Construct a K-length binary string from an array based on given conditions

Given an array arr[] consisting of N integers, and an integer K, the task is to construct a binary string of length K satisfying the following conditions: 
 

1. The character at i^th index is '1' if a subset with sum i can be formed from the array.
2. Otherwise, the character at i^th index is '0'.

Examples:

Input: arr[] = {1, 4}, K = 5
Output: 10011
Explanation: 
Character at 1st index can be made by '1' considering the subset {1}. 
Character at 4th index can be made by '1' considering the subset {4}. 
Character at 5th index can be made by '1' considering the subset {1, 4}.

Input: arr[] = {1, 6, 1}, K = 8
Output: 11000111

Approach: The idea is to use a greedy approach to solve this problem. Below are the steps:

• Initialize a bitset, say bit[], of size 10⁵ + 5 and set bit[0] = 1.
• Traverse through the array and for each array element arr[i], update bit as bit |= bit << arr[i] to have bit p if p can be obtained as a subset sum.
• At i^th iteration, bit[i] stores the initial sum and after performing bit << arr[i], all bits are shifted by arr[i]. Therefore, bit p becomes p + arr[i].
• Finally, bit | (bit << arr[i]) merges these two cases, whether to consider the i^th position or not.
• Iterate from 1 to K and print every value bit[i] as the required binary string.

Below is the implementation of the above approach:

// C++ program for the above approach

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

// To construct the
// required binary string
bitset<100003> bit;

// Function to construct binary string
// according to the given conditions
void constructBinaryString(int arr[],
                           int N, int K)
{
    // Initialize with 1
    bit[0] = 1;

    // Traverse the array
    for (int i = 0; i < N; i++) {

        // To check if the i-th integer
        // needs to be considered or not
        bit |= bit << arr[i];
    }

    // Print the binary string
    for (int i = 1; i <= K; i++) {
        cout << bit[i];
    }
}

// Driver Code
int main()
{
    // Given array
    int arr[] = { 1, 6, 1 };

    // Size of the array
    int N = sizeof(arr) / sizeof(arr[0]);

    // Given K
    int K = 8;

    constructBinaryString(arr, N, K);
}

// Java code to implement the approach
import java.util.Arrays;

class BinaryString {
  public static void main(String[] args) {

    // Given array
    int[] arr = {1, 6, 1};

    // Size of the array
    int N = arr.length;

    // Given K
    int K = 8;

    constructBinaryString(arr, N, K);
  }

  // Function to construct binary string
  // according to the given conditions
  static void constructBinaryString(int[] arr, int N, int K) {
    // Initialize with 1
    int bit = 1;

    // Traverse the array
    for (int i = 0; i < N; i++) {
      // To check if the i-th integer
      // needs to be considered or not
      bit |= (bit << arr[i]);
    }

    // Print the binary string
    String binaryString = Integer.toBinaryString(bit);
    System.out.println(binaryString.substring(1, K));
  }
}

// This code is contributed by phasing17

# Python program for the above approach

# To construct the
# required binary string
#bit = [0]*100003

# Function to construct binary string
# according to the given conditions
def constructBinaryString(arr,N, K):
    
    # Initialize with 1
    bit = 1
    
    # Traverse the array
    for i in range(0, N):
      
        # To check if the i-th eger
        # needs to be considered or not
        bit |= bit << arr[i]
    
    # Print the binary string
    #for i in range(1,K):
    #    print(bit[i])
    bit = bin(bit).replace("0b", "")
    print(bit[1:K + 1])
    
# Driver Code
# Given array
arr = [1, 6, 1]

# Size of the array
N = len(arr)

# Given K
K = 8

constructBinaryString(arr, N, K)

# This code is contributed by shubhamsingh10

using System;
using System.Linq;

namespace BinaryString {
  class Program {
    static void Main(string[] args)
    {
      // Given array
      int[] arr = { 1, 6, 1 };

      // Size of the array
      int N = arr.Length;

      // Given K
      int K = 8;

      ConstructBinaryString(arr, N, K);
    }

    // Function to construct binary string
    // according to the given conditions
    static void ConstructBinaryString(int[] arr, int N,
                                      int K)
    {
      // Initialize with 1
      int bit = 1;

      // Traverse the array
      for (int i = 0; i < N; i++) {
        // To check if the i-th integer
        // needs to be considered or not
        bit |= (bit << arr[i]);
      }

      // Print the binary string
      string binaryString = Convert.ToString(bit, 2);
      Console.WriteLine(binaryString.Substring(1, K));
    }
  }

}

// This code is contributed by phasing17.

// JavaScript program for the above approach

// To construct the
// required binary string
//bit = [0]*100003

// Function to construct binary string
// according to the given conditions
function constructBinaryString(arr,N, K)
{
    
    // Initialize with 1
    let bit = 1
    
    // Traverse the array
    for (var i = 0; i < N; i++)
      
        // To check if the i-th eger
        // needs to be considered or not
        bit |= (bit << arr[i])
    
    // Print the binary string
    //for i in range(1,K):
    //    print(bit[i])
    bit = bit.toString(2)
    console.log(bit.substring(1, K + 1))
}
    
// Driver Code
// Given array
let arr = [1, 6, 1]

// Size of the array
let N = arr.length

// Given K
let K = 8

constructBinaryString(arr, N, K)

// This code is contributed by phasing17

11000111

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