Given an array of n positive integers. Write a program to find the sum of maximum sum subsequence of the given array such that the integers in the subsequence are sorted in increasing order. For example, if input is {1, 101, 2, 3, 100, 4, 5}, then output should be 106 (1 + 2 + 3 + 100), if the input array is {3, 4, 5, 10}, then output should be 22 (3 + 4 + 5 + 10) and if the input array is {10, 5, 4, 3}, then output should be 10
Solution: This problem is a variation of the standard Longest Increasing Subsequence (LIS) problem. We need a slight change in the Dynamic Programming solution of LIS problem. All we need to change is to use sum as a criteria instead of a length of increasing subsequence.
Following are the Dynamic Programming solution to the problem :Â Â
C++
/* Dynamic Programming implementation of Maximum Sum Increasing Subsequence (MSIS) problem */#include <bits/stdc++.h> using namespace std;   /* maxSumIS() returns the maximum sum of increasing subsequence in arr[] of size n */int maxSumIS(int arr[], int n) {     int i, j, max = 0;     int msis[n];       /* Initialize msis values     for all indexes */    for ( i = 0; i < n; i++ )         msis[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] &&                 msis[i] < msis[j] + arr[i])                 msis[i] = msis[j] + arr[i];       /* Pick maximum of     all msis values */    for ( i = 0; i < n; i++ )         if ( max < msis[i] )             max = msis[i];       return max; }   // Driver Code int main() {     int arr[] = {1, 101, 2, 3, 100, 4, 5};     int n = sizeof(arr)/sizeof(arr[0]);     cout << "Sum of maximum sum increasing "            "subsequence is " << maxSumIS( arr, n ) << endl;     return 0; }   // This is code is contributed by rathbhupendra |
C
/* Dynamic Programming implementation of Maximum Sum Increasing Subsequence (MSIS) problem */#include<stdio.h>   /* maxSumIS() returns the maximum    sum of increasing subsequence    in arr[] of size n */int maxSumIS(int arr[], int n) {     int i, j, max = 0;     int msis[n];       /* Initialize msis values        for all indexes */    for ( i = 0; i < n; i++ )         msis[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] &&                 msis[i] < msis[j] + arr[i])                 msis[i] = msis[j] + arr[i];       /* Pick maximum of        all msis values */    for ( i = 0; i < n; i++ )         if ( max < msis[i] )             max = msis[i];       return max; }   // Driver Code int main() {     int arr[] = {1, 101, 2, 3, 100, 4, 5};     int n = sizeof(arr)/sizeof(arr[0]);     printf("Sum of maximum sum increasing "             "subsequence is %d\n",               maxSumIS( arr, n ) );     return 0; } |
Java
/* Dynamic Programming Java    implementation of Maximum Sum    Increasing Subsequence (MSIS)    problem */class GFG {     /* maxSumIS() returns the     maximum sum of increasing     subsequence in arr[] of size n */    static int maxSumIS(int arr[], int n)     {         int i, j, max = 0;         int msis[] = new int[n];           /* Initialize msis values            for all indexes */        for (i = 0; i < n; i++)             msis[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] &&                     msis[i] < msis[j] + arr[i])                     msis[i] = msis[j] + arr[i];           /* Pick maximum of all            msis values */        for (i = 0; i < n; i++)             if (max < msis[i])                 max = msis[i];           return max;     }       // Driver code     public static void main(String args[])     {         int arr[] = new int[]{1, 101, 2, 3, 100, 4, 5};         int n = arr.length;         System.out.println("Sum of maximum sum "+                             "increasing subsequence is "+                               maxSumIS(arr, n));     } }   // This code is contributed // by Rajat Mishra |
Python3
# Dynamic Programming based Python # implementation of Maximum Sum # Increasing Subsequence (MSIS) # problem   # maxSumIS() returns the maximum # sum of increasing subsequence # in arr[] of size n def maxSumIS(arr, n):     max = 0    msis = [0 for x in range(n)]       # Initialize msis values     # for all indexes     for i in range(n):         msis[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                msis[i] < msis[j] + arr[i]):                 msis[i] = msis[j] + arr[i]       # Pick maximum of     # all msis values     for i in range(n):         if max < msis[i]:             max = msis[i]       return max  # Driver Code arr = [1, 101, 2, 3, 100, 4, 5] n = len(arr) print("Sum of maximum sum increasing " +                      "subsequence is " +                  str(maxSumIS(arr, n)))   # This code is contributed # by Bhavya Jain |
C#
// Dynamic Programming C# implementation // of Maximum Sum Increasing Subsequence // (MSIS) problem using System; class GFG {           // maxSumIS() returns the     // maximum sum of increasing     // subsequence in arr[] of size n     static int maxSumIS( int []arr, int n )     {         int i, j, max = 0;         int []msis = new int[n];           /* Initialize msis values            for all indexes */        for ( i = 0; i < n; i++ )             msis[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] &&                     msis[i] < msis[j] + arr[i])                     msis[i] = msis[j] + arr[i];           /* Pick maximum of all            msis values */        for ( i = 0; i < n; i++ )             if ( max < msis[i] )                 max = msis[i];           return max;     }           // Driver Code     public static void Main()     {         int []arr = new int[]{1, 101, 2, 3, 100, 4, 5};         int n = arr.Length;         Console.WriteLine("Sum of maximum sum increasing "+                                         " subsequence is "+         maxSumIS(arr, n));     } }   // This code is contributed by Sam007 |
PHP
<?php // Dynamic Programming implementation // of Maximum Sum Increasing // Subsequence (MSIS) problem   // maxSumIS() returns the maximum // sum of increasing subsequence // in arr[] of size n function maxSumIS($arr, $n) {     $max = 0;     $msis= array($n);       // Initialize msis values     // for all indexes     for($i = 0; $i < $n; $i++ )         $msis[$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] &&                 $msis[$i] < $msis[$j] + $arr[$i])                 $msis[$i] = $msis[$j] + $arr[$i];       // Pick maximum of all msis values     for($i = 0;$i < $n; $i++ )         if ($max < $msis[$i] )             $max = $msis[$i];       return $max; }       // Driver Code     $arr = array(1, 101, 2, 3, 100, 4, 5);     $n = count($arr);     echo "Sum of maximum sum increasing subsequence is "                                    .maxSumIS( $arr, $n );           // This code is contributed by Sam007 ?> |
Javascript
<script>   // Dynamic Programming implementation // of Maximum Sum Increasing Subsequence // (MSIS) problem   // maxSumIS() returns the maximum // sum of increasing subsequence // in arr[] of size n function maxSumIS(arr, n) {     let i, j, max = 0;     let msis = new Array(n);       // Initialize msis values     // for all indexes     for(i = 0; i < n; i++)         msis[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] &&                 msis[i] < msis[j] + arr[i])                 msis[i] = msis[j] + arr[i];       // Pick maximum of     // all msis values     for(i = 0; i < n; i++)         if (max < msis[i])             max = msis[i];       return max; }   // Driver Code let arr = [ 1, 101, 2, 3, 100, 4, 5 ]; let n = arr.length; document.write("Sum of maximum sum increasing " +                "subsequence is " + maxSumIS(arr, n));                  // This code is contributed by rishavmahato348   </script> |
Output
Sum of maximum sum increasing subsequence is 106
Time Complexity: O(n^2)Â
Space Complexity O(n)Â
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