Subset Sum problem

Subset sum problem is to find subset of elements that are selected from a given set whose sum adds up to a given number K. We are considering the set contains non-negative values. It is assumed that the input set is unique (no duplicates are presented).

Exhaustive Search Algorithm for Subset Sum
One way to find subsets that sum to K is to consider all possible subsets. A power set contains all those subsets generated from a given set. The size of such a power set is 2^N.

Algorithm for sum of subset

Algorithm SUB_SET_PROBLEM, (i,sum, W, remSum)
// Description: Solve sub of subset problem using backtracking

//Input: W: Number for which subset is to be computed

i: Item index
 sum: Sum of integers selected so far
 remSum: Size of remaining problem i.e. (W - sum)
// Output: Solution tuple X

if FEASIBLE SUB_SET() == 1 then
if (sum == W) then 
print X [1...i]
end
else
X [i + 1] <- 1
SUB SET PROBLEM(i +1. sum + w[i] + 1 ,W, remSum - w * [i] + 1 )
X[i+1] <- 0 //Exclude the " item
SUB_SET_PROBLEM( i + 1, sum, W, remSum- w[i] + 1 )
end
Function FEASIBLE_SUB_SET()

if (sum+remSum≥ W) AND (sum == W )
or ( sum + w[i] + 1 <= W) then
return 0
end

return 1

First recursive call represents the case when the current item is selected, and hence the problem size is reduced by w[i]. 

Second recursive call represents the case when we do not select the current item.

Complexity Analysis: It is intuitive to derive the complexity of sum of subset problem. In state space tree, at level i, the tree has 2′ nodes. So given n items, total number of nodes in tree would be 1 + 2 + 2 ^ 2 + 2 ^ 3 +..2^ n

T(n)=1+2+2^ 2 +2^ 3 +. 0.2 ^ n = 2 ^ (n + 1) – 1 = O(2 ^ n)

 Thus, sum of sub set problem runs in exponential order.

Backtracking Algorithm for Subset Sum
Using exhaustive search we consider all subsets irrespective of whether they satisfy given constraints or not. Backtracking can be used to make a systematic consideration of the elements to be selected.
Assume given set of 4 elements, say w[1] … w[4]. Tree diagrams can be used to design backtracking algorithms. The following tree diagram depicts approach of generating variable sized tuple.

In the above tree, a node represents function call and a branch represents candidate element. The root node contains 4 children. In other words, root considers every element of the set as different branch. The next level sub-trees correspond to the subsets that includes the parent node. The branches at each level represent tuple element to be considered. For example, if we are at level 1, tuple_vector[1] can take any value of four branches generated. If we are at level 2 of left most node, tuple_vector[2] can take any value of three branches generated, and so on…

For example the left most child of root generates all those subsets that include w[1]. Similarly the second child of root generates all those subsets that includes w[2] and excludes w[1].

As we go down along depth of tree we add elements so far, and if the added sum is satisfying explicit constraints, we will continue to generate child nodes further. Whenever the constraints are not met, we stop further generation of sub-trees of that node, and backtrack to previous node to explore the nodes not yet explored. In many scenarios, it saves considerable amount of processing time.

The tree should trigger a clue to implement the backtracking algorithm (try yourself). It prints all those subsets whose sum add up to given number. We need to explore the nodes along the breadth and depth of the tree. Generating nodes along breadth is controlled by loop and nodes along the depth are generated using recursion (post order traversal). Pseudo code given below, 

if(subset is satisfying the constraint)
    print the subset
    exclude the current element and consider next element
else
    generate the nodes of present level along breadth of tree and
    recur for next levels

Following is the implementation of subset sum using variable size tuple vector. Note that the following program explores all possibilities similar to exhaustive search. It is to demonstrate how backtracking can be used. See next code to verify, how we can optimize the backtracking solution.

The power of backtracking appears when we combine explicit and implicit constraints, and we stop generating nodes when these checks fail. We can improve the above algorithm by strengthening the constraint checks and presorting the data. By sorting the initial array, we need not to consider rest of the array, once the sum so far is greater than target number. We can backtrack and check other possibilities.

Similarly, assume the array is presorted and we found one subset. We can generate next node excluding the present node only when inclusion of next node satisfies the constraints. Given below is optimized implementation (it prunes the subtree if it is not satisfying constraints).

C++

#include <bits/stdc++.h>
using namespace std;

#define ARRAYSIZE(a) (sizeof(a))/(sizeof(a[0]))
static int total_nodes;

// prints subset found
void printSubset(int A[], int size)
{
    for(int i = 0; i < size; i++)
    {
        cout<<" "<< A[i];
    }
    cout<<"\n";
}

// qsort compare function
int comparator(const void *pLhs, const void *pRhs)
{
    int *lhs = (int *)pLhs;
    int *rhs = (int *)pRhs;
    return *lhs > *rhs;
}

// inputs
// s            - set vector
// t            - tuplet vector
// s_size       - set size
// t_size       - tuplet size so far
// sum          - sum so far
// ite          - nodes count
// target_sum   - sum to be found
void subset_sum(int s[], int t[],
                int s_size, int t_size,
                int sum, int ite,
                int const target_sum)
{
    total_nodes++;

    if( target_sum == sum )
    {
        // We found sum
        printSubset(t, t_size);

        // constraint check
        if( ite + 1 < s_size && sum - s[ite] + s[ite + 1] <= target_sum )
        {

            // Exclude previous added item and consider next candidate
            subset_sum(s, t, s_size, t_size - 1, sum - s[ite], ite + 1, target_sum);
        }
        return;
    }
    else
    {

        // constraint check
        if( ite < s_size && sum + s[ite] <= target_sum )
        {

            // generate nodes along the breadth
            for( int i = ite; i < s_size; i++ )
            {
                t[t_size] = s[i];
                if( sum + s[i] <= target_sum )
                {

                    // consider next level node (along depth)
                    subset_sum(s, t, s_size, t_size + 1, sum + s[i], i + 1, target_sum);
                }
            }
        }
    }
}

// Wrapper that prints subsets that sum to target_sum
void generateSubsets(int s[], int size, int target_sum)
{
    int *tuplet_vector = (int *)malloc(size * sizeof(int));
    int total = 0;

    // sort the set
    qsort(s, size, sizeof(int), &comparator);
    for( int i = 0; i < size; i++ )
    {
        total += s[i];
    }
    if( s[0] <= target_sum && total >= target_sum )
    {
        subset_sum(s, tuplet_vector, size, 0, 0, 0, target_sum);
    }
    free(tuplet_vector);
}

// Driver code
int main()
{
    int weights[] = {15, 22, 14, 26, 32, 9, 16, 8};
    int target = 53;
    int size = ARRAYSIZE(weights);
    generateSubsets(weights, size, target);
    cout << "Nodes generated " << total_nodes;
    return 0;
}

//This code is contributed by shivanisinghss2110

C

#include <stdio.h>
#include <stdlib.h>

#define ARRAYSIZE(a) (sizeof(a))/(sizeof(a[0]))

static int total_nodes;

// prints subset found
void printSubset(int A[], int size)
{
    for(int i = 0; i < size; i++)
    {
        printf("%*d", 5, A[i]);
    }

    printf("\n");
}

// qsort compare function
int comparator(const void *pLhs, const void *pRhs)
{
    int *lhs = (int *)pLhs;
    int *rhs = (int *)pRhs;

    return *lhs > *rhs;
}

// inputs
// s            - set vector
// t            - tuplet vector
// s_size       - set size
// t_size       - tuplet size so far
// sum          - sum so far
// ite          - nodes count
// target_sum   - sum to be found
void subset_sum(int s[], int t[],
                int s_size, int t_size,
                int sum, int ite,
                int const target_sum)
{
    total_nodes++;

    if( target_sum == sum )
    {
        // We found sum
        printSubset(t, t_size);

        // constraint check
        if( ite + 1 < s_size && sum - s[ite] + s[ite+1] <= target_sum )
        {
            // Exclude previous added item and consider next candidate
            subset_sum(s, t, s_size, t_size-1, sum - s[ite], ite + 1, target_sum);
        }
        return;
    }
    else
    {
        // constraint check
        if( ite < s_size && sum + s[ite] <= target_sum )
        {
            // generate nodes along the breadth
            for( int i = ite; i < s_size; i++ )
            {
                t[t_size] = s[i];

                if( sum + s[i] <= target_sum )
                {
                    // consider next level node (along depth)
                    subset_sum(s, t, s_size, t_size + 1, sum + s[i], i + 1, target_sum);
                }
            }
        }
    }
}

// Wrapper that prints subsets that sum to target_sum
void generateSubsets(int s[], int size, int target_sum)
{
    int *tuplet_vector = (int *)malloc(size * sizeof(int));

    int total = 0;

    // sort the set
    qsort(s, size, sizeof(int), &comparator);

    for( int i = 0; i < size; i++ )
    {
        total += s[i];
    }

    if( s[0] <= target_sum && total >= target_sum )
    {

        subset_sum(s, tuplet_vector, size, 0, 0, 0, target_sum);

    }

    free(tuplet_vector);
}

int main()
{
    int weights[] = {15, 22, 14, 26, 32, 9, 16, 8};
    int target = 53;

    int size = ARRAYSIZE(weights);

    generateSubsets(weights, size, target);

    printf("Nodes generated %d\n", total_nodes);

    return 0;
}

Java

/*package whatever //do not write package name here */

import java.io.*;
import java.util.Arrays;

public class GFG {

  public static int total_nodes = 0;

  // prints subset found
  static void printSubset(int[] A, int size){
    for(int i = 0; i < size; i++){
      System.out.print(" " + A[i]);
    }
    System.out.println();
  }

  // inputs
  // s            - set vector
  // t            - tuplet vector
  // s_size       - set size
  // t_size       - tuplet size so far
  // sum          - sum so far
  // ite          - nodes count
  // target_sum   - sum to be found
  static void subset_sum(int[] s, int[] t, int s_size, int t_size, int sum, int ite, int target_sum){
    total_nodes++;

    if( target_sum == sum ){

      // We found sum
      printSubset(t, t_size);

      // constraint check
      if( ite + 1 < s_size && sum - s[ite] + s[ite + 1] <= target_sum ){

        // Exclude previous added item and consider next candidate
        subset_sum(s, t, s_size, t_size - 1, sum - s[ite], ite + 1, target_sum);
      }
      return;
    }
    else{

      // constraint check
      if( ite < s_size && sum + s[ite] <= target_sum ){

        // generate nodes along the breadth
        for( int i = ite; i < s_size; i++ ){

          t[t_size] = s[i];
          if( sum + s[i] <= target_sum ){

            // consider next level node (along depth)
            subset_sum(s, t, s_size, t_size + 1, sum + s[i], i + 1, target_sum);
          }
        }
      }
    }
  }

  // Wrapper that prints subsets that sum to target_sum
  static void generateSubsets(int[] s, int size, int target_sum)
  {
    int[] tuplet_vector = new int[size];
    int total = 0;

    // sort the set
    Arrays.sort(s);

    for(int i = 0; i < size; i++ ){
      total = total + s[i];
    }

    if( s[0] <= target_sum && total >= target_sum ){
      subset_sum(s, tuplet_vector, size, 0, 0, 0, target_sum);
    }
  }

  public static void main(String[] args) {

    int[] weights = {15, 22, 14, 26, 32, 9, 16, 8};
    int target = 53;
    int size = weights.length;
    generateSubsets(weights, size, target);
    System.out.println("Nodes generated " + total_nodes);
  }
}

// The code is contributed by Gautam goel.

Python3

from typing import List

total_nodes = 0

# prints subset found
def print_subset(A:List[int], size:int)->None:
    for i in range(size):
        print(A[i], end=' ')
    print()

   # inputs
  # s            - set vector
  # t            - tuplet vector
  # s_size       - set size
  # t_size       - tuplet size so far
  # sum          - sum so far
  # ite          - nodes count
  # target_sum   - sum to be found
def subset_sum(s:List[int], t:List[int], s_size:int, t_size:int, sum_so_far:int, ite:int, target_sum:int)->None:
    global total_nodes
    total_nodes += 1
    if target_sum == sum_so_far:

      # We found sum
        print_subset(t,t_size)

        # constraint check
        if ite + 1 < s_size and sum_so_far - s[ite] + s[ite + 1] <= target_sum:

          # Exclude previous added item and consider next candidate
            subset_sum(s, t, s_size, t_size - 1, sum_so_far - s[ite], ite + 1, target_sum)
        return
    elif ite < s_size and sum_so_far + s[ite] <= target_sum:
        for i in range(ite, s_size):
            t[t_size] = s[i]
            if sum_so_far + s[i] <= target_sum:
                subset_sum(s, t, s_size, t_size + 1, sum_so_far + s[i], i + 1, target_sum)

# Wrapper that prints subsets that sum to target_sum
def generateSubsets(s:List[int], size:int, target_sum:int)->None:
    t = [0]*size
    total = 0

    # sort the set
    s = sorted(s)
    for i in range(size):
        total += s[i]
    if s[0] <= target_sum and total >= target_sum:
        subset_sum(s, t, size, 0, 0, 0, target_sum)

# Driver code
weights = [15, 22, 14, 26, 32, 9, 16, 8]
target = 53
size = len(weights)
generateSubsets(weights, size, target)
print("Nodes generated ",total_nodes)

# This code is contributed by hardikkhushwaha

C#

using System;
using System.Linq;

public class GFG {

  public static int total_nodes = 0;

  // prints subset found
  static void printSubset(int[] A, int size)
  {
    for (int i = 0; i < size; i++) {
      Console.Write(" " + A[i]);
    }
    Console.WriteLine();
  }

  // inputs
  // s            - set vector
  // t            - tuplet vector
  // s_size       - set size
  // t_size       - tuplet size so far
  // sum          - sum so far
  // ite          - nodes count
  // target_sum   - sum to be found
  static void subset_sum(int[] s, int[] t, int s_size,
                         int t_size, int sum, int ite,
                         int target_sum)
  {
    total_nodes++;

    if (target_sum == sum) {

      // We found sum
      printSubset(t, t_size);

      // constraint check
      if (ite + 1 < s_size
          && sum - s[ite] + s[ite + 1]
          <= target_sum) {

        // Exclude previous added item and consider
        // next candidate
        subset_sum(s, t, s_size, t_size - 1,
                   sum - s[ite], ite + 1,
                   target_sum);
      }
      return;
    }
    else {

      // constraint check
      if (ite < s_size
          && sum + s[ite] <= target_sum) {

        // generate nodes along the breadth
        for (int i = ite; i < s_size; i++) {

          t[t_size] = s[i];
          if (sum + s[i] <= target_sum) {

            // consider next level node (along
            // depth)
            subset_sum(s, t, s_size, t_size + 1,
                       sum + s[i], i + 1,
                       target_sum);
          }
        }
      }
    }
  }

  // Wrapper that prints subsets that sum to target_sum
  static void generateSubsets(int[] s, int size,
                              int target_sum)
  {
    int[] tuplet_vector = new int[size];
    int total = 0;

    // sort the set
    Array.Sort(s);

    for (int i = 0; i < size; i++) {
      total = total + s[i];
    }

    if (s[0] <= target_sum && total >= target_sum) {
      subset_sum(s, tuplet_vector, size, 0, 0, 0,
                 target_sum);
    }
  }

  static public void Main()
  {

    int[] weights = { 15, 22, 14, 26, 32, 9, 16, 8 };
    int target = 53;
    int size = weights.Length;
    generateSubsets(weights, size, target);
    Console.WriteLine("Nodes generated " + total_nodes);
  }
}

// The code is contributed by akashish__

Javascript

// JavaScript code

// prints subset found
const printSubset = (A, size) => {
    for (let i = 0; i < size; i++) {
        console.log(A[i]);
    }
    console.log("\n");
}

// qsort compare function
const comparator = (lhs, rhs) => {
    return lhs - rhs;
}

let total_nodes = 0;

// inputs
// s            - set vector
// t            - tuplet vector
// s_size       - set size
// t_size       - tuplet size so far
// sum          - sum so far
// ite          - nodes count
// target_sum   - sum to be found
const subset_sum = (s, t, s_size, t_size, sum, ite, target_sum) => {
    total_nodes++;
    if (target_sum === sum) {

        // We found sum
        printSubset(t, t_size);

        // constraint check
        if (ite + 1 < s_size && sum - s[ite] + s[ite + 1] <= target_sum) {

            // Exclude previous added item and consider next candidate
            subset_sum(s, t, s_size, t_size - 1, sum - s[ite], ite + 1, target_sum);
        }
        return;
    } else {
        // constraint check
        if (ite < s_size && sum + s[ite] <= target_sum) {

            // generate nodes along the breadth
            for (let i = ite; i < s_size; i++) {
                t[t_size] = s[i];
                if (sum + s[i] <= target_sum) {

                    // consider next level node (along depth)
                    subset_sum(s, t, s_size, t_size + 1, sum + s[i], i + 1, target_sum);
                }
            }
        }
    }
}

// Wrapper that prints subsets that sum to target_sum
const generateSubsets = (s, size, target_sum) => {
    let tuplet_vector = Array(size);
    let total = 0;

    // sort the set
    s.sort(comparator);
    for (let i = 0; i < size; i++) {
        total += s[i];
    }
    if (s[0] <= target_sum && total >= target_sum) {
        subset_sum(s, tuplet_vector, size, 0, 0, 0, target_sum);
    }
}

// Driver Code
const weights = [15, 22, 14, 26, 32, 9, 16, 8];
const target = 53;
const size = weights.length;
generateSubsets(weights, size, target);
console.log("Nodes generated" +  total_nodes );

// This code is contributed by akashish__

Output:

8 9 14 22n 8 14 15 16n 15 16 22nNodes generated 68

Time Complexity:  O(2^(n/2))

Space Complexity: O(n)

As another approach, we can generate the tree in fixed size tuple analogs to binary pattern. We will kill the sub-trees when the constraints are not satisfied.
– – – Venki. 

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