140 lines
4.6 KiB
Java
140 lines
4.6 KiB
Java
<!-- GFM-TOC -->
|
||
* [汉诺塔](#汉诺塔)
|
||
* [哈夫曼编码](#哈夫曼编码)
|
||
<!-- GFM-TOC -->
|
||
|
||
|
||
# 汉诺塔
|
||
|
||
<div align="center"> <img src="pics/69d6c38d-1dec-4f72-ae60-60dbc10e9d15.png" width="300"/> </div><br>
|
||
|
||
有三个柱子，分别为 from、buffer、to。需要将 from 上的圆盘全部移动到 to 上，并且要保证小圆盘始终在大圆盘上。
|
||
|
||
这是一个经典的递归问题，分为三步求解：
|
||
|
||
① 将 n-1 个圆盘从 from -> buffer
|
||
|
||
<div align="center"> <img src="pics/f9240aa1-8d48-4959-b28a-7ca45c3e4d91.png" width="300"/> </div><br>
|
||
|
||
② 将 1 个圆盘从 from -> to
|
||
|
||
<div align="center"> <img src="pics/f579cab0-3d49-4d00-8e14-e9e1669d0f9f.png" width="300"/> </div><br>
|
||
|
||
③ 将 n-1 个圆盘从 buffer -> to
|
||
|
||
<div align="center"> <img src="pics/d02f74dd-8e33-4f3c-bf29-53203a06695a.png" width="300"/> </div><br>
|
||
|
||
如果只有一个圆盘，那么只需要进行一次移动操作。
|
||
|
||
从上面的讨论可以知道，a<sub>n</sub> = 2 * a<sub>n-1</sub> + 1，显然 a<sub>n</sub> = 2<sup>n</sup> - 1，n 个圆盘需要移动 2<sup>n</sup> - 1 次。
|
||
|
||
```java
|
||
public class Hanoi {
|
||
public static void move(int n, String from, String buffer, String to) {
|
||
if (n == 1) {
|
||
System.out.println("from " + from + " to " + to);
|
||
return;
|
||
}
|
||
move(n - 1, from, to, buffer);
|
||
move(1, from, buffer, to);
|
||
move(n - 1, buffer, from, to);
|
||
}
|
||
|
||
public static void main(String[] args) {
|
||
Hanoi.move(3, "H1", "H2", "H3");
|
||
}
|
||
}
|
||
```
|
||
|
||
```html
|
||
from H1 to H3
|
||
from H1 to H2
|
||
from H3 to H2
|
||
from H1 to H3
|
||
from H2 to H1
|
||
from H2 to H3
|
||
from H1 to H3
|
||
```
|
||
|
||
# 哈夫曼编码
|
||
|
||
根据数据出现的频率对数据进行编码，从而压缩原始数据。
|
||
|
||
例如对于一个文本文件，其中各种字符出现的次数如下：
|
||
|
||
- a : 10
|
||
- b : 20
|
||
- c : 40
|
||
- d : 80
|
||
|
||
可以将每种字符转换成二进制编码，例如将 a 转换为 00，b 转换为 01，c 转换为 10，d 转换为 11。这是最简单的一种编码方式，没有考虑各个字符的权值（出现频率）。而哈夫曼编码采用了贪心策略，使出现频率最高的字符的编码最短，从而保证整体的编码长度最短。
|
||
|
||
首先生成一颗哈夫曼树，每次生成过程中选取频率最少的两个节点，生成一个新节点作为它们的父节点，并且新节点的频率为两个节点的和。选取频率最少的原因是，生成过程使得先选取的节点位于树的更低层，那么需要的编码长度更长，频率更少可以使得总编码长度更少。
|
||
|
||
生成编码时，从根节点出发，向左遍历则添加二进制位 0，向右则添加二进制位 1，直到遍历到叶子节点，叶子节点代表的字符的编码就是这个路径编码。
|
||
|
||
<div align="center"> <img src="pics/8edc5164-810b-4cc5-bda8-2a2c98556377.jpg" width="300"/> </div><br>
|
||
|
||
```java
|
||
public class Huffman {
|
||
|
||
private class Node implements Comparable<Node> {
|
||
char ch;
|
||
int freq;
|
||
boolean isLeaf;
|
||
Node left, right;
|
||
|
||
public Node(char ch, int freq) {
|
||
this.ch = ch;
|
||
this.freq = freq;
|
||
isLeaf = true;
|
||
}
|
||
|
||
public Node(Node left, Node right, int freq) {
|
||
this.left = left;
|
||
this.right = right;
|
||
this.freq = freq;
|
||
isLeaf = false;
|
||
}
|
||
|
||
@Override
|
||
public int compareTo(Node o) {
|
||
return this.freq - o.freq;
|
||
}
|
||
}
|
||
|
||
public Map<Character, String> encode(Map<Character, Integer> frequencyForChar) {
|
||
PriorityQueue<Node> priorityQueue = new PriorityQueue<>();
|
||
for (Character c : frequencyForChar.keySet()) {
|
||
priorityQueue.add(new Node(c, frequencyForChar.get(c)));
|
||
}
|
||
while (priorityQueue.size() != 1) {
|
||
Node node1 = priorityQueue.poll();
|
||
Node node2 = priorityQueue.poll();
|
||
priorityQueue.add(new Node(node1, node2, node1.freq + node2.freq));
|
||
}
|
||
return encode(priorityQueue.poll());
|
||
}
|
||
|
||
private Map<Character, String> encode(Node root) {
|
||
Map<Character, String> encodingForChar = new HashMap<>();
|
||
encode(root, "", encodingForChar);
|
||
return encodingForChar;
|
||
}
|
||
|
||
private void encode(Node node, String encoding, Map<Character, String> encodingForChar) {
|
||
if (node.isLeaf) {
|
||
encodingForChar.put(node.ch, encoding);
|
||
return;
|
||
}
|
||
encode(node.left, encoding + '0', encodingForChar);
|
||
encode(node.right, encoding + '1', encodingForChar);
|
||
}
|
||
}
|
||
```
|
||
|
||
|
||
|
||
|
||
<img width="580px" src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/other/公众号海报1.png"></img>
|