><D7AA><EFBFBD><EFBFBD> 3- <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD> 4- <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ַ<EFBFBD><D6B7><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϲ<EFBFBD><CFB2>ڵ<EFBFBD><DAB5><EFBFBD>һ<EFBFBD><D2BB> key<65><79><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵܽڵ<DCBD><DAB5><EFBFBD>һ<EFBFBD><D2BB> key<65><79><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϲ<EFBFBD><CFB2>ڵ<EFBFBD><DAB5><EFBFBD> 2- <20>ڵ㣬<DAB5><E3A3AC>ô<EFBFBD><C3B4>û<EFBFBD>취<EFBFBD><ECB7A8><EFBFBD>ϲ<EFBFBD><CFB2>ڵ<EFBFBD><DAB5><EFBFBD> key <20>ˣ<EFBFBD><CBA3><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA>֤ɾ<D6A4><C9BE>·<EFBFBD><C2B7><EFBFBD>ϵ<EFBFBD><CFB5><EFBFBD><EFBFBD>нڵ㶼<DAB5><E3B6BC><EFBFBD><EFBFBD> 2- <20>ڵ㡣<DAB5><E3A1A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɾ<EFBFBD><C9BE><EFBFBD>Ĺ<EFBFBD><C4B9><EFBFBD><EFBFBD>У<EFBFBD><D0A3><EFBFBD>֤<EFBFBD><D6A4><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>֮һ<D6AE><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

1. <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ǰ<EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ㲻<DAB5><E3B2BB> 2- <20>ڵ㣬<DAB5><E3A3AC><EFBFBD>ɣ<EFBFBD>
2. <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ǰ<EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<D3BD><DAB5><EFBFBD> 2- <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵܽڵ㲻<DAB5><E3B2BB> 2- <20>ڵ㣬<DAB5><E3A3AC><EFBFBD>ֵܽڵ<DCBD><DAB5><EFBFBD>һ<EFBFBD><D2BB> key <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
3. <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ǰ<EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ<D3BD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵܽڵ㶼<DAB5><E3B6BC> 2- <20>ڵ㣬<DAB5><E3A3AC><EFBFBD><EFBFBD><EFBFBD>ӽڵ㡢<DAB5><E3A1A2><EFBFBD>ڵ<EFBFBD><DAB5>е<EFBFBD><D0B5><EFBFBD>С<EFBFBD><D0A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵܽڵ<DCBD><DAB5>ϲ<EFBFBD>Ϊһ<CEAA><D2BB> 4- <20>ڵ㡣

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/b001fa64-307c-49af-b4b2-2043fc26154e.png)

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>õ<EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С<EFBFBD><EFBFBD><EFBFBD><EFBFBD> 3- <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD> 4- <20>ڵ㣬ֱ<E3A3AC>Ӵ<EFBFBD><D3B4><EFBFBD>ɾ<EFBFBD><C9BE><EFBFBD><EFBFBD>Ȼ<EFBFBD><C8BB><EFBFBD>ٴ<EFBFBD>ͷ<EFBFBD>ֽ<EFBFBD><D6BD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1> 4- <20>ڵ㡣

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/70b66757-755c-4e17-a7b7-5ce808023643.png)

#### <20><><EFBFBD><EFBFBD>

һ<EFBFBD>Ŵ<EFBFBD>СΪ N <20>ĺ<EFBFBD><C4BA><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĸ߶Ȳ<DFB6><C8B2>ᳬ<EFBFBD><E1B3AC> 2lgN<67><4E><EFBFBD><EFBFBD><EEBBB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ӧ<EFBFBD><D3A6> 2-3 <20><><EFBFBD>й<EFBFBD><D0B9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ߵ<EFBFBD>·<EFBFBD><C2B7><EFBFBD>ڵ<EFBFBD>ȫ<EFBFBD><C8AB><EFBFBD><EFBFBD><EFBFBD><EFBFBD> 3- <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>඼<EFBFBD><E0B6BC> 2- <20>ڵ㡣

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>IJ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD>ʱ<EFBFBD>䶼<EFBFBD>Ƕ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ġ<EFBFBD>

## ɢ<>б<EFBFBD>

ɢ<EFBFBD>б<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>飬<EFBFBD><EFBFBD><EFBFBD>԰<EFBFBD>ɢ<EFBFBD>б<EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD><EFBFBD>ֵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD><EFBFBD>ͷ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>٣<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڳ<EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD>ֲ<EFBFBD><EFBFBD>ҺͲ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ķ<EFBFBD><EFBFBD>ű<EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>޷<EFBFBD>ͨ<EFBFBD><EFBFBD>ɢ<EFBFBD><EFBFBD>ֵ֪<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ<EFBFBD>С<EFBFBD><EFBFBD>ϵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD><EFBFBD>޷<EFBFBD>ʵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Բ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### ɢ<>к<EFBFBD><D0BA><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>СΪ M <20><>ɢ<EFBFBD>б<EFBFBD><D0B1><EFBFBD>ɢ<EFBFBD>к<EFBFBD><D0BA><EFBFBD><EFBFBD>ܹ<EFBFBD><DCB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><D7AA>Ϊ [0, M-1] <20>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ hash ֵ<><D6B5>

ɢ<EFBFBD>б<EFBFBD><EFBFBD>г<EFBFBD>ͻ<EFBFBD>Ĵ<EFBFBD><EFBFBD>ڣ<EFBFBD>Ҳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͬ<EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͬ<EFBFBD><EFBFBD> hash ֵ<><D6B5>

ɢ<EFBFBD>к<EFBFBD><EFBFBD><EFBFBD>Ӧ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

1. һ<><D2BB><EFBFBD>ԣ<EFBFBD><D4A3><EFBFBD><EFBFBD>ȵļ<C8B5>Ӧ<EFBFBD><D3A6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȵ<EFBFBD> hash ֵ<><D6B5>
2. <20><>Ч<EFBFBD>ԣ<EFBFBD><D4A3><EFBFBD><EFBFBD><EFBFBD>Ӧ<EFBFBD><D3A6><EFBFBD><EFBFBD><EFBFBD>㣬<EFBFBD>б<EFBFBD>Ҫ<EFBFBD>Ļ<EFBFBD><C4BB><EFBFBD><EFBFBD>԰<EFBFBD> hash ֵ<><D6B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD> hash <20><><EFBFBD><EFBFBD>ʱֱ<CAB1>ӷ<EFBFBD><D3B7>ء<EFBFBD>
3. <20><><EFBFBD><EFBFBD><EFBFBD>ԣ<EFBFBD><D4A3><EFBFBD><EFBFBD>м<EFBFBD><D0BC><EFBFBD> hash ֵӦ<D6B5><D3A6><EFBFBD><EFBFBD><EFBFBD>ȵطֲ<D8B7><D6B2><EFBFBD> [0, M-1] ֮<>䣬<EFBFBD><E4A3AC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA>ֱ<EFBFBD><D6B1>Ӱ<EFBFBD>쵽ɢ<ECB5BD>б<EFBFBD><D0B1><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܡ<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Խ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>е<EFBFBD> [0, M-1] ֮<>䣬<EFBFBD><E4A3AC><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> k<><6B><EFBFBD><EFBFBD><EFBFBD><EFBFBD> k%M <20>ȿɵõ<C9B5>һ<EFBFBD><D2BB> [0, M-1] ֮<><D6AE><EFBFBD><EFBFBD> hash ֵ<><D6B5>ע<EFBFBD><D7A2> M <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>޷<EFBFBD><DEB7><EFBFBD><EFBFBD>ü<EFBFBD><C3BC><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϣ<EFBFBD><CFA2><EFBFBD><EFBFBD><EFBFBD><EFBFBD> M Ϊ 10<sup>k</sup><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ôֻ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ü<EFBFBD><EFBFBD>ĺ<EFBFBD> k λ<><CEBB>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Խ<EFBFBD><EFBFBD><EFBFBD>ת<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʽ<EFBFBD><EFBFBD>Ȼ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ó<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڸ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Խ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʾ<EFBFBD>ɶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʽ<EFBFBD><EFBFBD>Ȼ<EFBFBD><EFBFBD>ʹ<EFBFBD>ö<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʽ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵ<EFBFBD><EFBFBD><EFBFBD>г<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>жಿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϵļ<EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD><EFBFBD><EFBFBD>ֶ<EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD> hash ֵ<><D6B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϲ<EFBFBD>ʱ<EFBFBD><CAB1>Ҫ<EFBFBD><D2AA>ÿ<EFBFBD><C3BF><EFBFBD><EFBFBD> hash ֵ<><D6B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͬ<EFBFBD><CDAC><EFBFBD><EFBFBD>Ҫ<EFBFBD>ĵ<EFBFBD>λ<EFBFBD><CEBB><EFBFBD><EFBFBD><EFBFBD>Խ<EFBFBD><D4BD>ü<EFBFBD><C3BC><EFBFBD><EFBFBD><EFBFBD> R <20><><EFBFBD>Ƶ<EFBFBD><C6B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD><C3BF><EFBFBD>ֶ<EFBFBD><D6B6><EFBFBD><EFBFBD>в<EFBFBD>ͬ<EFBFBD><CDAC>Ȩֵ<C8A8><D6B5>

<EFBFBD><EFBFBD><EFBFBD>磬<EFBFBD>ַ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>к<EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

```java
int hash = 0;
for(int i = 0; i < s.length(); i++)
    hash = (R * hash + s.charAt(i)) % M;
```

<EFBFBD>ٱ<EFBFBD><EFBFBD>磬ӵ<EFBFBD>ж<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ա<EFBFBD><EFBFBD><EFBFBD>Զ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĺ<EFBFBD>ϣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

```java
int hash = (((day * R + month) % M) * R + year) % M;
```

R <20><>ֵ<EFBFBD><D6B5><EFBFBD>Ǻ<EFBFBD><C7BA><EFBFBD>Ҫ<EFBFBD><D2AA>ͨ<EFBFBD><CDA8>ȡ 31<33><31>

Java <20>е<EFBFBD> hashCode() ʵ<><CAB5><EFBFBD><EFBFBD> hash <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĭ<EFBFBD><C4AC>ʹ<EFBFBD>ö<EFBFBD><C3B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڴ<EFBFBD><DAB4><EFBFBD>ֵַ<D6B7><D6B5><EFBFBD><EFBFBD>ʹ<EFBFBD><CAB9> hashCode() <20><><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1>Ӧ<EFBFBD><D3A6><EFBFBD><EFBFBD><EFBFBD>ϳ<EFBFBD><CFB3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD>á<EFBFBD><C3A1><EFBFBD>Ϊ<EFBFBD>ڴ<EFBFBD><DAB4><EFBFBD>ַ<EFBFBD><D6B7> 32 λ<><CEBB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֻ<EFBFBD><D6BB>Ҫ 31 λ<>ķǸ<C4B7><C7B8><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ӧ<EFBFBD><D3A6><EFBFBD><EFBFBD><EFBFBD>η<EFBFBD><CEB7><EFBFBD>λ֮<CEBB><D6AE><EFBFBD><EFBFBD>ʹ<EFBFBD>ó<EFBFBD><C3B3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

```java
int hash = (x.hashCode() & 0x7fffffff) % M;
```

ʹ<EFBFBD><EFBFBD> Java <20>Դ<EFBFBD><D4B4><EFBFBD> HashMap <20><><EFBFBD>Դ<EFBFBD><D4B4>Ĺ<EFBFBD>ϣ<EFBFBD><CFA3>ʵ<EFBFBD><CAB5>ʱ<EFBFBD><CAB1>ֻ<EFBFBD><D6BB>Ҫȥʵ<C8A5><CAB5> Key <20><><EFBFBD>͵<EFBFBD> hashCode() <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɣ<EFBFBD><C9A3><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD>Ͳ<EFBFBD><CDB2><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD> M <20>Ĵ<EFBFBD>С<EFBFBD>ȡ<EFBFBD>Java <20>涨 hashCode() <20>ܹ<EFBFBD><DCB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȷֲ<C8B7><D6B2><EFBFBD><EFBFBD><EFBFBD><EFBFBD>е<EFBFBD> 32 λ<><CEBB><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Java <20>е<EFBFBD> String<6E><67>Integer <20>ȶ<EFBFBD><C8B6><EFBFBD><EFBFBD><EFBFBD> hashCode() <20><><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5><EFBFBD><EFBFBD>һ<EFBFBD>㡣<EFBFBD><E3A1A3><EFBFBD><EFBFBD>չʾ<D5B9><CABE><EFBFBD>Զ<EFBFBD><D4B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʵ<EFBFBD><CAB5> hashCode()<29><>

```java
public class Transaction{
    private final String who;
    private final Date when;
    private final double amount;

    public int hashCode(){
        int hash = 17;
        hash = 31 * hash + who.hashCode();
        hash = 31 * hash + when.hashCode();
        hash = 31 * hash + ((Double) amount).hashCode();
        return hash;
    }
}
```

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>洢 hash ֵ<><D6B5>ͬ<EFBFBD>ļ<EFBFBD><C4BC><EFBFBD><EFBFBD>Ӷ<EFBFBD><D3B6><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͻ<EFBFBD><CDBB><EFBFBD><EFBFBD>ʱ<EFBFBD><CAB1><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ȳ<EFBFBD><C8B2><EFBFBD> Key <20><><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ȼ<EFBFBD><C8BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˳<EFBFBD><CBB3><EFBFBD><EFBFBD><EFBFBD>ҡ<EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/540133af-aaaf-4208-8f7f-33cb89ac9621.png)

<EFBFBD><EFBFBD><EFBFBD><EFBFBD> N <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>M <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD> (N>M)<29><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ϣ<EFBFBD><CFA3><EFBFBD><EFBFBD><EFBFBD>ܹ<EFBFBD><DCB9><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ե<EFBFBD><D4B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD><C3BF><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ<EFBFBD>С<EFBFBD><D0A1><EFBFBD><EFBFBD><EFBFBD><EFBFBD> N/M<><4D><EFBFBD><EFBFBD><EFBFBD><EFBFBD>δ<EFBFBD><CEB4><EFBFBD>еIJ<D0B5><C4B2>ҺͲ<D2BA><CDB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>ıȽϴ<C8BD><CFB4><EFBFBD>Ϊ \~N/M<><4D>

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>̽<EFBFBD>ⷨ<EFBFBD><E2B7A8>ɢ<EFBFBD>б<EFBFBD>

<EFBFBD><EFBFBD><EFBFBD><EFBFBD>̽<EFBFBD>ⷨʹ<EFBFBD>ÿ<EFBFBD>λ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͻ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͻ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ǰ̽<EFBFBD><EFBFBD>һ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>λ<EFBFBD><EFBFBD><EFBFBD>洢<EFBFBD><EFBFBD>ͻ<EFBFBD>ļ<EFBFBD><EFBFBD><EFBFBD>ʹ<EFBFBD><EFBFBD><EFBFBD>߳<EFBFBD>̽<EFBFBD>ⷨ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ<EFBFBD>С M Ӧ<><D3A6><EFBFBD><EFBFBD><EFBFBD>ڼ<EFBFBD><DABC>ĸ<EFBFBD><C4B8><EFBFBD> N<><4E>M>N)<29><>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/2b3410f1-9559-4dd1-bc3d-e3e572247be2.png)

```java
public class LinearProbingHashST<Key, Value> {
    private int N;
    private int M = 16;
    private Key[] keys;
    private Value[] vals;

    public LinearProbingHashST() {
        init();
    }

    public LinearProbingHashST(int M) {
        this.M = M;
        init();
    }

    private void init() {
        keys = (Key[]) new Object[M];
        vals = (Value[]) new Object[M];
    }

    private int hash(Key key) {
        return (key.hashCode() & 0x7fffffff) % M;
    }
}
```

#### <20><><EFBFBD><EFBFBD>

```java
public Value get(Key key) {
    for (int i = hash(key); keys[i] != null; i = (i + 1) % M) {
        if (keys[i].equals(key)) {
            return vals[i];
        }
    }
    return null;
}
```

#### <20><><EFBFBD><EFBFBD>

```java
public void put(Key key, Value val) {
    int i;
    for (i = hash(key); keys[i] != null; i = (i + 1) % M) {
        if (keys[i].equals(key)) {
            vals[i] = val;
            return;
        }
    }
    keys[i] = key;
    vals[i] = val;
    N++;
    resize();
}
```

#### ɾ<><C9BE>

ɾ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ӧ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҳ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڵļ<EFBFBD>ֵ<EFBFBD><EFBFBD><EFBFBD>²<EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD><EFBFBD>С<EFBFBD>

```java
public void delete(Key key) {
    if (!contains(key)) return;
    int i = hash(key);
    while (!key.equals(keys[i])) {
        i = (i + 1) % M;
    }
    keys[i] = null;
    vals[i] = null;
    i = (i + 1) % M;
    while (keys[i] != null) {
        Key keyToRedo = keys[i];
        Value valToRedo = vals[i];
        keys[i] = null;
        vals[i] = null;
        N--;
        put(keyToRedo, valToRedo);
        i = (i + 1) % M;
    }
    N--;
    resize();
}
```

#### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>С

<EFBFBD><EFBFBD><EFBFBD><EFBFBD>̽<EFBFBD>ⷨ<EFBFBD>ijɱ<EFBFBD>ȡ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ<EFBFBD>ij<EFBFBD><EFBFBD>ȣ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĿҲ<EFBFBD>о۴ء<EFBFBD><EFBFBD><EFBFBD><EFBFBD>۴غܳ<EFBFBD>ʱ<EFBFBD><EFBFBD><EFBFBD>ڲ<EFBFBD><EFBFBD>ҺͲ<EFBFBD><EFBFBD><EFBFBD>ʱҲ<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD>кܶ<EFBFBD><EFBFBD><EFBFBD>̽<EFBFBD>⡣

<EFBFBD><EFBFBD> = N/M<><4D><EFBFBD><EFBFBD> <20><> <20><>Ϊ<EFBFBD><CEAA><EFBFBD><EFBFBD><EFBFBD>ʡ<EFBFBD><CAA1><EFBFBD><EFBFBD><EFBFBD>֤<EFBFBD><D6A4><EFBFBD><EFBFBD><EFBFBD><EFBFBD> <20><> С<><D0A1> 1/2 ʱ̽<CAB1><CCBD><EFBFBD><EFBFBD>Ԥ<EFBFBD>ƴ<EFBFBD><C6B4><EFBFBD>ֻ<EFBFBD><D6BB> 1.5 <20><> 2.5 ֮<>䡣

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/0ddebc5c-7c24-46b1-98db-4fa5e54db16b.png)

Ϊ<EFBFBD>˱<EFBFBD>֤ɢ<EFBFBD>б<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܣ<EFBFBD>Ӧ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ<EFBFBD>С<EFBFBD><EFBFBD>ʹ<EFBFBD><EFBFBD> <20><> <20><> [1/4, 1/2] ֮<>䡣

```java
private void resize() {
    if (N >= M / 2) resize(2 * M);
    else if (N <= M / 8) resize(M / 2);
}

private void resize(int cap) {
    LinearProbingHashST<Key, Value> t = new LinearProbingHashST<>(cap);
    for (int i = 0; i < M; i++) {
        if (keys[i] != null) {
            t.put(keys[i], vals[i]);
        }
    }
    keys = t.keys;
    vals = t.vals;
    M = t.M;
}
```

<EFBFBD><EFBFBD>Ȼÿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>µ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>鶼<EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD>°<EFBFBD>ÿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵ<EFBFBD>Բ<EFBFBD><EFBFBD>뵽ɢ<EFBFBD>б<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ǵ<EFBFBD>̯<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ĽǶ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD>Ĵ<EFBFBD><EFBFBD><EFBFBD>ȴ<EFBFBD>Ǻ<EFBFBD>С<EFBFBD>ġ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ͼ<EFBFBD><EFBFBD><EFBFBD>Կ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ÿ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>鳤<EFBFBD>ȼӱ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ۼ<EFBFBD>ƽ<EFBFBD><EFBFBD>ֵ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> 1<><31><EFBFBD><EFBFBD>Ϊ<EFBFBD><CEAA><EFBFBD><EFBFBD>ÿ<EFBFBD><C3BF><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD>¼<EFBFBD><C2BC><EFBFBD>ɢ<EFBFBD><C9A2>ֵ<EFBFBD><D6B5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ƽ<EFBFBD><C6BD>ֵ<EFBFBD><D6B5><EFBFBD>½<EFBFBD><C2BD><EFBFBD>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/01658047-0d86-4a7a-a8ca-7ea20fa1fdde.png)

## Ӧ<><D3A6>

### <20><><EFBFBD>ַ<EFBFBD><D6B7>ű<EFBFBD>ʵ<EFBFBD>ֵıȽ<C4B1>

![](https://github.com/CyC2018/InterviewNotes/blob/master/pics/9ee83c8c-1165-476c-85a6-e6e434e5307a.jpg)

Ӧ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ȿ<EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҫ<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Բ<EFBFBD><EFBFBD><EFBFBD>ʱʹ<EFBFBD>ú<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

### Java <20>ķ<EFBFBD><C4B7>ű<EFBFBD>ʵ<EFBFBD><CAB5>

Java <20><> java.util.TreeMap <20><> java.util.HashMap <20>ֱ<EFBFBD><D6B1>ǻ<EFBFBD><C7BB>ں<EFBFBD><DABA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɢ<EFBFBD>б<EFBFBD><D0B1>ķ<EFBFBD><C4B7>ű<EFBFBD>ʵ<EFBFBD>֡<EFBFBD>

### <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>

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### ϡ<><CFA1><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>˷<EFBFBD>

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```java
import java.util.HashMap;

public class SparseVector {
    private HashMap<Integer, Double> hashMap;

    public SparseVector(double[] vector) {
        hashMap = new HashMap<>();
        for (int i = 0; i < vector.length; i++) {
            if (vector[i] != 0) {
                hashMap.put(i, vector[i]);
            }
        }
    }

    public double get(int i) {
        return hashMap.getOrDefault(i, 0.0);
    }

    public double dot(SparseVector other) {
        double sum = 0;
        for (int i : hashMap.keySet()) {
            sum += this.get(i) * other.get(i);
        }
        return sum;
    }
}
```