Find the missing element in an array of integers represented in binary format

Given N strings which represents all integers from 0 to N in binary format except any one. The task is to find the missing number. Input consists of an array of strings where array elements are represented in binary format.

Examples:

Input: arr[] = {“0000”, “0001”, “0010”, “0100”}
Output: 3

Input: arr[] = {“0000”, “0001”, “0010”, “0011”, “0100”, “0110”, “0111”, “1000”}
Output: 5

Approach:

  • An imbalance of 1’s and 0’s in the least significant bits of the numbers can be observed in the N integers given. Since one number is missing either a 0 or 1 from the LSB is missing. If the number which is missing has LSB = 0 then count(1) will be greater than equal to count(0). If LSB of missing number is 1 then count(1) is less than count(0).
  • From the step 1 one can easily determine the LSB of missing number.
  • Once determined, discard all the numbers having LSB different from that of the missing number, i.e., if the missing number has LSB = 0, then discard all the numbers with LSB = 1 and vice versa.
  • Continue the process from step 1 all over again and recur for the next LSB.
  • Continue with the above process till all the bits are traversed.

Below is the impplementation of the above approach:

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// C++ program to find the missing integer
// in N numbers when N bits are given
#include <bits/stdc++.h>
using namespace std;
  
class BitInteger {
private:
    bool* bits;
  
public:
    static const int INTEGER_SIZE = 32;
  
    BitInteger()
    {
        bits = new bool[INTEGER_SIZE];
    }
  
    // Constructor to convert an integer
    // variable into binary format
    BitInteger(int value)
    {
        bits = new bool[INTEGER_SIZE];
  
        for (int j = 0; j < INTEGER_SIZE; j++) {
  
            // The if statement will shift the
            // original value j times.
            // So that appropriate (INTEGER_SIZE - 1 -j)th
            // bits will be either 0/1.
            //  (INTEGER_SIZE - 1 -j)th bit for all
            // j = 0 to INTEGER_SIZE-1 corresponds
            // to  LSB to MSB respectively.
            if (((value >> j) & 1) == 1)
                bits[INTEGER_SIZE - 1 - j] = true;
            else
                bits[INTEGER_SIZE - 1 - j] = false;
        }
    }
    // Constructor to convert a
    // string into binary format.
    BitInteger(string str)
    {
        int len = str.length();
        int x = INTEGER_SIZE - len;
        bits = new bool[INTEGER_SIZE];
  
        // If len = 4. Then x = 32 - 4 = 28.
        // Hence iterate from
        // bit 28 to bit 32 and just
        // replicate the input string.
        int i = 0;
  
        for (int j = x; j <= INTEGER_SIZE && i < len; j++, i++) {
            if (str[i] == '1')
                bits[j] = true;
            else
                bits[j] = false;
        }
    }
  
    // this function fetches the kth bit
    int fetch(int k)
    {
        if (bits[k])
            return 1;
  
        return 0;
    }
  
    // this function will set a value
    // of bit indicated by k to given bitValue
    void set(int k, int bitValue)
    {
        if (bitValue == 0)
            bits[k] = false;
        else
            bits[k] = true;
    }
  
    // convert binary representation to integer
    int toInt()
    {
        int n = 0;
        for (int i = 0; i < INTEGER_SIZE; i++) {
            n = n << 1;
            if (bits[i])
                n = n | 1;
        }
        return n;
    }
};
  
// Function to find the missing number
int findMissingFunc(list<BitInteger>& myList, int column)
{
    // This means that we have processed
    // the entire 32 bit binary number.
    if (column < 0)
        return 0;
  
    list<BitInteger> oddIndices;
    list<BitInteger> evenIndices;
  
    for (BitInteger t : myList) {
  
        // Initially column = LSB. So
        // if LSB of the given number is 0,
        // then the number is even and
        // hence we add it to evenIndices list.
        // else if LSB = 0 then add it to oddIndices list.
        if (t.fetch(column) == 0)
            evenIndices.push_back(t);
        else
            oddIndices.push_back(t);
    }
  
    // Step 1 and Step 2 of the algorithm.
    // Here we determine the LSB bit of missing number.
  
    if (oddIndices.size() >= evenIndices.size())
  
        // LSB of the missing number is 0.
        // Hence it is an even number.
        // Step 3 and 4 of the algorithm
        // (discarding all odd numbers)
        return (findMissingFunc(evenIndices, column - 1)) << 1 | 0;
  
    else
        // LSB of the missing number is 1.
        // Hence it is an odd number.
        // Step 3 and 4 of the algorithm
        // (discarding all even numbers)
        return (findMissingFunc(oddIndices, column - 1)) << 1 | 1;
}
  
// Function to return the missing integer
int findMissing(list<BitInteger>& myList)
{
    // Initial call is with given array and LSB.
    return findMissingFunc(myList, BitInteger::INTEGER_SIZE - 1);
}
  
// Driver Code.
int main()
{
  
    // a corresponds to the input array which
    // is a list of binary numbers
    list<BitInteger> a = { BitInteger("0000"), BitInteger("0001"),
                           BitInteger("0010"), BitInteger("0100"),
                           BitInteger("0101") };
    int missing1 = findMissing(a);
    cout << missing1 << "\n";
  
    return 0;
}

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Output:

3

Time Complexity: O(N)



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