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Divide the array in K segments such that the sum of minimums is maximized
• Difficulty Level : Hard
• Last Updated : 19 May, 2021

Given an array a of size N and an integer K, the task is to divide the array into K segments such that sum of the minimum of K segments is maximized.
Examples:

Input: a[] = {5, 7, 4, 2, 8, 1, 6}, K = 3
Output:
Divide the array at indexes 0 and 1. Then the segments are {5}, {7}, {4, 2, 8, 1, 6}. Sum of the minimus is 5 + 7 + 1 = 13
Input: a[] = {6, 5, 3, 8, 9, 10, 4, 7, 10}, K = 4
Output: 27

Approach: The problem can be solved using Dynamic Programming. Try all the possible partitions that are possible using recursion. Let dp[i][k] be the maximum sum of minimums till index i with k partitions. Hence the possible states will be partition at every index from the index i till n. The maximum sum of minimums of all those states will be our answer. After writing this recurrence, we can use memorize.

Below is the implementation of the above approach:

## C++

 `// C++ program to find the sum of``// the minimum of all the segments``#include ``using` `namespace` `std;``const` `int` `MAX = 10;` `// Function to maximize the sum of the minimums``int` `maximizeSum(``int` `a[], ``int` `n, ``int` `ind,``                ``int` `k, ``int` `dp[MAX][MAX])``{` `    ``// If k segments have been divided``    ``if` `(k == 0) {``        ``// If we are at the end``        ``if` `(ind == n)``            ``return` `0;` `        ``// If we donot reach the end``        ``// then return a negative number``        ``// that cannot be the sum``        ``else``            ``return` `-1e9;``    ``}` `    ``// If at the end but``    ``// k segments are not formed``    ``else` `if` `(ind == n)``        ``return` `-1e9;` `    ``// If the state has not been visited yet``    ``else` `if` `(dp[ind][k] != -1)``        ``return` `dp[ind][k];` `    ``// If the state has not been visited``    ``else` `{``        ``int` `ans = 0;` `        ``// Get the minimum element in the segment``        ``int` `mini = a[ind];` `        ``// Iterate and try to break at every index``        ``// and create a segment``        ``for` `(``int` `i = ind; i < n; i++) {` `            ``// Find the minimum element in the segment``            ``mini = min(mini, a[i]);` `            ``// Find the sum of all the segments trying all``            ``// the possible combinations``            ``ans = max(ans, maximizeSum(``              ``a, n, i + 1, k - 1, dp) + mini);``        ``}` `        ``// Return the answer by``        ``// memoizing it``        ``return` `dp[ind][k] = ans;``    ``}``}` `// Driver Code``int` `main()``{``    ``int` `a[] = { 5, 7, 4, 2, 8, 1, 6 };``    ``int` `k = 3;``    ``int` `n = ``sizeof``(a) / ``sizeof``(a);` `    ``// Initialize dp array with -1``    ``int` `dp[MAX][MAX];``    ``memset``(dp, -1, ``sizeof` `dp);` `    ``cout << maximizeSum(a, n, 0, k, dp);` `    ``return` `0;``}`

## Java

 `// Java program to find the sum of``// the minimum of all the segments` `class` `GFG``{` `    ``static` `int` `MAX = ``10``;` `    ``// Function to maximize the``    ``// sum of the minimums``    ``public` `static` `int` `maximizeSum(``              ``int``[] a, ``int` `n, ``int` `ind,``                 ``int` `k, ``int``[][] dp)``    ``{``        ``// If k segments have been divided``        ``if` `(k == ``0``)``        ``{``            ``// If we are at the end``            ``if` `(ind == n)``                ``return` `0``;` `            ``// If we donot reach the end``            ``// then return a negative number``            ``// that cannot be the sum``            ``else``                ``return` `-``1000000000``;``        ``}` `        ``// If at the end but``        ``// k segments are not formed``        ``else` `if` `(ind == n)``            ``return` `-``1000000000``;` `        ``// If the state has not``        ``// been visited yet``        ``else` `if` `(dp[ind][k] != -``1``)``            ``return` `dp[ind][k];` `        ``// If the state has``        ``// not been visited``        ``else``        ``{``            ``int` `ans = ``0``;` `            ``// Get the minimum``            ``// element in the segment``            ``int` `mini = a[ind];` `            ``// Iterate and try to break``            ``// at every index``            ``// and create a segment``            ``for` `(``int` `i = ind; i < n; i++)``            ``{` `                ``// Find the minimum element``                ``// in the segment``                ``mini = Math.min(mini, a[i]);` `                ``// Find the sum of all the``                ``// segments trying all``                ``// the possible combinations``                ``ans = Math.max(ans,``                     ``maximizeSum(a, n, i + ``1``,``                           ``k - ``1``, dp) + mini);``            ``}` `            ``// Return the answer by``            ``// memoizing it``            ``return` `dp[ind][k] = ans;``        ``}``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int``[] a = { ``5``, ``7``, ``4``, ``2``, ``8``, ``1``, ``6` `};``        ``int` `k = ``3``;``        ``int` `n = a.length;` `        ``// Initialize dp array with -1``        ``int``[][] dp = ``new` `int``[MAX][MAX];``        ``for` `(``int` `i = ``0``; i < MAX; i++)``        ``{``            ``for` `(``int` `j = ``0``; j < MAX; j++)``                ``dp[i][j] = -``1``;``        ``}` `        ``System.out.println(``          ``maximizeSum(a, n, ``0``, k, dp));``    ``}``}` `// This code is contributed by``// sanjeev2552`

## Python3

 `# Python 3 program to find the sum of``# the minimum of all the segments``MAX` `=` `10` `# Function to maximize the``# sum of the minimums``def` `maximizeSum(a,n, ind, k, dp):``    ``# If k segments have been divided``    ``if` `(k ``=``=` `0``):``        ``# If we are at the end``        ``if` `(ind ``=``=` `n):``            ``return` `0` `        ``# If we donot reach the end``        ``# then return a negative number``        ``# that cannot be the sum``        ``else``:``            ``return` `-``1e9` `    ``# If at the end but``    ``# k segments are not formed``    ``elif` `(ind ``=``=` `n):``        ``return` `-``1e9` `    ``# If the state has been visited already``    ``elif` `(dp[ind][k] !``=` `-``1``):``        ``return` `dp[ind][k]` `    ``# If the state has not been visited``    ``else``:``        ``ans ``=` `0` `        ``# Get the minimum element``        ``# in the segment``        ``mini ``=` `a[ind]` `        ``# Iterate and try to break``        ``# at every index``        ``# and create a segment``        ``for` `i ``in` `range``(ind,n,``1``):``            ``# Find the minimum element``            ``# in the segment``            ``mini ``=` `min``(mini, a[i])` `            ``# Find the sum of all the``            ``# segments trying all``            ``# the possible combinations``            ``ans ``=` `max``(ans, maximizeSum(\``             ``a, n, i ``+` `1``, k ``-` `1``, dp) ``+` `mini)` `        ``# Return the answer by``        ``# memoizing it``        ``dp[ind][k] ``=` `ans``        ``return` `ans``        ` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``a ``=` `[``5``, ``7``, ``4``, ``2``, ``1``, ``6``]``    ``k ``=` `3``    ``n ``=` `len``(a)` `    ``# Initialize dp array with -1``    ``dp ``=` `[[``-``1` `for` `i ``in` `range``(``MAX``)]\``          ``for` `j ``in` `range``(``MAX``)]` `    ``print``(maximizeSum(a, n, ``0``, k, dp))` `# This code is contributed by``# Surendra_Gangwar`

## C#

 `// C# program to find the sum of``// the minimum of all the segments``using` `System;``    ` `class` `GFG``{``    ``static` `int` `MAX = 10;` `    ``// Function to maximize the sum of the minimums``    ``public` `static` `int` `maximizeSum(``             ``int``[] a, ``int` `n, ``int` `ind,``                  ``int` `k, ``int``[] dp)``    ``{``        ``// If k segments have been divided``        ``if` `(k == 0)``        ``{``            ``// If we are at the end``            ``if` `(ind == n)``                ``return` `0;` `            ``// If we donot reach the end``            ``// then return a negative number``            ``// that cannot be the sum``            ``else``                ``return` `-1000000000;``        ``}` `        ``// If at the end but``        ``// k segments are not formed``        ``else` `if` `(ind == n)``            ``return` `-1000000000;` `        ``// If the state has not``        ``// been visited yet``        ``else` `if` `(dp[ind, k] != -1)``            ``return` `dp[ind, k];` `        ``// If the state has not been visited``        ``else``        ``{``            ``int` `ans = 0;` `            ``// Get the minimum element``            ``// in the segment``            ``int` `mini = a[ind];` `            ``// Iterate and try to break``            ``// at every index``            ``// and create a segment``            ``for` `(``int` `i = ind; i < n; i++)``            ``{` `                ``// Find the minimum element``                ``// in the segment``                ``mini = Math.Min(mini, a[i]);` `                ``// Find the sum of all the``                ``// segments trying all``                ``// the possible combinations``                ``ans = Math.Max(ans,``                      ``maximizeSum(a, n, i + 1,``                           ``k - 1, dp) + mini);``            ``}` `            ``// Return the answer by``            ``// memoizing it``            ``return` `dp[ind,k] = ans;``        ``}``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``int``[] a = { 5, 7, 4, 2, 8, 1, 6 };``        ``int` `k = 3;``        ``int` `n = a.Length;` `        ``// Initialize dp array with -1``        ``int``[,] dp = ``new` `int``[MAX, MAX];``        ``for` `(``int` `i = 0; i < MAX; i++)``        ``{``            ``for` `(``int` `j = 0; j < MAX; j++)``                ``dp[i, j] = -1;``        ``}` `        ``Console.WriteLine(``          ``maximizeSum(a, n, 0, k, dp));``    ``}``}` `// This code is contributed by 29AjayKumar`

## Javascript

 ``
Output

`13`

Time Complexity: O(N * N * K)
Auxiliary Space: O(N * K)

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