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]);
    cout << "Min Sum is " << findMinSum(arr, n);
    return 0;
}

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. 
?>


Output:

Min Sum is 4

Time Complexity : O(n)
Auxiliary Space : O(n)

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 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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