Count substring of Binary string such that each character belongs to a palindrome of size greater than 1

Given binary string str, the task is to count the number of substrings of the given string str such that each character of the substring belongs to a palindromic substring of length at least 2.

Examples:

Input: S = “00111” 
Output: 6 
Explanation:
There are 6 such substrings in the given string such that each character belongs to a palindrome of size greater than 1 as {“00”, “0011”, “00111”, “11”, “111”, “11”}

Input: S = “0001011” 
Output: 15

Approach: The idea is to count the substrings in which every character doesn’t belong to a palindromic substring and then subtract this count from the total number of possible substrings of the string of size greater than 1. Below is the illustration of the steps:

• It can be observed that if we take the substring a₂a₃….a_k-1 (i.e. without starting and ending character), then each of its characters may belong to some palindrome.Proof:

If ai == ai-1 or ai == ai+1,
Then it belongs to a palindrome of length 2.

Otherwise, If ai != ai-1,
ai != ai+1 and ai+1 == ai-1,
Then, It belongs to a palindrome of size 3.

• Therefore, there are four patterns of substrings in which each character doesn’t belong to the palindrome: 
  1. “0111….11”
  2. “100…..00”
  3. “111….110”
  4. “000….001”
• Finally, subtract this count from the total number of substrings possible of length greater than 1.

Count = (N*(N – 1)/2) – (Count of the substrings in which each character doesn’t belongs to palindrome)

Below is the implementation of the above approach:

C++

// C++ implementation to find the
// substrings in binary string
// such that every character
// belongs to a palindrome

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

// Function to to find the
// substrings in binary string
// such that every character
// belongs to a palindrome
int countSubstrings(string s)
{
    int n = s.length();

    // Total substrings
    int answer = (n * (n - 1)) / 2;

    int cnt = 1;
    vector<int> v;

    // Loop to store the count of
    // continuous characters in
    // the given string
    for (int i = 1; i < n; i++) {

        if (s[i] == s[i - 1])
            cnt++;
        else {
            v.push_back(cnt);
            cnt = 1;
        }
    }

    if (cnt > 0)
        v.push_back(cnt);

    // Subtract non special
    // strings from answer
    for (int i = 0;
         i < v.size() - 1; i++) {
        answer -= (v[i]
                   + v[i + 1]
                   - 1);
    }

    return answer;
}

// Driver Code
int main()
{
    // Given string s
    string s = "00111";

    // Function Call
    cout << countSubstrings(s);
    return 0;
}

Java

// Java implementation to find the
// substrings in binary string
// such that every character
// belongs to a palindrome
import java.util.*;

class GFG{

// Function to to find the
// substrings in binary string
// such that every character
// belongs to a palindrome
public static int countSubstrings(String s)
{
    int n = s.length();

    // Total substrings
    int answer = (n * (n - 1)) / 2;

    int cnt = 1;
    Vector<Integer> v = new Vector<Integer>();

    // Loop to store the count of
    // continuous characters in
    // the given string
    for(int i = 1; i < n; i++)
    {
        if (s.charAt(i) == s.charAt(i - 1))
            cnt++;
        else
        {
            v.add(cnt);
            cnt = 1;
        }
    }
    if (cnt > 0)
        v.add(cnt);

    // Subtract non special
    // strings from answer
    for(int i = 0; i < v.size() - 1; i++)
    {
        answer -= (v.get(i) +
                   v.get(i + 1) - 1);
    }
    return answer;
}

// Driver code
public static void main(String[] args)
{

    // Given string s
    String s = "00111";

    // Function call
    System.out.print(countSubstrings(s));
}
}

// This code is contributed by divyeshrabadiya07

Python3

# Python3 implementation to find the
# substrings in binary string
# such that every character
# belongs to a palindrome

# Function to find the substrings in
# binary string such that every
# character belongs to a palindrome
def countSubstrings (s):

    n = len(s)

    # Total substrings
    answer = (n * (n - 1)) // 2

    cnt = 1
    v = []

    # Loop to store the count
    # of continuous characters
    # in the given string
    for i in range(1, n):
        if (s[i] == s[i - 1]):
            cnt += 1

        else:
            v.append(cnt)
            cnt = 1

    if (cnt > 0):
        v.append(cnt)

    # Subtract non special strings
    # from answer
    for i in range(len(v) - 1):
        answer -= (v[i] + v[i + 1] - 1)

    return answer

# Driver Code
if __name__ == '__main__':

    # Given string
    s = "00111"

    # Function call
    print(countSubstrings(s))

# This code is contributed by himanshu77

C#

// C# implementation to find the
// substrings in binary string
// such that every character
// belongs to a palindrome
using System;
using System.Collections.Generic;

class GFG{

// Function to to find the
// substrings in binary string
// such that every character
// belongs to a palindrome
public static int countSubstrings(String s)
{
    int n = s.Length;

    // Total substrings
    int answer = (n * (n - 1)) / 2;

    int cnt = 1;
    List<int> v = new List<int>();

    // Loop to store the count of
    // continuous characters in
    // the given string
    for(int i = 1; i < n; i++)
    {
        if (s[i] == s[i-1])
            cnt++;
        else
        {
            v.Add(cnt);
            cnt = 1;
        }
    }
    if (cnt > 0)
        v.Add(cnt);

    // Subtract non special
    // strings from answer
    for(int i = 0; i < v.Count - 1; i++)
    {
        answer -= (v[i] +
                   v[i + 1] - 1);
    }
    return answer;
}

// Driver code
public static void Main(String[] args)
{

    // Given string s
    String s = "00111";

    // Function call
    Console.Write(countSubstrings(s));
}
}

// This code contributed by sapnasingh4991

Javascript

<script>

// Javascript implementation to find the
// substrings in binary string
// such that every character
// belongs to a palindrome

// Function to to find the
// substrings in binary string
// such that every character
// belongs to a palindrome
function countSubstrings(s)
{
    var n = s.length;

    // Total substrings
    var answer = (n * (n - 1)) / 2;

    var cnt = 1;
    var v =[];

    // Loop to store the count of
    // continuous characters in
    // the given string
    for(var i = 1; i < n; i++)
    {
        if (s.charAt(i) == s.charAt(i - 1))
            cnt++;
        else
        {
            v.push(cnt);
            cnt = 1;
        }
    }
    if (cnt > 0)
        v.push(cnt);

    // Subtract non special
    // strings from answer
    for(var i = 0; i < v.length - 1; i++)
    {
        answer -= (v[i] +
                   v[i + 1] - 1);
    }
    return answer;
}

// Driver code
//Given string s
var s = "00111";

// Function call
document.write(countSubstrings(s));

// This code contributed by shikhasingrajput

</script>

Output:

6

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