算法导论Ch4

分治策略

最大子数组问题

A = [13,-3,-25,20,-3,-16,-23,18,20,-7,12,-5,-22,15,-4,7]

暴力解法：

分治方法：

将数组划分为两个规模尽量相等的子数组。最大子数组A[i..j]位置必然是：

完全在A[low,mid]中;

完全在A[mid+1,high]中;

跨越了中点，因此low<=i<=mid<=j<=high.

#!/usr/bin/env python # -*- coding:UTF-8 -*- import math   A = [13,-3,-25,20,-3,-16,-23,18,20,-7,12,-5,-22,15,-4,7]   def findMaxCrossingSubarray(A,low,mid,high):     leftSum = -float("inf")     SUM = 0     maxLeft = mid     for i in range(mid,low-1,-1):         SUM += A[i]         if SUM > leftSum:             leftSum = SUM             maxLeft = i                  rightSum = -float("inf")     SUM = 0     maxRight = mid+1     for j in range(mid+1,high+1,1):         SUM += A[j]         if SUM > rightSum:             rightSum = SUM             maxRight = j       return (maxLeft,maxRight,leftSum+rightSum)   def findMaximumSubarray(A,low,high):     if low == high:         return (low,high,A[low])     else:         mid = math.floor((low+high)/2)         (leftLow,leftHigh,leftSum)=findMaximumSubarray(A,low,mid)         (rightLow,rightHigh,rightSum)=findMaximumSubarray(A,mid+1,high)         (crossLow,crossHigh,crossSum)=findMaxCrossingSubarray(A,low,mid,high)           if leftSum >= rightSum and leftSum >= crossSum:             return (leftLow,leftHigh,leftSum)         elif rightSum >= leftSum and rightSum >= crossSum:             return (rightLow,rightHigh,rightSum)         else:             return (crossLow,crossHigh,crossSum)   if __name__ == '__main__':     result = findMaximumSubarray(A,0,len(A)-1)     print(result)

 

线性时间算法：

#!/usr/bin/env python # -*- coding:UTF-8 -*-   A = [13,-3,-25,20,-3,-16,-23,18,20,-7,12,-5,-22,15,-4,7]   ''' https://www.cnblogs.com/allzy/p/5162815.html 如果累加和出现小于0的情况，   则和最大的子序列肯定不可能包含前面的元素，   这时将累加和置0，从下个元素重新开始累加 ''' def findMaximumSubarrayLinearAlgo(A):     maxSum = 0     thisSum = 0     maxLeft = 0     maxRight = 0     for i in range(0,len(A)):         thisSum += A[i]         if thisSum > maxSum:             maxRight = i             maxSum = thisSum         elif thisSum < 0:             thisSum = 0             maxLeft = i+1     return (maxLeft,maxRight,maxSum) if maxSum >0 else None   if __name__ == '__main__':     result = findMaximumSubarrayLinearAlgo(A)     print(result)

 

后语：

从暴力解法到分治算法，是学以致用。

有线性时间算法，提示每个问题可能有更妙的解决办法。

矩阵乘法

常规方法：

Strassen算法：

通过中间过程的精妙优化，将运行时间的递归式由(1)变成(2)

    (1)

   (2)

 

后语：

就像书上说的，最初可能认为矩阵乘法都要花费时间，但通过看到Strassen算法，才看到算法的精妙。

求解递归式*

迭代法

递归树

主方法

Akra-Bazzi方法

 

转载于:https://www.cnblogs.com/sunnypoem/p/10864043.html