Non-decreasing subsequence of size k with minimum sum

Difficulty Level : Medium
Last Updated : 05 Dec, 2019

Given a sequence of n integers, you have to find out the non-decreasing subsequence of length k with minimum sum. If no sequence exists output -1.

Examples :

Input : [58 12 11 12 82 30 20 77 16 86], 
        k = 3
Output : 39
{11 + 12 + 16}

Input : [58 12 11 12 82 30 20 77 16 86], 
        k = 4
Output : 120
{11 + 12 + 20 + 77}

Input : [58 12 11 12 82 30 20 77 16 86], 
        k = 5
Output : 206

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Let solve(i, k) be the minimum sum of a subsequence of size k ending at index i. Then there would be two states:
1. Include current element. {solve(j, k-1) + a[i]}
2. Exclude current element. {solve(j, k)}
Our recurrence state would be:

 
dp[i][k] = min(solve(j, k-1) + a[i], solve(j, k)) 
  if a[i] >= a[j] for all 0 <= j <= i.

C++

// C++ program to find Non-decreasing sequence
// of size k with minimum sum
#include <bits/stdc++.h>
using namespace std;
const int MAX = 100;
const int inf = 2e9;
  
// Global table used for memoization
int dp[MAX][MAX];
  
void initialize()
{
    for (int i = 0; i < MAX; i++)
        for (int j = 0; j < MAX; j++)
            dp[i][j] = -1;
}
  
int solve(int arr[], int i, int k)
{
    // If already computed
    if (dp[i][k] != -1)
        return dp[i][k];
  
    // Corner cases
    if (i < 0)
        return inf;
    if (k == 1) {
        int ans = inf;
        for (int j = 0; j <= i; j++)
            ans = min(ans, arr[j]);
        return ans;
    }
  
    // Recursive computation.
    int ans = inf;
    for (int j = 0; j < i; j++)
        if (arr[i] >= arr[j])
            ans = min(ans, min(solve(arr, j, k),
                               solve(arr, j, k - 1) + arr[i]));
        else {
            ans = min(ans, solve(arr, j, k));
        }
  
    dp[i][k] = ans;
    return dp[i][k];
}
  
// Driver code
int main()
{
    initialize();
    int a[] = { 58, 12, 11, 12, 82, 30,
                20, 77, 16, 86 };
    int n = sizeof(a) / sizeof(a[0]);
    int k = 4;
    cout << solve(a, n - 1, k) << endl;
    return 0;
}

Java

// Java program to find Non-decreasing sequence
// of size k with minimum sum
import java.io.*;
import java.util.*;
  
class GFG {
    public static int MAX = 100;
    public static int inf = 1000000;
  
    // Table used for memoization
    public static int[][] dp = new int[MAX][MAX];
  
    // initialize
    static void initialize()
    {
        for (int i = 0; i < MAX; i++)
            for (int j = 0; j < MAX; j++)
                dp[i][j] = -1;
    }
  
    // Function to find non-decreasing sequence
    // of size k with minimum sum
    static int solve(int arr[], int i, int k)
    {
        // If already computed
        if (dp[i][k] != -1)
            return dp[i][k];
  
        // Corner cases
        if (i < 0)
            return inf;
        if (k == 1) {
            int ans = inf;
            for (int j = 0; j <= i; j++)
                ans = Math.min(ans, arr[j]);
            return ans;
        }
  
        // Recursive computation
        int ans = inf;
        for (int j = 0; j < i; j++)
            if (arr[i] >= arr[j])
                ans = Math.min(ans, Math.min(solve(arr, j, k), solve(arr, j, k - 1) + arr[i]));
            else
                ans = Math.min(ans, solve(arr, j, k));
  
        dp[i][k] = ans;
  
        return dp[i][k];
    }
  
    // driver program
    public static void main(String[] args)
    {
        initialize();
        int a[] = { 58, 12, 11, 12, 82, 30,
                    20, 77, 16, 86 };
        int n = a.length;
        int k = 4;
        System.out.println(solve(a, n - 1, k));
    }
}
  
// Contributed by Pramod Kumar

Python

# Python program to find Non-decreasing sequence
# of size k with minimum sum
   
# Global table used for memoization
dp = []
for i in xrange(10**2 + 1):
    temp = [-1]*(10**2 + 1)
    dp.append(temp)
   
def solve(a, i, k):
    if dp[i][k] != -1:  # Memoization
        return dp[i][k]
    elif i < 0: # out of bounds
        return float('inf')
   
    # when there is only one element
    elif k == 1:    
        return min(a[: i + 1])
   
    # Else two cases
    # 1 include current element 
    # solve(a, j, k-1) + a[i]
    # 2 ignore current element 
    # solve(a, j, k)
    else:  
        ans = float('inf')
        for j in xrange(i):
            if a[i] >= a[j]:
                ans = min(ans, solve(a, j, k), solve(a, j, k-1) + a[i])
            else:
                ans = min(ans, solve(a, j, k))
        dp[i][k] = ans
        return dp[i][k]
   
# Driver code
a = [58, 12, 11, 12, 82, 30, 20, 77, 16, 86]        
print solve(a, len(a)-1, 4)

C#

// C# program to find Non-decreasing sequence
// of size k with minimum sum
using System;
  
class GFG {
      
    public static int MAX = 100;
    public static int inf = 1000000;
  
    // Table used for memoization
    public static int[, ] dp = new int[MAX, MAX];
  
    // initialize
    static void initialize()
    {
        for (int i = 0; i < MAX; i++)
          for (int j = 0; j < MAX; j++)
            dp[i, j] = -1;
    }
  
    // Function to find non-decreasing 
    // sequence of size k with minimum sum
    static int solve(int[] arr, int i, int k)
    {
        int ans = 0;
          
        // If already computed
        if (dp[i, k] != -1)
            return dp[i, k];
  
        // Corner cases
        if (i < 0)
            return inf;
        if (k == 1)
        {
            ans = inf;
            for (int j = 0; j <= i; j++)
            ans = Math.Min(ans, arr[i]);
            return ans;
        }
  
        // Recursive computation
        ans = inf;
        for (int j = 0; j < i; j++)
            if (arr[i] >= arr[j])
                ans = Math.Min(ans, Math.Min(solve(arr, j, k),
                               solve(arr, j, k - 1) + arr[i]));
            else
                ans = Math.Min(ans, solve(arr, j, k));
  
        dp[i, k] = ans;
  
        return dp[i, k];
    }
  
    // driver program
    public static void Main()
    {
        initialize();
        int[] a = { 58, 12, 11, 12, 82, 30,
                          20, 77, 16, 86 };
        int n = a.Length;
        int k = 4;
        Console.WriteLine(solve(a, n - 1, k));
    }
}
  
// This code is contributed by vt_m

This article is contributed by Anuj Shah. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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