使用 D&C/递归的最大子数组

Question

我想用展示 (n log n) 的算法来实现最大子数组问题：

找到最大的连续子数组，或数组中连续元素的最大总和。

假设：并非所有元素都是负数

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我有一些可行的解决方案；问题在于重叠的中心数组，以及指定重叠子问题的适当索引，一些数组我得到了正确的答案，而另一些则没有。

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只是为了比较和检查正确性，我实现了一个称为 Kadane 算法的解决方案 （我相信复杂度是 Omega(n)）。

这是 Kandane 的算法（http://en.wikipedia.org/wiki/Maximum_subarray_problem）：

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 public static void Kadane(int array[]) {
        int max_ending_here = 0;
        for (int i = 0; i < array.length; i++) {
            max_ending_here = max_ending_here + array[i];
            max_so_far = Math.max(max_so_far, max_ending_here);
        }
        System.out.println("Kadane(int array []): " + max_so_far);
    }

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我的递归实现，使用分治法来比较子数组的最大值，然后对具有最大值的子数组进行递归调用，直到递归结束

public static void findMaxSubArray(int[] array, int lowIndex, int highIndex) {

        int mid = 0;
        int arrayLength = 0;
        int maxEndingHere = 0;

        if (array == null) {
            throw new NullPointerException();
        } else if 
                //base condition 
           (array.length < 2 || (highIndex==lowIndex)) {
            maxLowIndex = lowIndex;
            maxHighIndex = highIndex;
            System.out.println("findMaxSubArray(int[] array, int lowIndex, int highIndex)");
            System.out.println("global Max Range, low:" + maxLowIndex + " high: " + maxHighIndex);
            System.out.println("global Max Sum:" + globalMaximum);
        } else {
            System.out.println();
            int lowMidMax = 0;
            int midHighMax = 0;
            int centerMax = 0;
            //array length is always the highest index +1 
            arrayLength = highIndex + 1;

            //if even number of elements in array 
            if (array.length % 2 == 0) {
                mid = arrayLength / 2;
                System.out.println("low: " + lowIndex + " mid: " + mid);
                for (int i = lowIndex; i < mid; i++) {
                    System.out.print(array[i] + ",");
                }
                //calculate maximum contigous array encountered so far in low to mid indexes 
                for (int i = lowIndex; i < mid; i++) {
                    maxEndingHere = maxEndingHere + array[i];
                    if (maxEndingHere < 0) {
                        maxEndingHere = 0;
                    }

                    if (globalMaximum < maxEndingHere) {
                        //new maximum found 
                        globalMaximum = lowMidMax = maxEndingHere;
                        lowIndex = i;
                    }

                }
                //end low mid calc 

                for (int i = mid; i <= highIndex; i++) {
                    System.out.print(array[i] + ",");
                }
                System.out.println("mid: " + mid + " high: " + highIndex);
                //calculate maximum contigous array encountered so far in mid to high indexes 
                for (int i = mid; i <= highIndex; i++) {
                    maxEndingHere = maxEndingHere + array[i];
                    if (maxEndingHere < 0) {
                        maxEndingHere = 0;
                    }

                    if (globalMaximum < maxEndingHere) {
                        //new maximum found 
                        globalMaximum = midHighMax = maxEndingHere;
                        mid = i;
                    }

                }
//end mid high calc
                //calculate maximum contigous array encountered so far in center array
                int lowCenter = mid -1;
                int highCenter = highIndex -1;

                System.out.println("lowCenter: " + lowCenter + " highCenter: " + highCenter);
                for (int i = lowCenter; i < highCenter; i++) {
                    System.out.print(array[i] + ",");
                }
                //calculate maximum contigous array encountered so far in low to mid indexes 
                for (int i = lowCenter; i < highCenter; i++) {
                    maxEndingHere = maxEndingHere + array[i];
                    if (maxEndingHere < 0) {
                        maxEndingHere = 0;
                    }

                    if (globalMaximum < maxEndingHere) {
                        //new max found
                        globalMaximum = centerMax = maxEndingHere;
                        lowCenter = i;

                    }

                }
                //end center calc 
                //determine which range contains the maximum sub array 
                //if lowMidMax <= midHighMax and centerMax
                if (lowMidMax >= midHighMax && lowMidMax >= centerMax) {
                    maxLowIndex = lowIndex;
                    maxHighIndex = mid;
                    //recursive call
                    findMaxSubArray(array, lowIndex, mid);
                }
                if (midHighMax >= lowMidMax && midHighMax >= centerMax) {
                    maxLowIndex = mid;
                    maxHighIndex = highIndex;
                    //recursive call
                    findMaxSubArray(array, mid, highIndex);
                }
                if (centerMax >= midHighMax && centerMax >= lowMidMax) {
                    maxLowIndex = lowCenter;
                    maxHighIndex = highCenter;
                    //recursive call
                    findMaxSubArray(array, lowCenter, highCenter);
                }

            }//end if even parent array 
            //else if uneven array 
            else {
                mid = (int) Math.floor(arrayLength / 2);
                System.out.println("low: " + lowIndex + " mid: " + mid);
                for (int i = lowIndex; i < mid; i++) {
                    System.out.print(array[i] + ",");
                }
                //calculate maximum contigous array encountered so far in low to mid indexes 
                for (int i = lowIndex; i < mid; i++) {
                    maxEndingHere = maxEndingHere + array[i];
                    if (maxEndingHere < 0) {
                        maxEndingHere = 0;
                    }

                    if (globalMaximum < maxEndingHere) {
                        //new maximum found 
                        globalMaximum = lowMidMax = maxEndingHere;
                        lowIndex = i;
                    }

                }
                //end low mid calc
                System.out.println("mid+1: " + (mid + 1) + " high: " + highIndex);
                for (int i = mid + 1; i <= highIndex; i++) {
                    System.out.print(array[i] + ",");
                }
                //calculate maximum contigous array encountered so far in mid to high indexes 
                for (int i = mid + 1; i <= highIndex; i++) {
                    maxEndingHere = maxEndingHere + array[i];
                    if (maxEndingHere < 0) {
                        maxEndingHere = 0;
                    }

                    if (globalMaximum < maxEndingHere) {
                        //new maximum found 
                        globalMaximum = midHighMax = maxEndingHere;
                        mid = i;
                    }

                }
                //end mid high calc
                //calculate maximum contigous array encountered so far in center array
                int lowCenter =  mid;
                int highCenter = highIndex -1;

                System.out.println("lowCenter: " + lowCenter + " highCenter: " + highCenter);
                for (int i = lowCenter; i < highCenter; i++) {
                    System.out.print(array[i] + ",");
                }
                //calculate maximum contigous array encountered so far in low to mid indexes 
                for (int i = lowCenter; i < highCenter; i++) {
                    maxEndingHere = maxEndingHere + array[i];
                    if (maxEndingHere < 0) {
                        maxEndingHere = 0;
                    }

                    if (globalMaximum < maxEndingHere) {
                        //new max
                        globalMaximum = centerMax = maxEndingHere;
                        lowCenter = i;
                    }

                }
                //end center calc 

                //determine which range contains the maximum sub array 
                //if lowMidMax <= midHighMax and centerMax
                  if (lowMidMax >= midHighMax && lowMidMax >= centerMax) {
                    maxLowIndex = lowIndex;
                    maxHighIndex = mid;
                    //recursive call
                    findMaxSubArray(array, lowIndex, mid);
                }
                if (midHighMax >= lowMidMax && midHighMax >= centerMax) {
                    maxLowIndex = mid;
                    maxHighIndex = highIndex;
                    //recursive call
                    findMaxSubArray(array, mid, highIndex);
                }
                if (centerMax >= midHighMax && centerMax >= lowMidMax) {
                    maxLowIndex = lowCenter;
                    maxHighIndex = highCenter;
                    //recursive call
                    findMaxSubArray(array, lowCenter, highCenter);
                }


            }//end odd parent array length 
        }//end outer else array is ok to computed 

    }//end method

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结果：使用数组 subArrayProblem1 = {1, 2, 3, 4, 5, 6, 7, 8};

Kadane(int array []): 36 低：0 中：4 1,2,3,4,5,6,7,8,中：4 高：7 低中心：6 高中心：6

findMaxSubArray(int[] 数组, int lowIndex, int highIndex) 全局最大范围，低：7 高：7 全局最大总和：36 构建成功（总时间：0 秒）

问题虽然与 Kadane 相比全局最大总和是正确的，但低指数和高指数范围反映了最后一次递归调用。

结果：使用数组 subArrayProblem = {100, 113, 110, 85, 105, 102, 86, 63, 81, 101, 94, 106, 101, 79, 94, 90, 97}；

Kadane(int 数组 []): 1607 低：0 中：8 100,113,110,85,105,102,86,63,中+1:9 高:16 101,94,106,101,79,94,90,97,lowCenter: 16 highCenter: 15

findMaxSubArray(int[] 数组, int lowIndex, int highIndex) 全局最大范围，低：16 高：16 全局最大总和：1526

全局最大值不正确，注意区别其实是1个元素，也就是第81个元素

Answer 1

一种更简洁的方法来找到最大和的子数组（D/C 递归方式）：

// A Divide and Conquer based Java solution
// To find a subarray with the maximum sum

import java.util.Arrays;
import java.util.Scanner;

class MaxSubArray {

    private static Result maxCrossingSum(int arr[], int l, int m, int h) {

        int sum = 0;
        int maxLeft = 0;
        int leftSum = Integer.MIN_VALUE;
        for (int i = m; i >= l; i--) {
            sum = sum + arr[i];
            if (sum > leftSum) {
                leftSum = sum;
                maxLeft = i;
            }
        }

        sum = 0;
        int rightSum = Integer.MIN_VALUE;
        int maxRight = arr.length - 1;
        for (int i = m + 1; i <= h; i++) {
            sum = sum + arr[i];
            if (sum > rightSum) {
                rightSum = sum;
                maxRight = i;
            }
        }

        return new Result(maxLeft, maxRight, leftSum + rightSum);
    }

    private static Result maxSubArray(int[] A, int low, int high) {

        if (low == high) {
            return new Result(low, high, A[low]);
        }

        int mid = (low + high) / 2;

        Result leftSubArray = maxSubArray(A, low, mid);
        Result rightSubArray = maxSubArray(A, mid + 1, high);
        Result maxCrossingSubArray = maxCrossingSum(A, low, mid, high);

        int leftSum = leftSubArray.sum;
        int rightSum = rightSubArray.sum;
        int crossSum = maxCrossingSubArray.sum;

        if (leftSum > rightSum && leftSum > crossSum) {
            return new Result(leftSubArray.low, leftSubArray.high, leftSubArray.sum);
        } else if (rightSum > leftSum && rightSum > crossSum) {
            return new Result(rightSubArray.low, rightSubArray.high, rightSubArray.sum);
        } else {
            return new Result(maxCrossingSubArray.low, maxCrossingSubArray.high,
                maxCrossingSubArray.sum);
        }
    }

    public static class Result {

        public int low;
        public int high;
        public int sum;
        public Result(int low, int high, int sum) {
            this.low = low;
            this.high = high;
            this.sum = sum;
        }

    }

    /* Driver program to test maxSubArray */
    public static Result maxSubArray(int[] arr) {
        return maxSubArray(arr, 0, arr.length - 1);
    }

    public static void main(String[] args) {

        Scanner sc = new Scanner(System.in);
        String[] arrString = sc.nextLine().split(" ");

        int n = arrString.length;

        int[] arr = new int[n];
        for (int i = 0; i < n; i++) {
            arr[i] = Integer.parseInt(arrString[i]);
        }

        Result result = maxSubArray(arr);

        int[] subArray = new int[result.high - result.low + 1];
        int j = 0;
        for (int i = result.low; i <= result.high; i++) {
            subArray[j++] = arr[i];
        }

        System.out.println(Arrays.toString(subArray));
        System.out.println("Sum : " + result.sum);
    }
}

Answer 2

1. Kadane 算法的实现是错误的，它会在带有一些负数的数组上失败。 正确的应该是这样的：

 public static void Kadane(int array[]) {
        int max_ending_here = 0;
        for (int i = 0; i < array.length; i++) {
            max_ending_here = Math.max(array[i], max_ending_here + array[i]);
            max_so_far = Math.max(max_so_far, max_ending_here);
        }
        System.out.println("Kadane(int array []): " + max_so_far);
    }

您的代码中有很多错误，例如：

2.maxEndingHere 在计算答案之前应初始化为 0：

[lowIndex,mid)
[mid, highIndex]
[lowCenter, highCenter]

现在它只在第一次迭代之前被初始化。

3.lowCenter应该初始化为lowIndex

4.程序太长太复杂了……不知道有没有漏掉什么bug……

Answer 3

解决方案非常简单，pv 是一个变量，它向我们展示了从 k 其中 0

/**
 * @author : Yash M. Sawant
 */




#include <stdio.h>
#include <stdlib.h>
#include <time.h>

#define MAXLENGTH 17
#define MINVALUE -999



int main() {
    int i;

    int s[MAXLENGTH]; s[0] = 0;
    int c[MAXLENGTH] = {0, 13, -3, -25, 20, -3, -16, -23, 18, 20, -7, 12, -5, -22, 15, -4, 7};
    int pv = 0, p = -1, tp = -1;
    for(i = 1 ; i < MAXLENGTH ; i ++) {
        printf("%4d ", c[i]);
        if(s[i - 1] < pv + c[i]) {
            s[i] = pv + c[i];
            pv = pv + c[i];
            if(tp > p) {
                p = tp;
            }
        } else {
            s[i] = s[i - 1];
            pv = pv + c[i];
            if(pv < 0) {
                pv = 0; tp = i;
            }
        }
    }
    printf("\n");
    for(i = 0 ; i < MAXLENGTH ; i ++) {
        printf("%4d ", s[i]);
    }
    printf("\n");
    printf("Max Sub Array = %d and Starts at %d ", s[MAXLENGTH - 1], p);
    return 0;

}