【算法导论】最大子数组——递归

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。

　　

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

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
", 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;
}