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Add fft, add header info
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mbinary
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README.md
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README.md
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@ -12,27 +12,14 @@
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Some pictures and ideas are from `<<Introduction to Algotithm>>`
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# Notes
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* [docs](./docs)
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* [algorithm-general.md](./docs/algorithm-general.md)
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* [b-tree.md](./docs/b-tree.md)
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* [fib-heap.md](./docs/fib-heap.md)
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* [graph.md](./docs/graph.md)
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* [hashTable.md](./docs/hashTable.md)
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* [red-black-tree.md](./docs/red-black-tree.md)
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* [sort.md](./docs/sort.md)
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* [src](./docs/src)
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* [string-matching.md](./docs/string-matching.md)
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* [tree.md](./docs/tree.md)
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[Click here](./docs) to view notes
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# Index
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* [.](.)
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* [LICENSE](./LICENSE)
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* [README.md](./README.md)
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* [_config.yml](./_config.yml)
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* [backtracking](./backtracking)
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* [dataStructure](./dataStructure)
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* [allOone](./dataStructure/allOone)
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* [LRU](./dataStructure/LRU)
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* [bTree.py](./dataStructure/bTree.py)
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* [binaryHeap.py](./dataStructure/binaryHeap.py)
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* [binaryHeap1.py](./dataStructure/binaryHeap1.py)
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@ -41,6 +28,7 @@ Some pictures and ideas are from `<<Introduction to Algotithm>>`
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* [graph](./dataStructure/graph)
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* [hashTable.py](./dataStructure/hashTable.py)
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* [huffman](./dataStructure/huffman)
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* [insert_remove_getRandom.py](./dataStructure/insert_remove_getRandom.py)
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* [intervalTree.py](./dataStructure/intervalTree.py)
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* [leftHeap.py](./dataStructure/leftHeap.py)
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* [linkedList.py](./dataStructure/linkedList.py)
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@ -56,23 +44,40 @@ Some pictures and ideas are from `<<Introduction to Algotithm>>`
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* [winnerTree.py](./dataStructure/winnerTree.py)
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* [divideAndConquer](./divideAndConquer)
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* [min_distance_of_n_points.py](./divideAndConquer/min_distance_of_n_points.py)
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* [docs](./docs)
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* [README.md](./docs/README.md)
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* [_config.yml](./docs/_config.yml)
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* [algorithm-general.md](./docs/algorithm-general.md)
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* [b-tree.md](./docs/b-tree.md)
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* [dft.md](./docs/dft.md)
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* [fib-heap.md](./docs/fib-heap.md)
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* [graph.md](./docs/graph.md)
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* [hashTable.md](./docs/hashTable.md)
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* [red-black-tree.md](./docs/red-black-tree.md)
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* [sort.md](./docs/sort.md)
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* [src](./docs/src)
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* [string-matching.md](./docs/string-matching.md)
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* [tree.md](./docs/tree.md)
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* [dynamicProgramming](./dynamicProgramming)
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* [Vec2d.hs](./dynamicProgramming/Vec2d.hs)
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* [last-stone-weight.py](./dynamicProgramming/last-stone-weight.py)
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* [lcs.py](./dynamicProgramming/lcs.py)
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* [matrixChainMultiply.py](./dynamicProgramming/matrixChainMultiply.py)
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* [max-len-of-repeated-subarray.py](./dynamicProgramming/max-len-of-repeated-subarray.py)
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* [splitStripe.hs](./dynamicProgramming/splitStripe.hs)
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* [splitStripe.py](./dynamicProgramming/splitStripe.py)
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* [stoneGame.py](./dynamicProgramming/stoneGame.py)
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* [testVec2d.hs](./dynamicProgramming/testVec2d.hs)
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* [wildcard_matching.py](./dynamicProgramming/wildcard_matching.py)
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* [graph](./graph)
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* [cloneGraph.cpp](./graph/cloneGraph.cpp)
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* [dfs.py](./graph/dfs.py)
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* [isBipartGraph.py](./graph/isBipartGraph.py)
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* [math](./math)
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* [README.md](./math/README.md)
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* [convertWeight.py](./math/convertWeight.py)
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* [fastPow.py](./math/fastPow.py)
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* [fft.py](./math/fft.py)
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* [fibonacci](./math/fibonacci)
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* [numWeight](./math/numWeight)
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* [numberTheory](./math/numberTheory)
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* [numericalAnalysis](./math/numericalAnalysis)
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* [permute](./math/permute)
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@ -100,6 +105,7 @@ Some pictures and ideas are from `<<Introduction to Algotithm>>`
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* [README.md](./string/README.md)
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* [manacher.py](./string/manacher.py)
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* [markov.py](./string/markov.py)
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* [min-window-substring.py](./string/min-window-substring.py)
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* [rabin_karp.py](./string/rabin_karp.py)
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* [rotate.py](./string/rotate.py)
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* [src](./string/src)
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@ -1,3 +1,14 @@
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''' mbinary
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#########################################################################
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# File : lru_allone.py
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# Author: mbinary
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# Mail: zhuheqin1@gmail.com
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# Blog: https://mbinary.xyz
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# Github: https://github.com/mbinary
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# Created Time: 2019-05-23 23:50
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# Description:
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#########################################################################
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'''
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from allOne import allOne
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'''In this implementation, the lru doesn't use some funcs of allOne,
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''' mbinary
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#########################################################################
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# File : lru_orderedDict.py
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# Author: mbinary
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# Mail: zhuheqin1@gmail.com
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# Blog: https://mbinary.xyz
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# Github: https://github.com/mbinary
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# Created Time: 2019-05-23 20:13
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# Description:
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#########################################################################
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'''
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class LRUCache(object):
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def __init__(self, capacity):
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''' mbinary
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#########################################################################
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# File : binaryHeap1.py
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# Author: mbinary
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# Mail: zhuheqin1@gmail.com
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# Blog: https://mbinary.xyz
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# Github: https://github.com/mbinary
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# Created Time: 2019-04-16 09:41
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# Description:
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#########################################################################
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'''
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from functools import partial
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class heap:
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def __init__(self,lst,reverse = False):
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''' mbinary
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#########################################################################
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# File : circularQueue.py
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# Author: mbinary
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# Mail: zhuheqin1@gmail.com
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# Blog: https://mbinary.xyz
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# Github: https://github.com/mbinary
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# Created Time: 2019-04-16 09:41
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# Description:
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#########################################################################
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'''
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class MyCircularQueue:
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def __init__(self, k):
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''' mbinary
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#########################################################################
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# File : linkedList.py
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# Author: mbinary
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# Mail: zhuheqin1@gmail.com
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# Blog: https://mbinary.xyz
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# Github: https://github.com/mbinary
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# Created Time: 2019-04-16 09:41
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# Description:
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#########################################################################
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'''
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class node:
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def __init__(self,val,follow=None):
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self.val = val
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270
docs/dft.md
Normal file
270
docs/dft.md
Normal file
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@ -0,0 +1,270 @@
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<!-- TOC -->
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- [0.1. 定义](#01-定义)
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- [0.1.1. 连续](#011-连续)
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- [0.1.2. 离散](#012-离散)
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- [0.2. 性质](#02-性质)
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- [0.2.1. 分离性](#021-分离性)
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- [0.2.2. 位移定理](#022-位移定理)
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- [0.2.3. 周期性](#023-周期性)
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- [0.2.4. 共轭对称性](#024-共轭对称性)
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- [0.2.5. 旋转性](#025-旋转性)
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- [0.2.6. 加法定理](#026-加法定理)
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- [0.2.7. 平均值](#027-平均值)
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- [0.2.8. 相似性定理](#028-相似性定理)
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- [0.2.9. 卷积定理](#029-卷积定理)
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- [0.2.10. 相关定理](#0210-相关定理)
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- [0.2.11. Rayleigh 定理](#0211-rayleigh-定理)
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- [0.3. 快速傅里叶变换](#03-快速傅里叶变换)
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- [0.3.1. 复数中的单位根](#031-复数中的单位根)
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- [0.3.2. 快速傅里叶变换的计算](#032-快速傅里叶变换的计算)
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- [0.4. 代码](#04-代码)
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- [0.5. 参考](#05-参考)
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<!-- /TOC -->
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图像处理中, 为了方便处理，便于抽取特征，数据压缩等目的，常常要将图像进行变换。
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一般有如下变换方法
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1. 傅立叶变换Fourier Transform
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2. 离散余弦变换Discrete Cosine Transform
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3. 沃尔希-哈德玛变换Walsh-Hadamard Transform
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4. 斜变换Slant Transform
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5. 哈尔变换Haar Transform
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6. 离散K-L变换Discrete Karhunen-Leave Transform
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7. 奇异值分解SVD变换Singular-Value Decomposition
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8. 离散小波变换Discrete Wavelet Transform
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这篇文章介绍一下傅里叶变换
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## 0.1. 定义
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### 0.1.1. 连续
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积分形式
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如果一个函数的绝对值的积分存在，即
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|dt<\infty&space;)
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并且函数是连续的或者只有有限个不连续点，则对于 x 的任何值， 函数的傅里叶变换存在
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- 一维傅里叶变换
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=\int_{-\infty}&space;^\infty&space;h(t)e^{-j2\pi&space;ft}dt&space;)
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- 一维傅里叶逆变换
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=\int_{-\infty}&space;^\infty&space;h(t)e^{j2\pi&space;ft}dt&space;)
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同理多重积分
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### 0.1.2. 离散
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实际应用中，多用离散傅里叶变换 DFT.
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- 一维傅里叶变换
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=\sum_{x=0}&space;^{N-1}&space;f(x)e^{\frac{-2\pi&space;j}{N}&space;ux}&space;)
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- 一维傅里叶逆变换
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=\frac{1}{N}\sum_{u=0}&space;^{N-1}&space;F(u)e^{\frac{2\pi&space;j}{N}&space;ux}&space;)
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需要注意的是， 逆变换乘以  是为了**归一化**，这个系数可以随意改变， 即可以正变换乘以 , 逆变换就不乘，或者两者都乘以等系数。
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- 二维傅里叶变换
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=\frac{1}{N}\sum_{x=0}^{N-1}\sum_{y=0}&space;^{N-1}&space;f(x,y)e^{\frac{-2\pi&space;j}{N}&space;(ux+vy)}&space;)
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- 二维傅里叶逆变换
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=\frac{1}{N}\sum_{u=0}^{N-1}\sum_{v=0}&space;^{N-1}&space;F(u,v)e^{\frac{2\pi&space;j}{N}&space;(ux+vy)}&space;)
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幅度
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|&space;=&space;\sqrt{real(F)^2+imag(F)^2}&space;)
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相位
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}{real(F)}}&space;)
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对于图像的幅度谱显示，由于 |F(u,v)| 变换范围太大，一般显示 +1))
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用 `<=>` 表示傅里叶变换对
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<=>F(u)\\&space;f(x,y)<=>F(u,v)&space;)
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f,g,h 对应的傅里叶变换 F,G,H
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 表示  的共轭
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## 0.2. 性质
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### 0.2.1. 分离性
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=\sum_{y=0}&space;^{N-1}&space;f(x,y)e^{\frac{-2\pi&space;j}{N}&space;vy}\\&space;&F(u,v)=\frac{1}{N}\sum_{x=0}^{N-1}F(x,v)e^{\frac{-2\pi&space;j}{N}ux}&space;\end{aligned}&space;)
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进行多维变换时，可以依次对每一维进行变换。 下面在代码中就是这样实现的。
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### 0.2.2. 位移定理
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e^{\frac{2\pi&space;j}{N}(u_0x+v_0y)}&space;<=>F(u-u_0,v-v_0)&space;)
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<=>F(u,v)e^{\frac{-2\pi&space;j}{N}(ux_0+vy_0)}&space;)
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### 0.2.3. 周期性
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&space;=&space;F(u+N,v+N)&space;)
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### 0.2.4. 共轭对称性
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&space;=&space;F^*(-u,-v))
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a)偶分量函数在变换中产生偶分量函数;
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b)奇分量函数在变换中产生奇分量函数;
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c)奇分量函数在变换中引入系数-j;
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d)偶分量函数在变换中不引入系数.
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### 0.2.5. 旋转性
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if <=>F(\omega,\phi)&space;)
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then <=>F(\omega,\phi+t)&space;)
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### 0.2.6. 加法定理
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1.
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2.
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<=>aF[u,v]&space;)
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### 0.2.7. 平均值
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&space;=&space;\frac{1}{N}F(0,0)&space;)
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### 0.2.8. 相似性定理
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尺度变换
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<=>\frac{F(\frac{u}{a},\frac{v}{b})}{ab}&space;)
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### 0.2.9. 卷积定理
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卷积定义
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1d
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g(x-m)&space;)
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2d
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*g(x,y)&space;=&space;\frac{1}{MN}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}f(m,n)g(x-m,y-n)&space;)
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|
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卷积定理
|
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*g(x,y)&space;<=>&space;F(u,v)G(u,v)&space;)
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g(x,y)<=>F(u,v)*G(u,v)&space;)
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离散卷积
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用
|
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h[(k-i)T]&space;<=>&space;X(\frac{n}{NT})H(\frac{n}{NT})&space;)
|
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即两个周期为 N 的抽样函数， 他们的卷积的离散傅里叶变换等于他们的离散傅里叶变换的卷积
|
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|
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卷积的应用：
|
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去除噪声， 特征增强
|
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两个不同周期的信号卷积需要周期扩展的原因：如果直接进行傅里叶变换和乘积，会产生折叠误差(卷绕)。
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### 0.2.10. 相关定理
|
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下面用 表示相关。
|
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相关函数描述了两个信号之间的相似性，其相关性大小有相关系数衡量
|
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|
||||
- 相关函数的定义
|
||||
离散
|
||||
\quad&space;\infty&space;\quad&space;g(x,y)&space;=&space;\frac{1}{MN}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}f^*(m,n)g(x+m,y+n)&space;)
|
||||
连续
|
||||
&space;=&space;\int_{-\infty}^{\infty}x^*(\tau)&space;h(t+\tau)d\tau)
|
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- 定理
|
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\quad&space;\infty&space;\quad&space;g(x,y)<=>F^*(u,v)G(u,v)&space;)
|
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### 0.2.11. Rayleigh 定理
|
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能量变换
|
||||
对于有限区间非零函数 f(t), 其能量为
|
||||
|^2dt&space;)
|
||||
|
||||
其变换函数与原函数有相同的能量
|
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|^2dt&space;=&space;\int_{-\infty}^{\infty}|F(u)|^2dt&space;)
|
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## 0.3. 快速傅里叶变换
|
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由上面离散傅里叶变换的性质易知，直接计算 1维 dft 的时间复杂度维 )。
|
||||
|
||||
利用到单位根的对称性，快速傅里叶变换可以达到 )的时间复杂度。
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### 0.3.1. 复数中的单位根
|
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我们知道， 在复平面，复数 k可以表示成 ， 可以对应一个向量。即为幅角。
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在**单位圆**中 ，单位圆被分成  份， 由单位圆的对称性
|
||||
}&space;)
|
||||
现在记  ， 即被分成 n 份，幅度角为正且最小的向量称为 n 次单位向量， 记为，
|
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其余的 n-1 个向量分别为  ，它们可以由复数之间的乘法得来 )。
|
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单位根的性质
|
||||
1. 这个可以用 e 表示出来证明
|
||||
|
||||
2. 可以写成三角函数证明
|
||||
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||||
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容易看出 。
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对于 , 它事实上就是  。
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### 0.3.2. 快速傅里叶变换的计算
|
||||
下面的推导假设 ，以及代码实现 FFT 部分也是 如此。
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利用上面的对称性，
|
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将傅里叶计算进行奇偶分组
|
||||
&=\sum_{i=0}^{n-1}\omega_n&space;^{iu}&space;a^i\\&space;&=&space;\sum_{i=0}^{\frac{n}{2}-1}\omega_n&space;^{2iu}&space;a^{2i}+\sum_{i=0}^{\frac{n}{2}-1}\omega_n&space;^{(2i+1)u}&space;a^{2i+1}\\&space;&=\sum_{i=0}^{\frac{n}{2}-1}\omega_{\frac{n}{2}}&space;^{iu}&space;a^{2i}+\omega_n^u\sum_{i=0}^{\frac{n}{2}-1}\omega_{\frac{n}{2}}&space;^{iu}&space;a^{2i+1}\\&space;&&space;=&space;F_{even}(u)+\omega_n^u&space;F_{odd}(u)&space;\end{aligned}&space;)
|
||||
表示将 输入的次序中偶数点进行 Fourier 变换，  同理，这样就形成递推公式。
|
||||
现在还没有减少计算量，下面通过将分别计算的 奇项，偶项利用起来，只计算 前 项，后面的一半可以利用此结果马上算出来。每一次可以减少一半的计算量。
|
||||
|
||||
对于 
|
||||
&=F_{even}(\omega_{n}^{2i+n})+\omega_{n}^{i+\frac{n}{2}}\cdot&space;F_{odd}(\omega_{n}^{2i+n})\\&space;&=F_{even}(\omega_{\frac{n}{2}}^{i+\frac{n}{2}})+\omega_{\frac{n}{2}}^{i+\frac{n}{2}}\cdot&space;F_{odd}(\omega_{\frac{n}{2}}^{i+\frac{n}{2}})\\&space;&&space;=F_{even}(\omega_{\frac{n}{2}}^{i})-\omega_{\frac{n}{2}}^{i}\cdot&space;F_{odd}(\omega_{\frac{n}{2}}^{i})&space;\end{aligned}&space;)
|
||||
现在很清楚了，在每次计算 a[0..n-1] 的傅里叶变换F[0..n-1]，分别计算出奇 odd[0..n/2-1]，偶even[0..n/2-1]（可以递归地进行），
|
||||
那么傅里叶变换为：
|
||||
|
||||
|
||||
## 0.4. 代码
|
||||
下面是 python 实现
|
||||
一维用 FFT 实现， 不过 只实现了 2 的幂。/ 对于非 2 的幂，用 FFT 实现有点困难，还需要插值，所以我 用) 直接实现。
|
||||
|
||||
二维的 DFT利用 分离性，直接调用 一维 FFT。
|
||||
[GitHub](https://github.com/mbinary/algorithm)
|
||||
|
||||
```python
|
||||
import numpy as np
|
||||
|
||||
|
||||
def _fft(a, invert=False):
|
||||
N = len(a)
|
||||
if N == 1:
|
||||
return [a[0]]
|
||||
elif N & (N - 1) == 0: # O(nlogn), 2^k
|
||||
even = _fft(a[::2], invert)
|
||||
odd = _fft(a[1::2], invert)
|
||||
i = 2j if invert else -2j
|
||||
factor = np.exp(i * np.pi * np.arange(N // 2) / N)
|
||||
prod = factor * odd
|
||||
return np.concatenate([even + prod, even - prod])
|
||||
else: # O(n^2)
|
||||
w = np.arange(N)
|
||||
i = 2j if invert else -2j
|
||||
m = w.reshape((N, 1)) * w
|
||||
W = np.exp(m * i * np.pi / N)
|
||||
return np.concatenate(np.dot(W, a.reshape(
|
||||
(N, 1)))) # important, cannot use *
|
||||
|
||||
|
||||
def fft(a):
|
||||
'''fourier[a]'''
|
||||
n = len(a)
|
||||
if n == 0:
|
||||
raise Exception("[Error]: Invalid length: 0")
|
||||
return _fft(a)
|
||||
|
||||
|
||||
def ifft(a):
|
||||
'''invert fourier[a]'''
|
||||
n = len(a)
|
||||
if n == 0:
|
||||
raise Exception("[Error]: Invalid length: 0")
|
||||
return _fft(a, True) / n
|
||||
|
||||
|
||||
def fft2(arr):
|
||||
return np.apply_along_axis(fft, 0,
|
||||
np.apply_along_axis(fft, 1, np.asarray(arr)))
|
||||
|
||||
|
||||
def ifft2(arr):
|
||||
return np.apply_along_axis(ifft, 0,
|
||||
np.apply_along_axis(ifft, 1, np.asarray(arr)))
|
||||
|
||||
|
||||
def test(n=128):
|
||||
print('\nsequence length:', n)
|
||||
print('fft')
|
||||
li = np.random.random(n)
|
||||
print(np.allclose(fft(li), np.fft.fft(li)))
|
||||
|
||||
print('ifft')
|
||||
li = np.random.random(n)
|
||||
print(np.allclose(ifft(li), np.fft.ifft(li)))
|
||||
|
||||
print('fft2')
|
||||
li = np.random.random(n * n).reshape((n, n))
|
||||
print(np.allclose(fft2(li), np.fft.fft2(li)))
|
||||
|
||||
print('ifft2')
|
||||
li = np.random.random(n * n).reshape((n, n))
|
||||
print(np.allclose(ifft2(li), np.fft.ifft2(li)))
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
for i in range(1, 3):
|
||||
test(i * 16)
|
||||
```
|
||||
|
||||
## 0.5. 参考
|
||||
- [万寿红老师课件]()
|
||||
- [一小时学会快速傅里叶变换 Fast Fourier Transform](https://zhuanlan.zhihu.com/p/31584464)
|
||||
- [快速傅里叶变换（FFT）算法【详解】](https://www.cnblogs.com/ECJTUACM-873284962/p/6919424.html)
|
||||
|
||||
|
|
@ -1,3 +1,14 @@
|
|||
/* mbinary
|
||||
#########################################################################
|
||||
# File : cloneGraph.cpp
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-04-16 09:41
|
||||
# Description:
|
||||
#########################################################################
|
||||
*/
|
||||
class Solution {
|
||||
public:
|
||||
map<Node*,Node*> st;
|
||||
|
|
|
|||
11
math/fft.py
11
math/fft.py
|
|
@ -1,3 +1,14 @@
|
|||
''' mbinary
|
||||
#########################################################################
|
||||
# File : fft.py
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-06-11 12:48
|
||||
# Description:
|
||||
#########################################################################
|
||||
'''
|
||||
import numpy as np
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -1,3 +1,14 @@
|
|||
''' mbinary
|
||||
#########################################################################
|
||||
# File : nega.py
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-06-02 23:15
|
||||
# Description:
|
||||
#########################################################################
|
||||
'''
|
||||
def nega(n:int,base=-2:int)->:list:
|
||||
'''return list of num, the first is the highest digit'''
|
||||
if base>-2:
|
||||
|
|
|
|||
|
|
@ -1,3 +1,14 @@
|
|||
''' mbinary
|
||||
#########################################################################
|
||||
# File : test_token_scanner.py
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-04-16 09:41
|
||||
# Description:
|
||||
#########################################################################
|
||||
'''
|
||||
import unittest
|
||||
from token_scanner import gen_token
|
||||
|
||||
|
|
|
|||
|
|
@ -1,3 +1,14 @@
|
|||
''' mbinary
|
||||
#########################################################################
|
||||
# File : genExpr.py
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-04-16 09:41
|
||||
# Description:
|
||||
#########################################################################
|
||||
'''
|
||||
from random import randint
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -1,3 +1,14 @@
|
|||
''' mbinary
|
||||
#########################################################################
|
||||
# File : work_dispatch.py
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-04-16 09:41
|
||||
# Description:
|
||||
#########################################################################
|
||||
'''
|
||||
'''
|
||||
设有n件工作要分配给n个人去完成，将工作i分配给第j个人所需费用为c_ij 。试设计一个算法，为每个人分配1件不同的工作，并使总费用达到最小。
|
||||
'''
|
||||
|
|
|
|||
|
|
@ -1,3 +1,14 @@
|
|||
/* mbinary
|
||||
#########################################################################
|
||||
# File : quickSort.c
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-04-16 09:41
|
||||
# Description:
|
||||
#########################################################################
|
||||
*/
|
||||
int partition(int *arr,int i,int j)
|
||||
{
|
||||
int pivot = arr[j],p=i,q=j;
|
||||
|
|
|
|||
|
|
@ -1,3 +1,14 @@
|
|||
''' mbinary
|
||||
#########################################################################
|
||||
# File : codecogs.py
|
||||
# Author: mbinary
|
||||
# Mail: zhuheqin1@gmail.com
|
||||
# Blog: https://mbinary.xyz
|
||||
# Github: https://github.com/mbinary
|
||||
# Created Time: 2019-04-16 09:41
|
||||
# Description:
|
||||
#########################################################################
|
||||
'''
|
||||
import os
|
||||
import re
|
||||
import sys
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user