Non-decreasing subsequence of size k with minimum sum

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 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);     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];        // intialize     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];        // intialize     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

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Improved By : vt_m, valarMorghulis18