Print all subsets with given sum
Last Updated :
17 Mar, 2025
Given an array arr[] of non-negative integers and an integer target. The task is to print all subsets of the array whose sum is equal to the given target.
Note: If no subset has a sum equal to target, print -1.
Examples:
Input: arr[] = [5, 2, 3, 10, 6, 8], target = 10
Output: [ [5, 2, 3], [2, 8], [10] ]
Explanation: We need to find all subsets of arr[] that sum up to target = 10.
Subset [5, 2, 3] - Sum = 5 + 2 + 3 = 10
Subset [2, 8] - Sum = 2 + 8 = 10
Subset [10] - Sum = 10
Input: arr[] = [5, 7, 8], target = 3
Output: [ [-1] ]
Explanation: There are no subsets of the array that sum up to the target 3.
Input: arr[] = [35, 2, 8, 22], target = 0
Output: [ [ ] ]
Explanation: The empty subset is the only subset with a sum of 0.
Using Recursion - O(2^n) Time and O(n) Space
The idea is to use recursion to explore all possible subsets of the given array. We either include or exclude each element while keeping track of the remaining target sum. If we reach the end of the array and the target becomes 0, we store the valid subset. Otherwise, we backtrack and explore other possibilities.
C++
// C++ Code to find subsets with sum equal
// to target using recursion
#include <bits/stdc++.h>
using namespace std;
void findSubsets(vector<int>& arr, int index,
int target,vector<int>& curr,
vector<vector<int>>& result) {
if (index >= arr.size()) {
// If we reach the end and the target
// becomes 0, we found a valid subset
if (target == 0) {
result.push_back(curr);
return;
}
// Otherwise, we return as no valid
// subset is found
return;
}
// Include current element in subset
curr.push_back(arr[index]);
findSubsets(arr, index + 1,
target - arr[index], curr, result);
// Backtrack and exclude the current element
curr.pop_back();
findSubsets(arr,
index + 1, target, curr, result);
}
// Function to find all subsets summing to target
vector<vector<int>> perfectSum(vector<int>& arr,
int target) {
vector<vector<int>> result;
vector<int> curr;
findSubsets(arr, 0, target, curr, result);
return result;
}
// Function to print subsets in required format
void print2dArray(vector<vector<int>>& arr) {
if (arr.empty()) {
// No valid subsets found
cout << "-1\n";
return;
}
int row, col;
for (row = 0; row < arr.size(); row++) {
cout << "[";
for (col = 0; col < arr[row].size(); col++) {
cout << arr[row][col];
if (col != arr[row].size() - 1) {
cout << ", ";
}
}
cout << "]";
if(row < arr.size() - 1) cout << ", ";
}
}
int main() {
vector<int> arr = {5, 2, 3, 10, 6, 8};
int target = 10;
// Find subsets and print result
vector<vector<int>> result
= perfectSum(arr, target);
print2dArray(result);
return 0;
}
// Java Code to find subsets with sum equal
// to target using recursion
import java.util.ArrayList;
import java.util.List;
class GfG {
static void findSubsets(int[] arr, int index,
int target, List<Integer> curr,
List<List<Integer>> result) {
if (index >= arr.length) {
// If we reach the end and the target
// becomes 0, we found a valid subset
if (target == 0) {
result.add(new ArrayList<>(curr));
return;
}
// Otherwise, we return as no valid
// subset is found
return;
}
// Include current element in subset
curr.add(arr[index]);
findSubsets(arr, index + 1,
target - arr[index], curr, result);
// Backtrack and exclude the current element
curr.remove(curr.size() - 1);
findSubsets(arr,
index + 1, target, curr, result);
}
// Function to find all subsets summing to target
static List<List<Integer>> perfectSum(int[] arr,
int target) {
List<List<Integer>> result = new ArrayList<>();
List<Integer> curr = new ArrayList<>();
findSubsets(arr, 0, target, curr, result);
return result;
}
// Function to print subsets in required format
static void print2dArray(List<List<Integer>> arr) {
if (arr.isEmpty()) {
// No valid subsets found
System.out.println("-1");
return;
}
for (int row = 0; row < arr.size(); row++) {
System.out.print("[");
for (int col = 0; col < arr.get(row).size(); col++) {
System.out.print(arr.get(row).get(col));
if (col != arr.get(row).size() - 1) {
System.out.print(", ");
}
}
System.out.print("]");
if (row < arr.size() - 1) System.out.print(", ");
}
}
public static void main(String[] args) {
int[] arr = {5, 2, 3, 10, 6, 8};
int target = 10;
// Find subsets and print result
List<List<Integer>> result = perfectSum(arr, target);
print2dArray(result);
}
}
# Python Code to find subsets with sum equal
# to target using recursion
def findSubsets(arr, index, target, curr, result):
if index >= len(arr):
# If we reach the end and the target
# becomes 0, we found a valid subset
if target == 0:
result.append(curr[:])
return
# Otherwise, we return as no valid
# subset is found
return
# Include current element in subset
curr.append(arr[index])
findSubsets(arr, index + 1, target - arr[index], curr, result)
# Backtrack and exclude the current element
curr.pop()
findSubsets(arr, index + 1, target, curr, result)
# Function to find all subsets summing to target
def perfectSum(arr, target):
result = []
curr = []
findSubsets(arr, 0, target, curr, result)
return result
# Function to print subsets in required format
def print2dArray(arr):
if not arr:
# No valid subsets found
print("-1")
return
for row in range(len(arr)):
print("[", end="")
for col in range(len(arr[row])):
print(arr[row][col], end="")
if col != len(arr[row]) - 1:
print(", ", end="")
print("]", end="")
if row < len(arr) - 1:
print(", ", end="")
if __name__ == "__main__":
arr = [5, 2, 3, 10, 6, 8]
target = 10
# Find subsets and print result
result = perfectSum(arr, target)
print2dArray(result)
// C# Code to find subsets with sum equal
// to target using recursion
using System;
using System.Collections.Generic;
class GfG {
static void findSubsets(int[] arr, int index,
int target, List<int> curr,
List<List<int>> result) {
if (index >= arr.Length) {
// If we reach the end and the target
// becomes 0, we found a valid subset
if (target == 0) {
result.Add(new List<int>(curr));
return;
}
// Otherwise, we return as no valid
// subset is found
return;
}
// Include current element in subset
curr.Add(arr[index]);
findSubsets(arr, index + 1,
target - arr[index], curr, result);
// Backtrack and exclude the current element
curr.RemoveAt(curr.Count - 1);
findSubsets(arr,
index + 1, target, curr, result);
}
// Function to find all subsets summing to target
static List<List<int>> perfectSum(int[] arr,
int target) {
List<List<int>> result = new List<List<int>>();
List<int> curr = new List<int>();
findSubsets(arr, 0, target, curr, result);
return result;
}
// Function to print subsets in required format
static void print2dArray(List<List<int>> arr) {
if (arr.Count == 0) {
// No valid subsets found
Console.WriteLine("-1");
return;
}
for (int row = 0; row < arr.Count; row++) {
Console.Write("[");
for (int col = 0; col < arr[row].Count; col++) {
Console.Write(arr[row][col]);
if (col != arr[row].Count - 1) {
Console.Write(", ");
}
}
Console.Write("]");
if (row < arr.Count - 1) Console.Write(", ");
}
}
static void Main() {
int[] arr = {5, 2, 3, 10, 6, 8};
int target = 10;
// Find subsets and print result
List<List<int>> result = perfectSum(arr, target);
print2dArray(result);
}
}
// JavaScript Code to find subsets with sum equal
// to target using recursion
function findSubsets(arr, index, target, curr, result) {
if (index >= arr.length) {
// If we reach the end and the target
// becomes 0, we found a valid subset
if (target === 0) {
result.push([...curr]);
return;
}
// Otherwise, we return as no valid
// subset is found
return;
}
// Include current element in subset
curr.push(arr[index]);
findSubsets(arr, index + 1, target - arr[index], curr, result);
// Backtrack and exclude the current element
curr.pop();
findSubsets(arr, index + 1, target, curr, result);
}
// Function to find all subsets summing to target
function perfectSum(arr, target) {
let result = [];
let curr = [];
findSubsets(arr, 0, target, curr, result);
return result;
}
// Function to print subsets in required format
function print2dArray(arr) {
if (arr.length === 0) {
// No valid subsets found
console.log("-1");
return;
}
let output = arr.map(subset => "[" + subset.join(", ") + "]").join(", ");
console.log(output);
}
let arr = [5, 2, 3, 10, 6, 8];
let target = 10;
// Find subsets and print result
let result = perfectSum(arr, target);
print2dArray(result);
Output[5, 2, 3], [2, 8], [10]
Time Complexity: O(2^n), as each element has two choices, include or exclude.
Space Complexity: O(n), due to maximum recursion depth in the worst case scenario.
Using Dynamic Programming - O(2^n) Time and O(n*target) Space
The idea is to use Dynamic Programming (DP) to determine whether a subset with the given target sum exists. We build a dp table where dp[i][j] indicates if a sum j is possible using the first i elements. Once the table is constructed, we use recursion to backtrack and find all valid subsets. If 0s are present, we handle them separately to ensure all possible combinations are included.
Steps to implement the above idea:
- Initialize a dp table of size n × (target + 1) to track subset sum possibilities using boolean values.
- Set dp[i][0] = true for all i since a sum of 0 is always possible with an empty subset.
- Fill the dp table by checking whether a sum j can be formed by including or excluding arr[i].
- If dp[n-1][target] is false, return an empty result as no valid subset exists.
- Use recursion to backtrack through the dp table and find all subsets summing to target.
- Maintain a temporary list to store the current subset and push valid ones to the result list.
- Finally, print the result list in the required format, handling cases where no valid subset exists.
Below is an implementation of the above approach:
C++
// C++ Code to find subsets with sum equal to target
// using Dynamic Programming
#include <bits/stdc++.h>
using namespace std;
// Recursively finds all subsets with the given target
void findSubsets(vector<int>& arr, vector<vector<bool>>& dp,
int i, int target, vector<int>& curr,
vector<vector<int>>& res) {
// Base case: If target becomes 0
if (i == 0) {
if (target == 0) res.push_back(curr);
if (arr[0] == target) {
curr.push_back(arr[0]);
res.push_back(curr);
curr.pop_back();
}
return;
}
// Exclude current element
if (dp[i-1][target]) {
findSubsets(arr, dp, i-1, target, curr, res);
}
// Include current element if it does not exceed target
if (target >= arr[i] && dp[i-1][target-arr[i]]) {
curr.push_back(arr[i]);
findSubsets(arr, dp, i-1, target-arr[i], curr, res);
curr.pop_back();
}
}
// Returns all subsets with the given target sum
vector<vector<int>> perfectSum(vector<int>& arr,
int target) {
int n = arr.size();
if (n == 0 || target < 0) return {};
// DP table to store subset sum possibilities
vector<vector<bool>> dp(n, vector<bool>(target+1, false));
// Correct DP initialization for handling zeroes
dp[0][0] = true;
if (arr[0] <= target) dp[0][arr[0]] = true;
for (int i = 1; i < n; ++i) {
for (int j = 0; j <= target; ++j) {
dp[i][j] = dp[i-1][j]
|| (arr[i] <= j && dp[i-1][j-arr[i]]);
}
}
// If no subsets sum to target, return empty
if (!dp[n-1][target]) return {};
vector<vector<int>> res;
vector<int> curr;
findSubsets(arr, dp, n-1, target, curr, res);
return res;
}
// Function to print subsets in required format
void print2dArray(vector<vector<int>>& arr) {
if (arr.empty()) {
// No valid subsets found
cout << "-1\n";
return;
}
// Printing subsets in formatted output
for (int row = 0; row < arr.size(); row++) {
cout << "[";
for (int col = arr[row].size() - 1; col >= 0 ; col--) {
cout << arr[row][col];
if (col != 0) {
cout << ", ";
}
}
cout << "]";
if (row < arr.size() - 1) cout << ", ";
}
}
// Driver function
int main() {
vector<int> arr = {5, 2, 3, 10, 6, 8};
int target = 10;
vector<vector<int>> result = perfectSum(arr, target);
print2dArray(result);
return 0;
}
// Java Code to find subsets with sum equal to target
// using Dynamic Programming
import java.util.*;
class GfG {
// Recursively finds all subsets with the given target
static void findSubsets(int[] arr, boolean[][] dp,
int i, int target, List<Integer> curr,
List<List<Integer>> res) {
// Base case: If target becomes 0
if (i == 0) {
if (target == 0) res.add(new ArrayList<>(curr));
if (arr[0] == target) {
curr.add(arr[0]);
res.add(new ArrayList<>(curr));
curr.remove(curr.size() - 1);
}
return;
}
// Exclude current element
if (dp[i-1][target]) {
findSubsets(arr, dp, i-1, target, curr, res);
}
// Include current element if it does not exceed target
if (target >= arr[i] && dp[i-1][target-arr[i]]) {
curr.add(arr[i]);
findSubsets(arr, dp, i-1, target-arr[i], curr, res);
curr.remove(curr.size() - 1);
}
}
// Returns all subsets with the given target sum
static List<List<Integer>> perfectSum(int[] arr, int target) {
int n = arr.length;
if (n == 0 || target < 0) return new ArrayList<>();
// DP table to store subset sum possibilities
boolean[][] dp = new boolean[n][target + 1];
// Correct DP initialization for handling zeroes
dp[0][0] = true;
if (arr[0] <= target) dp[0][arr[0]] = true;
for (int i = 1; i < n; ++i) {
for (int j = 0; j <= target; ++j) {
dp[i][j] = dp[i-1][j]
|| (arr[i] <= j && dp[i-1][j-arr[i]]);
}
}
// If no subsets sum to target, return empty
if (!dp[n-1][target]) return new ArrayList<>();
List<List<Integer>> res = new ArrayList<>();
findSubsets(arr, dp, n-1, target, new ArrayList<>(), res);
return res;
}
// Function to print subsets in required format
static void print2dArray(List<List<Integer>> arr) {
if (arr.isEmpty()) {
// No valid subsets found
System.out.println("-1");
return;
}
// Printing subsets in formatted output
for (int row = 0; row < arr.size(); row++) {
System.out.print("[");
for (int col = arr.get(row).size() - 1; col >= 0; col--) {
System.out.print(arr.get(row).get(col));
if (col != 0) {
System.out.print(", ");
}
}
System.out.print("]");
if (row < arr.size() - 1) System.out.print(", ");
}
}
// Driver function
public static void main(String[] args) {
int[] arr = {5, 2, 3, 10, 6, 8};
int target = 10;
List<List<Integer>> result = perfectSum(arr, target);
print2dArray(result);
}
}
# Python Code to find subsets with sum equal to target
# using Dynamic Programming
# Recursively finds all subsets with the given target
def findSubsets(arr, dp, i, target, curr, res):
# Base case: If target becomes 0
if i == 0:
if target == 0:
res.append(curr[:])
if arr[0] == target:
curr.append(arr[0])
res.append(curr[:])
curr.pop()
return
# Exclude current element
if dp[i-1][target]:
findSubsets(arr, dp, i-1, target, curr, res)
# Include current element if it does not exceed target
if target >= arr[i] and dp[i-1][target-arr[i]]:
curr.append(arr[i])
findSubsets(arr, dp, i-1, target-arr[i], curr, res)
curr.pop()
# Returns all subsets with the given target sum
def perfectSum(arr, target):
n = len(arr)
if n == 0 or target < 0:
return []
# DP table to store subset sum possibilities
dp = [[False] * (target+1) for _ in range(n)]
# Correct DP initialization for handling zeroes
dp[0][0] = True
if arr[0] <= target:
dp[0][arr[0]] = True
for i in range(1, n):
for j in range(target+1):
dp[i][j] = dp[i-1][j] or (arr[i] <= j and dp[i-1][j-arr[i]])
# If no subsets sum to target, return empty
if not dp[n-1][target]:
return []
res = []
curr = []
findSubsets(arr, dp, n-1, target, curr, res)
return res
# Function to print subsets in required format
def print2dArray(arr):
if not arr:
# No valid subsets found
print("-1")
return
# Printing subsets in formatted output
for row in range(len(arr)):
print("[", end="")
for col in range(len(arr[row]) - 1, -1, -1):
print(arr[row][col], end="")
if col != 0:
print(", ", end="")
print("]", end="")
if row < len(arr) - 1:
print(", ", end="")
if __name__ == "__main__":
arr = [5, 2, 3, 10, 6, 8]
target = 10
result = perfectSum(arr, target)
print2dArray(result)
// C# Code to find subsets with sum equal to target
// using Dynamic Programming
using System;
using System.Collections.Generic;
class GfG {
// Recursively finds all subsets with the given target
static void findSubsets(int[] arr, bool[,] dp,
int i, int target, List<int> curr,
List<List<int>> res) {
// Base case: If target becomes 0
if (i == 0) {
if (target == 0) res.Add(new List<int>(curr));
if (arr[0] == target) {
curr.Add(arr[0]);
res.Add(new List<int>(curr));
curr.RemoveAt(curr.Count - 1);
}
return;
}
// Exclude current element
if (dp[i-1, target]) {
findSubsets(arr, dp, i-1, target, curr, res);
}
// Include current element if it does not exceed target
if (target >= arr[i] && dp[i-1, target-arr[i]]) {
curr.Add(arr[i]);
findSubsets(arr, dp, i-1, target-arr[i], curr, res);
curr.RemoveAt(curr.Count - 1);
}
}
// Returns all subsets with the given target sum
static List<List<int>> perfectSum(int[] arr, int target) {
int n = arr.Length;
if (n == 0 || target < 0) return new List<List<int>>();
// DP table to store subset sum possibilities
bool[,] dp = new bool[n, target + 1];
// Correct DP initialization for handling zeroes
dp[0, 0] = true;
if (arr[0] <= target) dp[0, arr[0]] = true;
for (int i = 1; i < n; ++i) {
for (int j = 0; j <= target; ++j) {
dp[i, j] = dp[i-1, j]
|| (arr[i] <= j && dp[i-1, j-arr[i]]);
}
}
// If no subsets sum to target, return empty
if (!dp[n-1, target]) return new List<List<int>>();
List<List<int>> res = new List<List<int>>();
findSubsets(arr, dp, n-1, target, new List<int>(), res);
return res;
}
// Function to print subsets in required format
static void print2dArray(List<List<int>> arr) {
if (arr.Count == 0) {
// No valid subsets found
Console.WriteLine("-1");
return;
}
// Printing subsets in formatted output
for (int row = 0; row < arr.Count; row++) {
Console.Write("[");
for (int col = arr[row].Count - 1; col >= 0; col--) {
Console.Write(arr[row][col]);
if (col != 0) {
Console.Write(", ");
}
}
Console.Write("]");
if (row < arr.Count - 1) Console.Write(", ");
}
}
// Driver function
static void Main() {
int[] arr = {5, 2, 3, 10, 6, 8};
int target = 10;
List<List<int>> result = perfectSum(arr, target);
print2dArray(result);
}
}
// JavaScript Code to find subsets with sum equal to target
// using Dynamic Programming
function findSubsets(arr, dp, i, target, curr, res) {
// Base case: If target becomes 0
if (i === 0) {
if (target === 0) res.push([...curr]);
if (arr[0] === target) {
curr.push(arr[0]);
res.push([...curr]);
curr.pop();
}
return;
}
// Exclude current element
if (dp[i - 1][target]) {
findSubsets(arr, dp, i - 1, target, curr, res);
}
// Include current element if it does not exceed target
if (target >= arr[i] && dp[i - 1][target - arr[i]]) {
curr.push(arr[i]);
findSubsets(arr, dp, i - 1, target - arr[i], curr, res);
curr.pop();
}
}
function perfectSum(arr, target) {
let n = arr.length;
if (n === 0 || target < 0) return [];
// DP table to store subset sum possibilities
let dp = Array.from({ length: n }, () => Array(target + 1).fill(false));
// Correct DP initialization for handling zeroes
dp[0][0] = true;
if (arr[0] <= target) dp[0][arr[0]] = true;
for (let i = 1; i < n; ++i) {
for (let j = 0; j <= target; ++j) {
dp[i][j] = dp[i - 1][j] || (arr[i] <= j && dp[i - 1][j - arr[i]]);
}
}
// If no subsets sum to target, return empty
if (!dp[n - 1][target]) return [];
let res = [];
let curr = [];
findSubsets(arr, dp, n - 1, target, curr, res);
return res;
}
function print2dArray(arr) {
if (arr.length === 0) {
// No valid subsets found
console.log("-1");
return;
}
// Printing subsets in formatted output
let output = "";
for (let row = 0; row < arr.length; row++) {
output += "[";
for (let col = arr[row].length - 1; col >= 0; col--) {
output += arr[row][col];
if (col !== 0) {
output += ", ";
}
}
output += "]";
if (row < arr.length - 1) output += ", ";
}
console.log(output);
}
// Driver Code
let arr = [5, 2, 3, 10, 6, 8];
let target = 10;
let result = perfectSum(arr, target);
print2dArray(result);
Output[5, 2, 3], [10], [2, 8]
Time Complexity: O(2^n), as DP table filling takes O(n * target), and subset generation takes O(2^n).
Space Complexity: O(n * target), as DP table uses O(n * target) space, and recursion stack takes O(n).
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Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
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