Count Substrings that can be Made of Length 1 by Replacing \"01\" or \"10\" with 1 or 0

In this problem, we will count substrings that we can make of length 1 by replacing the ?10' and ?01' substrings with ?1' or ?0' characters.

When any binary string contains an equal number of ?0' and ?1', we can always make it of length 1 by performing the replacement operations. So, the problem is solved by finding the substrings with equal numbers of ?0' and ?1'.

Problem statement ? We have given a binary string named bin_str and of length bin_len. We need to count the total number of substrings that we can make of length 1 by changing ?10' or ?01' to either ?1' or ?0'.

Sample examples

Input

bin_str = "01010";

Output

6

Explanation ? The substrings are 01, 01, 0101, 10, 10, and 1010.

Input

bin_str = "01";

Output

1

Explanation ? We can replace 01 with ?1' or ?0' to make it of length 1.

Input

bin_str = "00000";

Output

0

Explanation ? No substring exists, which we can make of length 1 after replacement.

Approach 1

The problem can be solved by counting the number of substrings having equal numbers of ?1' and ?0.' We will find all substrings of the string and count ?1' and ?0' in the string. If the count of ?1' and ?0' are the same, we will increment the ?ans' by 1.

Algorithm

Step 1 ? Initialize the ?cnt' with 0 to store the number of valid substrings.

Step 2 ? Also, initialize the ?cnt0' and ?cnt1' with 0 to store the count of ?1' and ?0'.

Step 3 ? Start traversing the binary string.

Step 4 ? Use a nested loop to traverse from index p to the last index.

Step 5 ? If the current character is ?0', increment ?cnt0' by 1. Otherwise, increment ?cnt1' by 1.

Step 6 ? If cnt0 and cnt1 are equal, increment the ?cnt' value by 1.

Step 7 ? Return the ?cnt' value.

Example

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

int totalSubstrs(string &bin_str, int &bin_len) {
    // To store a number of valid substrings
    int cnt = 0;
    // Couting number of substrings having equal 1's and 0's
    for (int p = 0; p < bin_len; p++) {
        // To store count of 1's and 0's in substring
        int cnt0 = 0, cnt1 = 0;
        for (int q = p; q < bin_len; q++) {
            if (bin_str[q] == '0') {
                cnt0++;
            } else {
                cnt1++;
            }
            // When the substring has an equal number of 0's and 1's
            if (cnt0 == cnt1) {
                cnt++;
            }
        }
    }
    return cnt;
}
int main() {
    string bin_str = "01010";
    int bin_len = bin_str.length();
    cout << "The number of substrings according to the given problem statement is " << totalSubstrs(bin_str, bin_len);
    return 0;
}

Output

The number of substrings according to the given problem statement is 6

Time complexity ? O(N*N) to traverse all substrings of the binary string.

Space complexity ? O(1), as we don't use any extra space.

Approach 2

In this approach, we will use the prefix sum technique to count the number of substrings having equal numbers of ?1' and ?0'. We will keep track of the difference of the count of ?1' and ?0' at every index.

If any two index difference of count of ?1' and ?0' is the same, we can take the substring between both indices.

Algorithm

Step 1 ? Initialize the ?cnt' and ?diff' with 0 to store the count of valid substrings and the difference of count of ?1' and ?0'.

Step 2 ? Also, initialize the ?prefixSum' list with 0 to store the difference.

Step 3 ? Start traversing the string. If the current character is ?0', increment the ?diff' value by 1. Otherwise, decrement the ?diff' value by ?1'.

Step 4 ? If ?diff' is ?0', increment the ?cnt' by 1.

Step 5 ? If the value in the ?prefixSum' array at the ?diff' index is greater than 0, add prefixSum[diff] to the ?cnt'.

Step 6 ? Increment the value of the ?prefixSum' list at the ?diff' index by 1.

Step 7 ? Return the ?cnt' value.

Example

#include <iostream>
#include <vector>
using namespace std;

int totalSubstrs(string bin_str, int bin_len) {
    int cnt = 0;
    int diff = 0; // To store the difference of count of 1's and 0's
    vector<int> prefixSum(bin_len, 0); 
    for (char ch : bin_str) {
        // For the '0' character
        if (ch == '0') {
            diff++;
        } else if (ch == '1') {
            // For '1' character
            diff--;
        }
        if (diff == 0) {
            // When the number of '0' and '1' are equal
            cnt++;
        }
        if (prefixSum[diff] > 0) {
            cnt += prefixSum[diff];
        }
        prefixSum[diff]++;
    }
    return cnt;
}
int main() {
    string bin_str = "01010";
    int bin_len = bin_str.length();
    cout << "The number of substrings according to the given problem statement is - " << totalSubstrs(bin_str, bin_len);
    return 0;
}

Output

The number of substrings according to the given problem statement is 6

Time complexity ? O(N) to get the prefix sum of the difference of count of ?1' and ?0'.

Space complexity ? O(N) to store the difference in the prefix sum list.

The problem is similar to finding the substrings with equal numbers of ?1' and ?0'. We have also used the prefix sum technique to solve the problem by storing the difference of the count of ?1' and ?0' in the prefix array. The second approach is time optimized, but it may take more memory for larger binary strings.