Count ways to partition a string such that both parts have equal distinct characters

Difficulty Level : Medium
Last Updated : 03 Aug, 2020

Given a string S, the task is to count the number of ways to partition the string into two non-empty parts such that both the parts have the same number of distinct characters.

Examples:

Input: S = “aaaa”
Output: 3
Explanation:
There can be three ways to partition the string –
{{a, aaa}, {aa, aa}, {aaa, a}}

Input: S = “ababa”
Output: 2
Explanation:
There can be two ways to partition the string –
{{ab, aba}, {aba, ba}}

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive Approach: The idea is to choose the every possible partition point of the string and for each partition compute the distinct characters in the partitioned string. If the count of distinct characters in both the partitioned string is equal then increment the number of ways by 1.

Below is the implementation of the above approach:

C++

// C++ implementation to count the 
// number of ways to partition the  
// string such that each partition  
// have same number of distinct  
// characters in the string 
  
#include <bits/stdc++.h> 
  
using namespace std; 
  
// Function to count the distinct 
// characters in the string 
int distinctChars(string s) 
{ 
    // Frequency of each character 
    int freq[26] = { 0 }; 
    int count = 0; 
      
    // Loop to count the frequency 
    // of each character of the string 
    for (int i = 0; i < s.length(); i++) 
        freq[s[i] - 'a']++; 
  
    // If frequency is greater than 0 
    // then the character occured 
    for (int i = 0; i < 26; i++) { 
        if (freq[i] > 0) 
            count++; 
    } 
  
    return count; 
} 
  
// Function to count the number  
// of ways to partition the string 
// such that each partition have  
// same number of distinct character 
int waysToSplit(string s) 
{ 
    int n = s.length(); 
    int answer = 0; 
      
    // Loop to choose the partition 
    // index for the string 
    for (int i = 1; i < n; i++) { 
          
        // Divide in two parts 
        string left = s.substr(0, i); 
        string right = s.substr(i, n - i); 
  
        // Check whether number of distinct 
        // characters are equal 
        if (distinctChars(left) ==  
             distinctChars(right)) 
            answer++; 
    } 
    return answer; 
} 
  
// Driver Code 
int main() 
{ 
    string s = "ababa"; 
  
    cout << waysToSplit(s); 
    return 0; 
} 

Java

// Java implementation to count the 
// number of ways to partition the  
// String such that each partition  
// have same number of distinct  
// characters in the String 
class GFG{ 
   
// Function to count the distinct 
// characters in the String 
static int distinctChars(String s) 
{ 
    // Frequency of each character 
    int freq[] = new int[26]; 
    int count = 0; 
       
    // Loop to count the frequency 
    // of each character of the String 
    for (int i = 0; i < s.length(); i++) 
        freq[s.charAt(i) - 'a']++; 
   
    // If frequency is greater than 0 
    // then the character occured 
    for (int i = 0; i < 26; i++) { 
        if (freq[i] > 0) 
            count++; 
    } 
   
    return count; 
} 
   
// Function to count the number  
// of ways to partition the String 
// such that each partition have  
// same number of distinct character 
static int waysToSplit(String s) 
{ 
    int n = s.length(); 
    int answer = 0; 
       
    // Loop to choose the partition 
    // index for the String 
    for (int i = 1; i < n; i++) { 
           
        // Divide in two parts 
        String left = s.substring(0, i); 
        String right = s.substring(i, n); 
   
        // Check whether number of distinct 
        // characters are equal 
        if (distinctChars(left) ==  
             distinctChars(right)) 
            answer++; 
    } 
    return answer; 
} 
   
// Driver Code 
public static void main(String[] args) 
{ 
    String s = "ababa"; 
   
    System.out.print(waysToSplit(s)); 
} 
} 
  
// This code is contributed by sapnasingh4991 

Python3

# Python3 implementation to count the  
# number of ways to partition the  
# string such that each partition  
# have same number of distinct  
# characters in the string  
  
# Function to count the distinct  
# characters in the string  
def distinctChars(s) :  
  
    # Frequency of each character  
    freq = [0]*26;  
    count = 0;  
      
    # Loop to count the frequency  
    # of each character of the string  
    for i in range(len(s)) :  
        freq[ord(s[i]) - ord('a')] += 1;  
  
    # If frequency is greater than 0  
    # then the character occured  
    for i in range(26) : 
        if (freq[i] > 0) : 
            count += 1;  
  
    return count;  
  
# Function to count the number  
# of ways to partition the string  
# such that each partition have  
# same number of distinct character  
def waysToSplit(s) :  
    n = len(s);  
    answer = 0;  
      
    # Loop to choose the partition  
    # index for the string  
    for i in range(1, n) : 
          
        # Divide in two parts  
        left = s[0 : i];  
        right = s[i : n ];  
  
        # Check whether number of distinct  
        # characters are equal  
        if (distinctChars(left) == distinctChars(right)) : 
            answer += 1;  
      
    return answer;  
  
# Driver Code  
if __name__ == "__main__" :  
  
    s = "ababa";  
  
    print(waysToSplit(s));  
      
# This code is contributed by Yash_R 

C#

// C# implementation to count the 
// number of ways to partition the  
// String such that each partition  
// have same number of distinct  
// characters in the String 
using System; 
  
class GFG{ 
   
// Function to count the distinct 
// characters in the String 
static int distinctChars(string s) 
{ 
    // Frequency of each character 
    int []freq = new int[26]; 
    int count = 0; 
       
    // Loop to count the frequency 
    // of each character of the String 
    for (int i = 0; i < s.Length; i++) 
        freq[s[i] - 'a']++; 
   
    // If frequency is greater than 0 
    // then the character occured 
    for (int i = 0; i < 26; i++) { 
        if (freq[i] > 0) 
            count++; 
    } 
   
    return count; 
} 
   
// Function to count the number  
// of ways to partition the String 
// such that each partition have  
// same number of distinct character 
static int waysToSplit(string s) 
{ 
    int n = s.Length; 
    int answer = 0; 
       
    // Loop to choose the partition 
    // index for the String 
    for (int i = 1; i < n; i++) { 
           
        // Divide in two parts 
        string left = s.Substring(0, i); 
        string right = s.Substring(i, n-i); 
   
        // Check whether number of distinct 
        // characters are equal 
        if (distinctChars(left) ==  
             distinctChars(right)) 
            answer++; 
    } 
    return answer; 
} 
   
// Driver Code 
public static void Main(string[] args) 
{ 
    string s = "ababa"; 
   
    Console.WriteLine(waysToSplit(s)); 
} 
} 
  
// This code is contributed by Yash_R 

Output:

Time complexity: O(N²)
Auxiliary Space: O(26)

Efficient Approach: The idea is to precompute the distinct character for every possible substring with the help of hash-map for visited characters and Prefix and suffix sum arrays for the distinct characters from start to current index of the string. Below is the illustration of the steps:

Traverse the string for each possible indices and compute the count of distinct characters from start to that index.
- If the current index character is visited for the first time, then increment the count of the distinct characters from the previous index by 1.
```
if (visited[s[i]] == False)
    prefix[i] = prefix[i-1] + 1
```
- If the current index character is visited earlier, then count of distinct characters from starting index to current index will be equal to last index distinct characters. That is –
```
if (visited[s[i]] == True)
    prefix[i] = prefix[i-1]
```
Finally, Traverse the string and for each index count of distinct characters for each partitioned string can be calculated as –
```
Count in left partitioned string 
    = prefix[i]

Count in right partitioned string
    = prefix[L] - prefix[i+1] 
```

Below is the implementation of the above approach:

C++

// C++ implementation to count the 
// number of ways to partition the  
// string such that each partition  
// have same number of distinct  
// characters in the string 
  
#include <bits/stdc++.h> 
  
using namespace std; 
  
// Function to count the number  
// of ways to partition the string 
// such that each partition have  
// same number of distinct character 
int waysToSplit(string s) 
{ 
    int n = s.length(); 
    int answer = 0; 
      
    // Prefix and suffix array for  
    // distinct character from  
    // start and end 
    int prefix[n] = { 0 }; 
    int suffix[n] = { 0 }; 
  
    // To check whether a character  
    // has appeared till ith index 
    int seen[26] = { 0 }; 
  
    // Calculating prefix array 
    for (int i = 0; i < n; i++) { 
  
        int prev = (i - 1 >= 0 ?  
              prefix[i - 1] : 0); 
  
        // Character appears for  
        // the first time in string 
        if (seen[s[i] - 'a'] == 0) { 
            prefix[i] += (prev + 1); 
        } 
        else
            prefix[i] = prev; 
  
        // Character is visited 
        seen[s[i] - 'a'] = 1; 
    } 
      
    // Resetting seen for  
    // suffix calculation 
    memset(seen, 0, sizeof(seen)); 
  
    // Calculating the suffix array 
    suffix[n - 1] = 0; 
    for (int i = n - 1; i >= 1; i--) { 
        int prev = suffix[i]; 
  
        // Character appears  
        // for the first time 
        if (seen[s[i] - 'a'] == 0) { 
            suffix[i - 1] += (prev + 1); 
        } 
        else
            suffix[i - 1] = prev; 
  
        // This character  
        // has now appeared 
        seen[s[i] - 'a'] = 1; 
    } 
      
    // Loop to calculate the number 
    // partition points in the string 
    for (int i = 0; i < n; i++) { 
        // Check whether number of  
        // distinct characters are equal 
        if (prefix[i] == suffix[i]) 
            answer++; 
    } 
    return answer; 
} 
  
// Driver Code 
int main() 
{ 
    string s = "ababa"; 
  
    cout << waysToSplit(s); 
    return 0; 
} 

Java

// Java implementation to count the  
// number of ways to partition the  
// string such that each partition  
// have same number of distinct  
// characters in the string  
class GFG { 
      
    // Function to count the number  
    // of ways to partition the string  
    // such that each partition have  
    // same number of distinct character  
    static int waysToSplit(String s)  
    {  
        int n = s.length();  
        int answer = 0;  
          
        // Prefix and suffix array for  
        // distinct character from  
        // start and end  
        int prefix[] = new int[n] ;  
        int suffix[] = new int[n];  
      
        // To check whether a character  
        // has appeared till ith index  
        int seen[] = new int[26]; 
      
        // Calculating prefix array  
        for (int i = 0; i < n; i++) {  
      
            int prev = (i - 1 >= 0 ?  
                prefix[i - 1] : 0);  
      
            // Character appears for  
            // the first time in string  
            if (seen[s.charAt(i) - 'a'] == 0) {  
                prefix[i] += (prev + 1);  
            }  
            else
                prefix[i] = prev;  
      
            // Character is visited  
            seen[s.charAt(i)- 'a'] = 1;  
        }  
          
        // Resetting seen for  
        // suffix calculation  
        for(int i = 0; i <26; i++) 
            seen[i] = 0; 
      
        // Calculating the suffix array  
        suffix[n - 1] = 0;  
        for (int i = n - 1; i >= 1; i--) {  
            int prev = suffix[i];  
      
            // Character appears  
            // for the first time  
            if (seen[s.charAt(i) - 'a'] == 0) {  
                suffix[i - 1] += (prev + 1);  
            }  
            else
                suffix[i - 1] = prev;  
      
            // This character  
            // has now appeared  
            seen[s.charAt(i)- 'a'] = 1;  
        }  
          
        // Loop to calculate the number  
        // partition points in the string  
        for (int i = 0; i < n; i++) { 
   
            // Check whether number of  
            // distinct characters are equal  
            if (prefix[i] == suffix[i])  
                answer++;  
        }  
        return answer;  
    }  
      
    // Driver Code  
    public static void main (String[] args) 
    {  
        String s = "ababa";  
      
        System.out.println(waysToSplit(s));  
    }  
  
} 
  
// This code is contributed by Yash_R 

Python3

# Python3 implementation to count the  
# number of ways to partition the  
# string such that each partition  
# have same number of distinct  
# characters in the string  
  
# Function to count the number  
# of ways to partition the string  
# such that each partition have  
# same number of distinct character  
def waysToSplit(s) :  
  
    n = len(s); 
    answer = 0;  
      
    # Prefix and suffix array for  
    # distinct character from  
    # start and end  
    prefix = [ 0 ]*n;  
    suffix = [0 ]*n;  
  
    # To check whether a character  
    # has appeared till ith index  
    seen = [ 0 ]*26;  
  
    # Calculating prefix array  
    for i in range(n) : 
  
        prev = prefix[i - 1] if (i - 1 >= 0 ) else  0;  
  
        # Character appears for  
        # the first time in string  
        if (seen[ord(s[i]) - ord('a')] == 0) : 
            prefix[i] += (prev + 1);  
          
        else : 
            prefix[i] = prev;  
  
        # Character is visited  
        seen[ord(s[i]) - ord('a')] = 1;  
      
    # Resetting seen for  
    # suffix calculation  
    seen = [0]*len(seen);  
  
    # Calculating the suffix array  
    suffix[n - 1] = 0;  
    for i in range(n - 1, 0, -1) : 
        prev = suffix[i];  
  
        # Character appears  
        # for the first time  
        if (seen[ord(s[i]) - ord('a')] == 0) : 
            suffix[i - 1] += (prev + 1);  
          
        else : 
            suffix[i - 1] = prev;  
  
        # This character  
        # has now appeared  
        seen[ord(s[i]) - ord('a')] = 1;  
      
    # Loop to calculate the number  
    # partition points in the string  
    for i in range(n) : 
        # Check whether number of  
        # distinct characters are equal  
        if (prefix[i] == suffix[i]) : 
            answer += 1;  
      
    return answer;  
  
  
# Driver Code  
if __name__ == "__main__" :  
  
    s = "ababa";  
  
    print(waysToSplit(s));  
  
# This code is contributed by Yash_R 

C#

// C# implementation to count the  
// number of ways to partition the  
// string such that each partition  
// have same number of distinct  
// characters in the string  
using System; 
  
public class GFG { 
      
    // Function to count the number  
    // of ways to partition the string  
    // such that each partition have  
    // same number of distinct character  
    static int waysToSplit(string s)  
    {  
        int n = s.Length;  
        int answer = 0;  
          
        // Prefix and suffix array for  
        // distinct character from  
        // start and end  
        int []prefix = new int[n] ;  
        int []suffix = new int[n];  
      
        // To check whether a character  
        // has appeared till ith index  
        int []seen = new int[26]; 
      
        // Calculating prefix array  
        for (int i = 0; i < n; i++) {  
      
            int prev = (i - 1 >= 0 ? prefix[i - 1] : 0);  
      
            // Character appears for  
            // the first time in string  
            if (seen[s[i] - 'a'] == 0) {  
                prefix[i] += (prev + 1);  
            }  
            else
                prefix[i] = prev;  
      
            // Character is visited  
            seen[s[i]- 'a'] = 1;  
        }  
          
        // Resetting seen for  
        // suffix calculation  
        for(int i = 0; i <26; i++) 
            seen[i] = 0; 
      
        // Calculating the suffix array  
        suffix[n - 1] = 0;  
        for (int i = n - 1; i >= 1; i--) {  
            int prev = suffix[i];  
      
            // Character appears  
            // for the first time  
            if (seen[s[i] - 'a'] == 0) {  
                suffix[i - 1] += (prev + 1);  
            }  
            else
                suffix[i - 1] = prev;  
      
            // This character  
            // has now appeared  
            seen[s[i]- 'a'] = 1;  
        }  
          
        // Loop to calculate the number  
        // partition points in the string  
        for (int i = 0; i < n; i++) { 
   
            // Check whether number of  
            // distinct characters are equal  
            if (prefix[i] == suffix[i])  
                answer++;  
        }  
        return answer;  
    }  
      
    // Driver Code  
    public static void Main (string[] args) 
    {  
        string s = "ababa";  
      
        Console.WriteLine(waysToSplit(s));  
    }  
  
} 
  
// This code is contributed by Yash_R 

Output:

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

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My Personal Notes

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Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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