Maximum size subset with given sum using Backtracking

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Given an array arr[] consisting of N integers and an integer K, the task is to find the length of the longest subsequence with a sum equal to K.
Examples:

Input: arr[] = {-4, -2, -2, -1, 6}, K = 0
Output: 3
Explanation:
The longest subsequence is of length 3 which is {-4, -2, 6} having sum 0.
Input: arr[] = {-3, 0, 1, 1, 2}, K = 1
Output: 5
Explanation: The longest subsequence is of length 5 which is {-3, 0, 1, 1, 2} having sum 1.

Naive Approach: The simplest approach to solve the problem is to generate all the possible subsequences of different lengths and check if their sum is equal to K. Out of all these subsequences with sum K, find the subsequence with the longest length.
Time complexity: O(2^N)
Recursive & Backtracking Approach: The basic approach of this problem is to sort the vector and find the sum of all the possible subsequences and pick up the subsequence with the maximum length having the given sum. This can be done using Recursion and Backtracking.
Follow the steps below to solve this problem:

Sort the given array/vector.
Initialize a global variable max_length to 0, which stores the maximum length subset.
For every index i in the array, call the recursion function to find out all the possible subsets with elements in the range [i, N-1] having sum K.
Every time a subset with sum K is found, check if its size is greater than the current max_length value. If yes, then update the value of max_length.
After all the possible subset sums are computed, return the max_length.

Below is the implementation of the above approach:

C++

// C++ Program to implement the
// above approach
#include <bits/stdc++.h>
using namespace std;
 
// Initialise maximum possible
// length of subsequence
int max_length = 0;
 
// Store elements to compare
// max_length with its size
// and change the value of
// max_length accordingly
vector<int> store;
 
// Store the elements of the
// longest subsequence
vector<int> ans;
 
// Function to find the length
// of longest subsequence
void find_max_length(
    vector<int>& arr,
    int index, int sum, int k)
{
    sum = sum + arr[index];
    store.push_back(arr[index]);
    if (sum == k) {
        if (max_length < store.size()) {
            // Update max_length
            max_length = store.size();
 
            // Store the subsequence
            // elements
            ans = store;
        }
    }
 
    for (int i = index + 1;
         i < arr.size(); i++) {
        if (sum + arr[i] <= k) {
 
            // Recursively proceed
            // with obtained sum
            find_max_length(arr, i,
                            sum, k);
 
            // popping elements
            // from back
            // of vector store
            store.pop_back();
        }
 
        // if sum > 0 then we don't
        // required thatsubsequence
        // so return and continue
        // with earlier elements
        else
            return;
    }
 
    return;
}
 
int longestSubsequence(vector<int> arr,
                       int n, int k)
{
 
    // Sort the given array
    sort(arr.begin(), arr.end());
 
    // Traverse the array
    for (int i = 0; i < n; i++) {
        // If max_length is already
        // greater than or equal
        // than remaining length
        if (max_length >= n - i)
            break;
 
        store.clear();
 
        find_max_length(arr, i, 0, k);
    }
 
    return max_length;
}
 
// Driver code
int main()
{
    vector<int> arr{ -3, 0, 1, 1, 2 };
    int n = arr.size();
    int k = 1;
 
    cout << longestSubsequence(arr,
                               n, k);
 
    return 0;
}

Java

// Java Program to implement the
// above approach
import java.util.*;
class GFG{
  
// Initialise maximum possible
// length of subsequence
static int max_length = 0;
  
// Store elements to compare
// max_length with its size
// and change the value of
// max_length accordingly
static Vector<Integer> store = new Vector<Integer>();
  
// Store the elements of the
// longest subsequence
static Vector<Integer> ans = new Vector<Integer>();
  
// Function to find the length
// of longest subsequence
static void find_max_length(
    int []arr,
    int index, int sum, int k)
{
    sum = sum + arr[index];
    store.add(arr[index]);
    if (sum == k) 
    {
        if (max_length < store.size()) 
        {
            // Update max_length
            max_length = store.size();
  
            // Store the subsequence
            // elements
            ans = store;
        }
    }
  
    for (int i = index + 1;
             i < arr.length; i++) 
    {
        if (sum + arr[i] <= k) 
        {
  
            // Recursively proceed
            // with obtained sum
            find_max_length(arr, i,
                            sum, k);
  
            // popping elements
            // from back
            // of vector store
            store.remove(store.size() - 1);
        }
  
        // if sum > 0 then we don't
        // required thatsubsequence
        // so return and continue
        // with earlier elements
        else
            return;
    }
    return;
}
  
static int longestSubsequence(int []arr,
                                 int n, int k)
{
  
    // Sort the given array
    Arrays.sort(arr);
  
    // Traverse the array
    for (int i = 0; i < n; i++) 
    {
        // If max_length is already
        // greater than or equal
        // than remaining length
        if (max_length >= n - i)
            break;
  
        store.clear();
  
        find_max_length(arr, i, 0, k);
    }
    return max_length;
}
  
// Driver code
public static void main(String[] args)
{
    int []arr = { -3, 0, 1, 1, 2 };
    int n = arr.length;
    int k = 1;
  
    System.out.print(longestSubsequence(arr,
                                           n, k));
}
}
 
// This code is contributed by Princi Singh

Python3

# Python3 Program to implement the
# above approach
# Initialise maximum possible
# length of subsequence
max_length = 0
 
# Store elements to compare
# max_length with its size
# and change the value of
# max_length accordingly
store = []
 
# Store the elements of the
# longest subsequence
ans = []
 
# Function to find the length
# of longest subsequence
def find_max_length(arr, index, sum, k):  
    global max_length
    sum = sum + arr[index]
    store.append(arr[index])
    if (sum == k):
        if (max_length < len(store)):
            # Update max_length
            max_length = len(store)
 
            # Store the subsequence
            # elements
            ans = store
 
    for i in range ( index + 1, len(arr)):
        if (sum + arr[i] <= k):
 
            # Recursively proceed
            # with obtained sum
            find_max_length(arr, i,
                            sum, k)
 
            # popping elements
            # from back
            # of vector store
            store.pop()
        
        # if sum > 0 then we don't
        # required thatsubsequence
        # so return and continue
        # with earlier elements
        else:
            return
    return
 
def longestSubsequence(arr, n, k):
 
    # Sort the given array
    arr.sort()
 
    # Traverse the array
    for i in range (n):
       
        # If max_length is already
        # greater than or equal
        # than remaining length
        if (max_length >= n - i):
            break
 
        store.clear()
        find_max_length(arr, i, 0, k)
    
    return max_length
 
# Driver code
if __name__ == "__main__":
   
    arr = [-3, 0, 1, 1, 2]
    n = len(arr)
    k = 1
    print (longestSubsequence(arr, n, k))
     
# This code is contributed by Chitranayal

C#

// C# program to implement the
// above approach
using System;
using System.Collections.Generic;
class GFG{
 
// Initialise maximum possible
// length of subsequence
static int max_length = 0;
 
// Store elements to compare
// max_length with its size
// and change the value of
// max_length accordingly
static List<int> store = new List<int>();
 
// Store the elements of the
// longest subsequence
static List<int> ans = new List<int>();
 
// Function to find the length
// of longest subsequence
static void find_max_length(int []arr,
                            int index, 
                            int sum, int k)
{
    sum = sum + arr[index];
    store.Add(arr[index]);
     
    if (sum == k) 
    {
        if (max_length < store.Count) 
        {
             
            // Update max_length
            max_length = store.Count;
 
            // Store the subsequence
            // elements
            ans = store;
        }
    }
 
    for(int i = index + 1;
            i < arr.Length; i++) 
    {
        if (sum + arr[i] <= k) 
        {
 
            // Recursively proceed
            // with obtained sum
            find_max_length(arr, i,
                            sum, k);
 
            // popping elements
            // from back
            // of vector store
            store.RemoveAt(store.Count - 1);
        }
 
        // If sum > 0 then we don't
        // required thatsubsequence
        // so return and continue
        // with earlier elements
        else
            return;
    }
    return;
}
 
static int longestSubsequence(int []arr,
                              int n, int k)
{
 
    // Sort the given array
    Array.Sort(arr);
 
    // Traverse the array
    for(int i = 0; i < n; i++) 
    {
         
        // If max_length is already
        // greater than or equal
        // than remaining length
        if (max_length >= n - i)
            break;
 
        store.Clear();
 
        find_max_length(arr, i, 0, k);
    }
    return max_length;
}
 
// Driver code
public static void Main(String[] args)
{
    int []arr = { -3, 0, 1, 1, 2 };
    int n = arr.Length;
    int k = 1;
 
    Console.Write(longestSubsequence(arr,
                                     n, k));
}
}
 
// This code is contributed by gauravrajput1 

Javascript

<script>
// Javascript Program to implement the
// above approach
 
// Initialise maximum possible
// length of subsequence
let max_length = 0;
 
// Store elements to compare
// max_length with its size
// and change the value of
// max_length accordingly
let store = [];
 
// Store the elements of the
// longest subsequence
let ans = [];
 
// Function to find the length
// of longest subsequence
function find_max_length(arr,index,sum,k)
{
    sum = sum + arr[index];
    store.push(arr[index]);
    if (sum == k)
    {
        if (max_length < store.length)
        {
            // Update max_length
            max_length = store.length;
   
            // Store the subsequence
            // elements
            ans = store;
        }
    }
   
    for (let i = index + 1;
             i < arr.length; i++)
    {
        if (sum + arr[i] <= k)
        {
   
            // Recursively proceed
            // with obtained sum
            find_max_length(arr, i,
                            sum, k);
   
            // popping elements
            // from back
            // of vector store
            store.pop();
        }
   
        // if sum > 0 then we don't
        // required thatsubsequence
        // so return and continue
        // with earlier elements
        else
            return;
    }
    return;
}
 
function longestSubsequence(arr, n, k)
{
 
    // Sort the given array
    arr.sort(function(a,b){return a-b;});
   
    // Traverse the array
    for (let i = 0; i < n; i++)
    {
     
        // If max_length is already
        // greater than or equal
        // than remaining length
        if (max_length >= n - i)
            break;
   
        store=[];
   
        find_max_length(arr, i, 0, k);
    }
    return max_length;
}
 
// Driver code
let arr = [-3, 0, 1, 1, 2 ];
let n = arr.length;
let k = 1;
document.write(longestSubsequence(arr,n, k));
 
// This code is contributed by avanitrachhadiya2155
</script>

Output:

Time Complexity: O(N³)
Auxiliary Space: O(N)
Dynamic Programming Approach: Refer to this article for a further optimized approach to solve the problem.

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