最大和减子序列

给定一个由N个正整数组成的数组。任务是找到给定数组的最大总和递减子序列(MSDS)的总和，以使子序列中的整数按递减顺序排序。
例子：

Input: arr[] = {5, 4, 100, 3, 2, 101, 1} 
Output: 106 
100 + 3 + 2 + 1 = 106
Input: arr[] = {10, 5, 4, 3} 
Output: 22 
10 + 5 + 4 + 3 = 22 
 

此问题是最长递减子序列问题的一种变体。上述问题的最佳子结构为：
假设arr [0..n-1]为输入数组， MSDS [i]为以索引i结尾的MSDS的最大和，以使arr [i]为MSDS的最后一个元素。
然后，MSDS [i]可以写成：

MSDS[i] = a[i] + max( MSDS[j] ) where i > j > 0 and arr[j] > arr[i] or, 
MSDS[i] = a[i], if no such j exists.

要查找给定数组的MSDS，我们需要返回max(MSDS [i])，其中n> i> 0。
下面是上述方法的实现：

C++

// CPP code to return the maximum sum
// of decreasing subsequence in arr[]
#include 
using namespace std;
 
// function to return the maximum
// sum of decreasing subsequence
// in arr[]
int maxSumDS(int arr[], int n)
{
    int i, j, max = 0;
    int MSDS[n];
 
    // Initialize msds values
    // for all indexes
    for (i = 0; i < n; i++)
        MSDS[i] = arr[i];
 
    // Compute maximum sum values
    // in bottom up manner
    for (i = 1; i < n; i++)
        for (j = 0; j < i; j++)
            if (arr[i] < arr[j] && MSDS[i] < MSDS[j] + arr[i])
                MSDS[i] = MSDS[j] + arr[i];
 
    // Pick maximum of all msds values
    for (i = 0; i < n; i++)
        if (max < MSDS[i])
            max = MSDS[i];
 
    return max;
}
 
// Driver Code
int main()
{
    int arr[] = { 5, 4, 100, 3, 2, 101, 1 };
     
    int n = sizeof(arr) / sizeof(arr[0]);
 
    cout << "Sum of maximum sum decreasing subsequence is: "
         << maxSumDS(arr, n);
    return 0;
}

Java

// Java code to return the maximum sum
// of decreasing subsequence in arr[]
import java.io.*;
import java.lang.*;
 
class GfG {
     
    // function to return the maximum
    // sum of decreasing subsequence
    // in arr[]
    public static int maxSumDS(int arr[], int n)
    {
        int i, j, max = 0;
        int[] MSDS = new int[n];
     
        // Initialize msds values
        // for all indexes
        for (i = 0; i < n; i++)
            MSDS[i] = arr[i];
     
        // Compute maximum sum values
        // in bottom up manner
        for (i = 1; i < n; i++)
            for (j = 0; j < i; j++)
                if (arr[i] < arr[j] &&
                    MSDS[i] < MSDS[j] + arr[i])
                    MSDS[i] = MSDS[j] + arr[i];
     
        // Pick maximum of all msds values
        for (i = 0; i < n; i++)
            if (max < MSDS[i])
                max = MSDS[i];
     
        return max;
    }
     
    // Driver Code
    public static void main(String argc[])
    {
        int arr[] = { 5, 4, 100, 3, 2, 101, 1 };
         
        int n = 7;
     
        System.out.println("Sum of maximum sum"
               + " decreasing subsequence is: "
                           + maxSumDS(arr, n));
    }
}
 
// This code os contributed by Sagar Shukla.

Python3

# Python3 code to return the maximum sum
# of decreasing subsequence in arr[]
 
# Function to return the maximum
# sum of decreasing subsequence
# in arr[]
def maxSumDS(arr, n):
     
    i, j, max = (0, 0, 0)
     
    MSDS=[0 for i in range(n)]
  
    # Initialize msds values
    # for all indexes
    for i in range(n):
        MSDS[i] = arr[i]
  
    # Compute maximum sum values
    # in bottom up manner
    for i in range(1, n):
        for j in range(i):
            if (arr[i] < arr[j] and
                MSDS[i] < MSDS[j] + arr[i]):
                MSDS[i] = MSDS[j] + arr[i]
  
    # Pick maximum of all msds values
    for i in range(n):
        if (max < MSDS[i]):
            max = MSDS[i]
 
    return max
     
if __name__ == "__main__":
     
    arr=[5, 4, 100, 3,
         2, 101, 1]
    n=len(arr)
    print("Sum of maximum sum decreasing subsequence is: ",
           maxSumDS(arr, n))
 
# This code is contributed by Rutvik_56

C#

// C# code to return the
// maximum sum of decreasing
// subsequence in arr[]
using System;
 
class GFG
{
     
    // function to return the
    // maximum sum of decreasing
    // subsequence in arr[]
    public static int maxSumDS(int []arr,
                               int n)
    {
        int i, j, max = 0;
        int[] MSDS = new int[n];
     
        // Initialize msds values
        // for all indexes
        for (i = 0; i < n; i++)
            MSDS[i] = arr[i];
     
        // Compute maximum sum values
        // in bottom up manner
        for (i = 1; i < n; i++)
            for (j = 0; j < i; j++)
                if (arr[i] < arr[j] &&
                    MSDS[i] < MSDS[j] + arr[i])
                    MSDS[i] = MSDS[j] + arr[i];
     
        // Pick maximum of
        // all msds values
        for (i = 0; i < n; i++)
            if (max < MSDS[i])
                max = MSDS[i];
     
        return max;
    }
     
    // Driver Code
    static public void Main ()
    {
        int []arr = {5, 4, 100,
                     3, 2, 101, 1};
        int n = 7;
        Console.WriteLine("Sum of maximum sum" +
                " decreasing subsequence is: " +
                              maxSumDS(arr, n));
    }
}
 
// This code is contributed by m_kit

输出：

Sum of maximum sum decreasing subsequence is: 106

时间复杂度： O(N ² )
辅助空间： O(N)