# Find minimum sum such that one of every three consecutive elements is taken

Given an array of **n** non-negative numbers, the task is to find the minimum sum of elements (picked from the array) such that at least one element is picked out of every 3 consecutive elements in the array.

**Examples :**

Input : arr[] = {1, 2, 3} Output : 1 Input : arr[] = {1, 2, 3, 6, 7, 1} Output : 4 We pick 3 and 1 (3 + 1 = 4) Note that there are following subarrays of three consecutive elements {1, 2, 3}, {2, 3, 6}, {3, 6, 7} and {6, 7, 1} We have picked one element from every subarray. Input : arr[] = {1, 2, 3, 6, 7, 1, 8, 6, 2, 7, 7, 1} Output : 7 The result is obtained as sum of 3 + 1 + 2 + 1

Let sum(i) be the minimum possible sum when arr[i] is part of a solution sum (not necessarily result) and is last picked element. Then our result is minimum of sum(n-1), sum(n-2) and sum(n-3) [We must pick at least one of the last three elements].

We can recursively compute sum(i) as sum of arr[i] and minimum(sum(i-1), sum(i-2), sum(i-3)). Since there are overlapping subproblems in recursive structure of problem, we can use Dynamic Programming to solve this problem.

Below is the implementation of above idea.

## C++

`// A Dynamic Programming based C++ program to find minimum` `// possible sum of elements of array such that an element` `// out of every three consecutive is picked.` `#include <iostream>` `using` `namespace` `std;` `// A utility function to find minimum of 3 elements` `int` `minimum(` `int` `a, ` `int` `b, ` `int` `c)` `{` ` ` `return` `min(min(a, b), c);` `}` `// Returns minimum possible sum of elements such that an` `// element out of every three consecutive elements is` `// picked.` `int` `findMinSum(` `int` `arr[], ` `int` `n)` `{` ` ` `// Create a DP table to store results of subproblems.` ` ` `// sum[i] is going to store minimum possible sum when` ` ` `// arr[i] is part of the solution.` ` ` `int` `sum[n];` ` ` `// When there are less than or equal to 3 elements` ` ` `sum[0] = arr[0];` ` ` `sum[1] = arr[1];` ` ` `sum[2] = arr[2];` ` ` `// Iterate through all other elements` ` ` `for` `(` `int` `i = 3; i < n; i++)` ` ` `sum[i] = arr[i] + minimum(sum[i - 3], sum[i - 2], sum[i - 1]);` ` ` `return` `minimum(sum[n - 1], sum[n - 2], sum[n - 3]);` `}` `// Driver code` `int` `main()` `{` ` ` `int` `arr[] = { 1, 2, 3, 20, 2, 10, 1 };` ` ` `int` `n = ` `sizeof` `(arr) / ` `sizeof` `(arr[0]);` ` ` `printf` `(` `"Min Sum is %d"` `, findMinSum(arr, n));` ` ` `return` `0;` `}` |

## C

`// A Dynamic Programming based C++ program to find minimum` `// possible sum of elements of array such that an element` `// out of every three consecutive is picked.` `#include <stdio.h>` `// Find minimum between two numbers.` `int` `min(` `int` `num1, ` `int` `num2)` `{` ` ` `return` `(num1 > num2) ? num2 : num1;` `}` `// A utility function to find minimum of 3 elements` `int` `minimum(` `int` `a, ` `int` `b, ` `int` `c)` `{` ` ` `return` `min(min(a, b), c);` `}` `// Returns minimum possible sum of elements such that an` `// element out of every three consecutive elements is` `// picked.` `int` `findMinSum(` `int` `arr[], ` `int` `n)` `{` ` ` `// Create a DP table to store results of subproblems.` ` ` `// sum[i] is going to store minimum possible sum when` ` ` `// arr[i] is part of the solution.` ` ` `int` `sum[n];` ` ` `// When there are less than or equal to 3 elements` ` ` `sum[0] = arr[0];` ` ` `sum[1] = arr[1];` ` ` `sum[2] = arr[2];` ` ` `// Iterate through all other elements` ` ` `for` `(` `int` `i = 3; i < n; i++)` ` ` `sum[i] = arr[i] + minimum(sum[i - 3], sum[i - 2], sum[i - 1]);` ` ` `return` `minimum(sum[n - 1], sum[n - 2], sum[n - 3]);` `}` `// Driver code` `int` `main()` `{` ` ` `int` `arr[] = { 1, 2, 3, 20, 2, 10, 1 };` ` ` `int` `n = ` `sizeof` `(arr) / ` `sizeof` `(arr[0]);` ` ` `printf` `(` `"Min Sum is %d"` `, findMinSum(arr, n));` ` ` `return` `0;` `}` `// This code is contributed by Aditya Kumar (adityakumar129)` |

## Java

`// A Dynamic Programming based java program to` `// find minimum possible sum of elements of array` `// such that an element out of every three` `// consecutive is picked.` `import` `java.io.*;` `class` `GFG ` `{` ` ` `// A utility function to find minimum of` ` ` `// 3 elements` ` ` `static` `int` `minimum(` `int` `a, ` `int` `b, ` `int` `c)` ` ` `{` ` ` `return` `Math. min(Math.min(a, b), c);` ` ` `}` ` ` ` ` `// Returns minimum possible sum of elements such` ` ` `// that an element out of every three consecutive` ` ` `// elements is picked.` ` ` `static` `int` `findMinSum(` `int` `arr[], ` `int` `n)` ` ` `{` ` ` `// Create a DP table to store results of` ` ` `// subproblems. sum[i] is going to store` ` ` `// minimum possible sum when arr[i] is` ` ` `// part of the solution.` ` ` `int` `sum[] =` `new` `int` `[n];` ` ` ` ` `// When there are less than or equal to` ` ` `// 3 elements` ` ` `sum[` `0` `] = arr[` `0` `];` ` ` `sum[` `1` `] = arr[` `1` `];` ` ` `sum[` `2` `] = arr[` `2` `];` ` ` ` ` `// Iterate through all other elements` ` ` `for` `(` `int` `i = ` `3` `; i < n; i++)` ` ` `sum[i] = arr[i] + minimum(sum[i - ` `3` `], ` ` ` `sum[i - ` `2` `], sum[i - ` `1` `]);` ` ` ` ` `return` `minimum(sum[n - ` `1` `], sum[n - ` `2` `], sum[n - ` `3` `]);` ` ` `}` ` ` ` ` `// Driver code` ` ` `public` `static` `void` `main (String[] args) ` ` ` `{` ` ` `int` `arr[] = {` `1` `, ` `2` `, ` `3` `, ` `20` `, ` `2` `, ` `10` `, ` `1` `};` ` ` `int` `n = arr.length;` ` ` `System.out.println(` `"Min Sum is "` `+ findMinSum(arr, n));` ` ` ` ` `}` `}` `// This code is contributed by vt_m` |

## Python3

`# A Dynamic Programming based python 3 program to` `# find minimum possible sum of elements of array` `# such that an element out of every three` `# consecutive is picked.` `# A utility function to find minimum of` `# 3 elements` `def` `minimum(a, b, c):` ` ` `return` `min` `(` `min` `(a, b), c);` `# Returns minimum possible sum of elements such` `# that an element out of every three consecutive` `# elements is picked.` `def` `findMinSum(arr,n):` ` ` `# Create a DP table to store results of` ` ` `# subproblems. sum[i] is going to store` ` ` `# minimum possible sum when arr[i] is` ` ` `# part of the solution.` ` ` `sum` `=` `[]` ` ` `# When there are less than or equal to` ` ` `# 3 elements` ` ` `sum` `.append(arr[` `0` `])` ` ` `sum` `.append(arr[` `1` `])` ` ` `sum` `.append(arr[` `2` `])` ` ` ` ` `# Iterate through all other elements` ` ` `for` `i ` `in` `range` `(` `3` `, n):` ` ` `sum` `.append( arr[i] ` `+` `minimum(` `sum` `[i` `-` `3` `],` ` ` `sum` `[i` `-` `2` `], ` `sum` `[i` `-` `1` `]))` ` ` `return` `minimum(` `sum` `[n` `-` `1` `], ` `sum` `[n` `-` `2` `], ` `sum` `[n` `-` `3` `])` `# Driver code` `arr ` `=` `[` `1` `, ` `2` `, ` `3` `, ` `20` `, ` `2` `, ` `10` `, ` `1` `]` `n ` `=` `len` `(arr)` `print` `( ` `"Min Sum is "` `,findMinSum(arr, n))` `# This code is contributed by Sam007` |

## C#

`// A Dynamic Programming based C# program to` `// find minimum possible sum of elements of array` `// such that an element out of every three` `// consecutive is picked.` `using` `System;` `class` `GFG` `{` ` ` `// A utility function to find minimum of` ` ` `// 3 elements` ` ` `static` `int` `minimum(` `int` `a, ` `int` `b, ` `int` `c)` ` ` `{` ` ` `return` `Math. Min(Math.Min(a, b), c);` ` ` `}` ` ` ` ` `// Returns minimum possible sum of elements such` ` ` `// that an element out of every three consecutive` ` ` `// elements is picked.` ` ` `static` `int` `findMinSum(` `int` `[]arr, ` `int` `n)` ` ` `{` ` ` `// Create a DP table to store results of` ` ` `// subproblems. sum[i] is going to store` ` ` `// minimum possible sum when arr[i] is` ` ` `// part of the solution.` ` ` `int` `[]sum =` `new` `int` `[n];` ` ` ` ` `// When there are less than or equal to` ` ` `// 3 elements` ` ` `sum[0] = arr[0];` ` ` `sum[1] = arr[1];` ` ` `sum[2] = arr[2];` ` ` ` ` `// Iterate through all other elements` ` ` `for` `(` `int` `i = 3; i < n; i++)` ` ` `sum[i] = arr[i] + minimum(sum[i - 3], ` ` ` `sum[i - 2], sum[i - 1]);` ` ` ` ` `return` `minimum(sum[n - 1], sum[n - 2], sum[n - 3]);` ` ` `}` ` ` ` ` `// Driver code` ` ` `public` `static` `void` `Main () ` ` ` `{` ` ` `int` `[]arr = {1, 2, 3, 20, 2, 10, 1};` ` ` `int` `n = arr.Length;` ` ` `Console.WriteLine(` `"Min Sum is "` `+ findMinSum(arr, n));` ` ` ` ` `}` `}` `//This code is contributed by Sam007` |

## PHP

`<?php` `// A Dynamic Programming based ` `// PHP program to find minimum ` `// possible sum of elements of` `// array such that an element ` `// out of every three consecutive` `// is picked.` `// function to find minimum of` `// 3 elements` `function` `minimum(` `$a` `, ` `$b` `, ` `$c` `)` `{` ` ` `return` `min(min(` `$a` `, ` `$b` `), ` `$c` `);` `}` `// Returns minimum possible sum` `// of elements such that an` `// element out of every three` `// consecutive elements is picked.` `function` `findMinSum(` `$arr` `, ` `$n` `)` `{` ` ` ` ` `// Create a DP table to store results of` ` ` `// subproblems. sum[i] is going to store` ` ` `// minimum possible sum when arr[i] is` ` ` `// part of the solution.` ` ` `$sum` `[` `$n` `] = 0;` ` ` `// When there are less ` ` ` `// than or equal to` ` ` `// 3 elements` ` ` `$sum` `[0] = ` `$arr` `[0];` ` ` `$sum` `[1] = ` `$arr` `[1];` ` ` `$sum` `[2] = ` `$arr` `[2];` ` ` `// Iterate through all other elements` ` ` `for` `(` `$i` `= 3; ` `$i` `< ` `$n` `; ` `$i` `++)` ` ` `$sum` `[` `$i` `] = ` `$arr` `[` `$i` `] + minimum(` `$sum` `[` `$i` `- 3],` ` ` `$sum` `[` `$i` `- 2], ` `$sum` `[` `$i` `- 1]);` ` ` `return` `minimum(` `$sum` `[` `$n` `- 1], ` `$sum` `[` `$n` `- 2], ` ` ` `$sum` `[` `$n` `- 3]);` `}` ` ` `// Driver code` ` ` `$arr` `= ` `array` `(1, 2, 3, 20, 2, 10, 1);` ` ` `$n` `= sizeof(` `$arr` `);` ` ` `echo` `"Min Sum is "` `, findMinSum(` `$arr` `, ` `$n` `);` ` ` `// This code is contributed by nitin mittal. ` `?>` |

## Javascript

`<script>` `// A Dynamic Programming based Javascript program to` `// find minimum possible sum of elements of array` `// such that an element out of every three` `// consecutive is picked.` `// A utility function to find minimum of` `// 3 elements` `function` `minimum(a, b, c)` `{` ` ` `return` `Math.min(Math.min(a, b), c);` `}` `// Returns minimum possible sum of elements such` `// that an element out of every three consecutive` `// elements is picked.` `function` `findMinSum(arr, n)` `{` ` ` `// Create a DP table to store results of` ` ` `// subproblems. sum[i] is going to store` ` ` `// minimum possible sum when arr[i] is` ` ` `// part of the solution.` ` ` `var` `sum= Array(n).fill(0);` ` ` `// When there are less than or equal to` ` ` `// 3 elements` ` ` `sum[0] = arr[0];` ` ` `sum[1] = arr[1];` ` ` `sum[2] = arr[2];` ` ` `// Iterate through all other elements` ` ` `for` `(` `var` `i=3; i<n; i++)` ` ` `sum[i] = arr[i] +` ` ` `minimum(sum[i-3], sum[i-2], sum[i-1]);` ` ` `return` `minimum(sum[n-1], sum[n-2], sum[n-3]);` `}` `// Driver code` `var` `arr = [1, 2, 3, 20, 2, 10, 1];` `var` `n = arr.length;` `document.write( ` `"Min Sum is "` `+ findMinSum(arr, n));` `</script>` |

**Output**

Min Sum is 4

**Time Complexity : **O(n) **Auxiliary Space : **O(n), since n extra space has been taken.

This problem and solution are contributed by **Ayush Saluja**. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.