【算法导论】最大子数组——递归

你的名字 2021-03-28 13:56 454阅读 0赞

1.描述：找出数组A的和最大的非空连续子数组，我们称这样的连续子数组为最大子数组。

![1213881-20180902155520931-2115318469.png][]

2. 用分治策略来求解。

a. 假设我们要求A的子数组A\[low, high\]的最大子数组。根据分治策略，我们先将A\[low,high\] 平分

b. 那么 A\[low,highj\]的子数组A\[i,j\]只有三种可能

a）完全位于A\[low, mid\]; 此时 low <= i <= j <= mid

b)  完全位于A\[nid+1, high\]中，此时 mid + 1 <= i <= j <= high

c)  跨越了中点mid， 此时 low <= i  <= mid < j < high

![1213881-20180902160447285-387423155.png][]

3.  伪代码

FIND-MAXIMUM-SUBARRAY(A, low, high)
        if high == low
            return (low, high. A[low])   //只有一个元素
        else 
            mid = (low + high)/2     //向下取整
            (left-low, left-high, left-sum) = FIND-MAXIMUM-SUBARRAY(A, low, mid)
            (right-low, right-high, right-sum) = FIND-MAXIMUM-SUBARRAY(A, mid + 1, high)
            (cross-low, cross-high, cross-sum) = FIND-MAX-CROSSING-SUBARRAY(A, low, mid, high)
            if left-sum >= right-sum and left-sum >= cross-sum
                return (left-low, left-high, left-sum)
            else if right-sum >= left-sum and right-sum >= cross-sum
                 return (right-low, right-high, right-sum)
            return (cross-low, cross-high, cross-sum)
    
    
    FIND-MAX-CROSSING-SUBARRAY(A, low, mid, high)
        left-sum = -∞
        sum = 0
        for i = mid downto low
            sum = sum + A[i]
            if sum > left-sum
                left-sum = sum
                max-left = i
        right-sum = -∞
        sum =0;
        for j = mid + 1 to high
            sum = sum + A[j]
            if sum > right-sum
                right-sum = sum
                max-right = j
        return (max-left, max-right, left-sum + right-sum)

4. 分析

我之前说过，所有的比较最后都是两个数比较。把最大子数组通过分治策略最后都是一个元素，这时候就是直接返回这个数，交给上一层。

这时候数组有两个数，子数组就到了2所说的比较三种情况，再一层层向上递交结果

5. 代码实现

java

public class MaxArray {
    
        private static class Result {
            int low;
            int high;
            int sum;
    
            public Result(int low, int high, int sum) {
                this.low = low;
                this.high = high;
                this.sum = sum;
            }
        }
    
    
        static Result findMaximumSubarray(int[] A, int low, int high) {
            if (low == high) {
                return new Result(low, high, A[low]);
            } else {
                int mid = (low + high)/2;
                Result leftResult = findMaximumSubarray(A, low, mid);
                Result rightResult = findMaximumSubarray(A, mid+1, high);
                Result crossResult = findMaxCrossingSubarray(A, low, mid, high);
                if (leftResult.sum >= rightResult.sum && leftResult.sum >= crossResult.sum)
                    return leftResult;
                else if (rightResult.sum >= leftResult.sum && rightResult.sum >= crossResult.sum)
                    return rightResult;
                else return crossResult;
            }
    
        }
    
        static Result findMaxCrossingSubarray(int[] A, int low, int mid, int high) {
    
            //向左试探
            int leftSum = Integer.MIN_VALUE;   //哨兵
            int maxLeft = mid;
            int sum = 0;
            for (int i = mid; i >= low; i--) {
                sum += A[i];
                if (sum > leftSum) {
                    leftSum = sum;
                    maxLeft = i;
                }
            }
            //向右试探
            int rightSum = Integer.MIN_VALUE;
            int maxRight = mid + 1;
            sum = 0;
            for (int j = mid + 1; j <= high; j++) {
                sum += A[j];
                if (sum > rightSum) {
                    rightSum = sum;
                    maxRight = j;
                }
            }
            //将两边的结果合起来
            return new Result(maxLeft, maxRight, leftSum + rightSum);
        }
    
        public static void main(String[] args) {
            int[] A = {-1, 5, 6, 9, 10, -9, -8, 100, -200};
            Result result = findMaximumSubarray(A, 0,  A.length-1);
            System.out.println(result.low + "," + result.high + " " + result.sum);
        }
    }

python

def find_maximum_subarray(nums, low, high):
        if low == high:
            return {
        "low": low, "high": high, "sum": nums[low]}
        else:
            mid = int((low + high) / 2)
            left_result = find_maximum_subarray(nums, low, mid)
            right_result = find_maximum_subarray(nums, mid + 1, high)
            cross_result = find_max_crossing_subarray(nums, low, mid, high)
            if left_result["sum"] >= right_result["sum"] and left_result["sum"] >= cross_result["sum"]:
                return left_result
            else:
                if right_result["sum"] >= left_result["sum"] and right_result["sum"] >= cross_result["sum"]:
                    return right_result
                else:
                    return cross_result
    
    
    def find_max_crossing_subarray(nums, low, mid, high):
        left_sum = -float('inf')
        total = 0
        max_left = mid
        for i in range(mid, low-1, -1):
            total += nums[i]
            if total > left_sum:
                left_sum = total
                max_left = i
    
        rigth_sum = -float('inf')
        total = 0
        max_right = mid + 1
        for j in range(mid+1, high+1):
            total += nums[j]
            if total > rigth_sum:
                rigth_sum = total
                max_right = j
    
        return {
        "low": max_left, "high": max_right, "sum": left_sum + rigth_sum}
    
    
    if __name__ == "__main__":
        numss = [-1, 5, 6, 9, 10, -9, -8, 100, -200]
        result = find_maximum_subarray(numss, 0, len(numss)-1)
        print(result)

再分享个python用切片的方法

def find_maximum_subarray_slice(nums):
        max_sum = -float('inf')
        result = {}
        for i in range(len(nums)+1):
            for j in range(i, len(nums)+1):
                total = sum(nums[i:j])
                if total > max_sum:
                    max_sum = total
                    result["low"] = i
                    result["high"] = j-1
        result["sum"] = max_sum
        return result

C语言

typedef struct{
        int low;
        int high;
        int sum;
    }result;
    
    result find_maximum_subarray(int [], int, int);
    result find_max_crossing_subarray(int [], int, int, int);
    
    
    
    int main()
    {
        int arr[] = {-1, 5, 6, 9, 10, -9, -8, 100, -200};
        result re = find_maximum_subarray(arr, 0, 8);
        printf("%d, %d, %d\n", re.low, re.high, re.sum);
        return 0;
    }
    
    result find_maximum_subarray(int arr[], int low, int high)
    {
        if (low == high)
        {
            result re;
            re.low = low;
            re.high = high;
            re.sum = arr[low];
            return re;    
        }
        else
        {
            int mid = (low + high) / 2;
            result left_result = find_maximum_subarray(arr, low, mid);
            result right_result = find_maximum_subarray(arr, mid + 1, high);
            result cross_result = find_max_crossing_subarray(arr, low, mid, high);
            if(left_result.sum >= right_result.sum && left_result.sum >= cross_result.sum)
                return left_result;
            else if(right_result.sum >= left_result.sum && right_result.sum >= cross_result.sum)
                return right_result;
            else return cross_result;
        }
    }
    
    result find_max_crossing_subarray(int arr[], int low, int mid, int high)
    {
        int left_sum = -((unsigned)(~0) >> 1); //设置哨兵 
        int sum = 0;
        int i, max_left;
        for(i = mid; i >= low; i--) 
        {
            sum += arr[i];
            if(sum > left_sum)
            {
                left_sum = sum;
                max_left = i;
            }
        }
        
        int right_sum = -((unsigned)(~0) >> 1);
        sum = 0;
        int j, max_right;
        for(j = mid+1; j <= high; j++)
        {
            sum += arr[j];
            if(sum > right_sum)
            {
                right_sum = sum;
                max_right = j;
            }
        }
        
        result re;
        re.low = max_left;
        re.high = max_right;
        re.sum = left_sum + right_sum;
        return re;
    }

[1213881-20180902155520931-2115318469.png]: /images/20210328/1616939742912.png
[1213881-20180902160447285-387423155.png]: /images/20210328/1616939763223.png