# Find the smallest positive integer value that cannot be represented as sum of any subset of a given array

Given an array of positive numbers, find the smallest positive integer value that cannot be represented as the sum of elements of any subset of a given set.

The expected time complexity is O(nlogn).

Examples:

Input: arr[] = {1, 10, 3, 11, 6, 15}; Output: 2 Input: arr[] = {1, 1, 1, 1}; Output: 5 Input: arr[] = {1, 1, 3, 4}; Output: 10 Input: arr[] = {1, 2, 5, 10, 20, 40}; Output: 4 Input: arr[] = {1, 2, 3, 4, 5, 6}; Output: 22

A **Simple Solution** is to start from value 1 and check all values one by one if they can sum to values in the given array. This solution is very inefficient as it reduces to the subset sum problem which is a well-known NP-Complete Problem.

Using a simple loop, we can solve this problem in O(N log N) time. Let the input array be arr[0..n-1]. We first sort the array in non-decreasing order. Let the smallest element that cannot be represented by elements at indexes from 0 to (i-1) be ‘res’. We initialize ‘res’ as 1 (smallest possible answer) and traverse the given array from i=0. There are the following two possibilities when we consider element at index i:* 1) We decide that ‘res’ is the final result*: If arr[i] is greater than ‘res’, then we found the gap which is ‘res’ because the elements after arr[i] are also going to be greater than ‘res’.

*: If arr[i] is not greater than ‘res’, the value of ‘res’ is incremented by arr[i] so ‘res’ = ‘res’+arr[i] (why? If elements from 0 to (i-1) can represent 1 to ‘res-1’, then elements from 0 to i can represent from 1 to ‘res + arr[i] – 1’ by adding arr[i] to all subsets that represent 1 to ‘res-1’ which we have already determined to be represented)*

**2) The value of ‘res’ is incremented after considering arr[i]**Following is the implementation of the above idea.

## C++

`// C++ program to find the smallest positive value that cannot be` `// represented as sum of subsets of a given sorted array` `#include<iostream>` `#include<vector>` `#include<algorithm>` `using` `namespace` `std;` `// Returns the smallest number that cannot be represented as sum` `// of subset of elements from set represented by sorted array arr[0..n-1]` `long` `long` `smallestpositive(vector<` `long` `long` `> arr, ` `int` `n) {` ` ` `long` `long` `int` `res = 1; ` `// Initialize result` ` ` ` ` `sort(arr.begin(), arr.end());` ` ` `// Traverse the array and increment 'res' if arr[i] is` ` ` `// smaller than or equal to 'res'.` ` ` `for` `(` `int` `i = 0; i < n && arr[i] <= res; i++)` ` ` `res = res + arr[i];` ` ` ` ` `return` `res;` ` ` `}` `// Driver program to test above function` `int` `main() {` ` ` `vector<` `long` `long` `> arr1 = {1, 3, 4, 5};` ` ` `cout << smallestpositive(arr1, arr1.size()) << endl;` ` ` ` ` `vector<` `long` `long` `> arr2 = {1, 2, 6, 10, 11, 15};` ` ` `cout << smallestpositive(arr2, arr2.size()) << endl;` ` ` ` ` `vector<` `long` `long` `> arr3 = {1, 1, 1, 1};` ` ` `cout << smallestpositive(arr3, arr3.size()) << endl;` ` ` ` ` `vector<` `long` `long` `> arr4 = {1, 1, 3, 4};` ` ` `cout << smallestpositive(arr4, arr4.size()) << endl;` ` ` ` ` `return` `0;` ` ` `}` |

## Java

`// Java program to find the smallest positive value that cannot be` `// represented as sum of subsets of a given sorted array` `import` `java.util.Arrays;` `class` `FindSmallestInteger ` `{` ` ` `// Returns the smallest number that cannot be represented as sum` ` ` `// of subset of elements from set represented by sorted array arr[0..n-1]` ` ` `int` `findSmallest(` `int` `arr[], ` `int` `n) ` ` ` `{` ` ` `int` `res = ` `1` `; ` `// Initialize result` ` ` `// sort the input array` ` ` `Arrays.sort(arr);` ` ` ` ` `// Traverse the array and increment 'res' if arr[i] is` ` ` `// smaller than or equal to 'res'.` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++)` ` ` `{` ` ` `if` `(arr[i] > res){` ` ` `return` `res;` ` ` `}` ` ` `else` `{` ` ` `res+=arr[i];` ` ` `}` ` ` `}` ` ` ` ` `return` `res;` ` ` `}` ` ` `// Driver program to test above functions` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{` ` ` `FindSmallestInteger small = ` `new` `FindSmallestInteger();` ` ` `int` `arr1[] = {` `1` `, ` `3` `, ` `4` `, ` `5` `};` ` ` `int` `n1 = arr1.length;` ` ` `System.out.println(small.findSmallest(arr1, n1));` ` ` `int` `arr2[] = {` `1` `, ` `2` `, ` `6` `, ` `10` `, ` `11` `, ` `15` `};` ` ` `int` `n2 = arr2.length;` ` ` `System.out.println(small.findSmallest(arr2, n2));` ` ` `int` `arr3[] = {` `1` `, ` `1` `, ` `1` `, ` `1` `};` ` ` `int` `n3 = arr3.length;` ` ` `System.out.println(small.findSmallest(arr3, n3));` ` ` `int` `arr4[] = {` `1` `, ` `1` `, ` `3` `, ` `4` `};` ` ` `int` `n4 = arr4.length;` ` ` `System.out.println(small.findSmallest(arr4, n4));` ` ` `}` `}` `// This code has been contributed by Mukul Sharma (msharma04) ` |

## Python3

`# Python3 program to find the smallest` `# positive value that cannot be` `# represented as sum of subsets ` `# of a given sorted array` `# Returns the smallest number ` `# that cannot be represented as sum` `# of subset of elements from set` `# represented by sorted array arr[0..n-1]` `def` `findSmallest(arr, n):` ` ` `res ` `=` `1` `#Initialize result` ` ` `# Traverse the array and increment` ` ` `# 'res' if arr[i] is smaller than` ` ` `# or equal to 'res'.` ` ` `for` `i ` `in` `range` `(` `0` `, n ):` ` ` `if` `arr[i] <` `=` `res:` ` ` `res ` `=` `res ` `+` `arr[i]` ` ` `else` `:` ` ` `break` ` ` `return` `res` `# Driver program to test above function` `arr1 ` `=` `[` `1` `, ` `3` `, ` `4` `, ` `5` `]` `n1 ` `=` `len` `(arr1)` `print` `(findSmallest(arr1, n1))` `arr2` `=` `[` `1` `, ` `2` `, ` `6` `, ` `10` `, ` `11` `, ` `15` `]` `n2 ` `=` `len` `(arr2)` `print` `(findSmallest(arr2, n2))` `arr3` `=` `[` `1` `, ` `1` `, ` `1` `, ` `1` `]` `n3 ` `=` `len` `(arr3)` `print` `(findSmallest(arr3, n3))` `arr4 ` `=` `[` `1` `, ` `1` `, ` `3` `, ` `4` `]` `n4 ` `=` `len` `(arr4)` `print` `(findSmallest(arr4, n4))` `# This code is.contributed by Smitha Dinesh Semwal` |

## C#

`// C# program to find the smallest` `// positive value that cannot be` `// represented as sum of subsets ` `// of a given sorted array` `using` `System;` `class` `GFG {` ` ` ` ` `// Returns the smallest number that` ` ` `// cannot be represented as sum` ` ` `// of subset of elements from set ` ` ` `// represented by sorted array` ` ` `// arr[0..n-1]` ` ` `static` `int` `findSmallest(` `int` `[]arr, ` `int` `n) ` ` ` `{` ` ` `// Initialize result` ` ` `int` `res = 1;` ` ` `// Traverse the array and ` ` ` `// increment 'res' if arr[i] is` ` ` `// smaller than or equal to 'res'.` ` ` `for` `(` `int` `i = 0; i < n && ` ` ` `arr[i] <= res; i++)` ` ` `res = res + arr[i];` ` ` `return` `res;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `Main() ` ` ` `{` ` ` `int` `[]arr1 = {1, 3, 4, 5};` ` ` `int` `n1 = arr1.Length;` ` ` `Console.WriteLine(findSmallest(arr1, n1));` ` ` `int` `[]arr2 = {1, 2, 6, 10, 11, 15};` ` ` `int` `n2 = arr2.Length;` ` ` `Console.WriteLine(findSmallest(arr2, n2));` ` ` `int` `[]arr3 = {1, 1, 1, 1};` ` ` `int` `n3 = arr3.Length;` ` ` `Console.WriteLine(findSmallest(arr3, n3));` ` ` `int` `[]arr4 = {1, 1, 3, 4};` ` ` `int` `n4 = arr4.Length;` ` ` `Console.WriteLine(findSmallest(arr4, n4));` ` ` `}` `}` `// This code is contributed by Sam007` |

## PHP

`<?php` `// PHP program to find the smallest` `// positive value that cannot be` `// represented as sum of subsets` `// of a given sorted array` `// Returns the smallest number that` `// cannot be represented as sum of ` `// subset of elements from set` `// represented by sorted array ` `// arr[0..n-1]` `function` `findSmallest(` `$arr` `, ` `$n` `)` `{` ` ` ` ` `// Initialize result` ` ` `$res` `= 1; ` ` ` ` ` `// Traverse the array and ` ` ` `// increment 'res' if arr[i] is` ` ` `// smaller than or equal to 'res'.` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `$n` `and` `$arr` `[` `$i` `] <= ` `$res` `; ` `$i` `++)` ` ` `$res` `= ` `$res` `+ ` `$arr` `[` `$i` `];` ` ` ` ` `return` `$res` `;` `}` `// Driver Code` `$arr1` `= ` `array` `(1, 3, 4, 5);` `$n1` `= ` `count` `(` `$arr1` `);` `echo` `findSmallest(` `$arr1` `, ` `$n1` `),` `"\n"` `;` `$arr2` `= ` `array` `(1, 2, 6, 10, 11, 15);` `$n2` `= ` `count` `(` `$arr2` `);` `echo` `findSmallest(` `$arr2` `, ` `$n2` `),` `"\n"` `;` `$arr3` `= ` `array` `(1, 1, 1, 1);` `$n3` `= ` `count` `(` `$arr3` `);` `echo` `findSmallest(` `$arr3` `, ` `$n3` `),` `"\n"` `;` `$arr4` `= ` `array` `(1, 1, 3, 4);` `$n4` `= ` `count` `(` `$arr4` `);` `echo` `findSmallest(` `$arr4` `, ` `$n4` `);` `// This code is contributed by anuj_67.` `?>` |

## Javascript

`<script>` `// javascript program to find the smallest positive value that cannot be` `// represented as sum of subsets of a given sorted array` ` ` `// Returns the smallest number that cannot be represented as sum` ` ` `// of subset of elements from set represented by sorted array arr[0..n-1]` ` ` `function` `findSmallest(arr , n)` ` ` `{` ` ` `var` `res = 1; ` `// Initialize result` ` ` `// Traverse the array and increment 'res' if arr[i] is` ` ` `// smaller than or equal to 'res'.` ` ` `for` `(i = 0; i < n && arr[i] <= res; i++)` ` ` `res = res + arr[i];` ` ` `return` `res;` ` ` `}` ` ` `// Driver program to test above functions` ` ` `var` `arr1 = [ 1, 3, 4, 5 ];` ` ` `var` `n1 = arr1.length;` ` ` `document.write(findSmallest(arr1, n1)+` `"<br/>"` `);` ` ` `var` `arr2 = [ 1, 2, 6, 10, 11, 15 ];` ` ` `var` `n2 = arr2.length;` ` ` `document.write(findSmallest(arr2, n2)+` `"<br/>"` `);` ` ` `var` `arr3 = [ 1, 1, 1, 1 ];` ` ` `var` `n3 = arr3.length;` ` ` `document.write(findSmallest(arr3, n3)+` `"<br/>"` `);` ` ` `var` `arr4 = [ 1, 1, 3, 4 ];` ` ` `var` `n4 = arr4.length;` ` ` `document.write(findSmallest(arr4, n4)+` `"<br/>"` `);` `// This code is contributed by aashish1995` `</script>` |

**Output**

2 4 5 10

Output:

2 4 5 10

The Time Complexity of the above program is O(nlogn).

The Space Complexity is O(1) in best case for heap sort.

This article is contributed by **Anushka Raghuwanshi**. Please write comments if you find anything incorrect or share more information about the topic discussed above.