Program to find sum of beauty of all substrings in Python

Given a string, we need to find the sum of beauty of all its substrings. The beauty of a string is the difference between the frequencies of the most frequent and least frequent characters.

For example, if the string is "abaacc", the frequency of 'a' is 3 and 'b' is 1, so beauty = 3 - 1 = 2.

Problem Example

If the input string is "xxyzy", the substrings with non-zero beauty are:

• "xxy" ? beauty = 2 - 1 = 1
• "xxyz" ? beauty = 2 - 1 = 1
• "xxyzy" ? beauty = 2 - 1 = 1
• "xyzy" ? beauty = 2 - 1 = 1
• "yzy" ? beauty = 2 - 1 = 1

Total sum = 1 + 1 + 1 + 1 + 1 = 5

Algorithm

We use a nested loop approach to generate all possible substrings and calculate their beauty:

1. Initialize result as 0
2. For each starting position i, iterate through all ending positions j
3. Extract substring from i to j and count character frequencies
4. Calculate beauty as max frequency - min frequency
5. Add beauty to the result

Implementation

from collections import Counter

def solve(s):
    res = 0
    for i in range(len(s)):
        for j in range(i + 2, len(s) + 1):
            c = Counter(s[i:j])
            v = list(c.values())
            if v:  # Check if substring is not empty
                res += (max(v) - min(v))
    return res

s = "xxyzy"
print(solve(s))

5

Step-by-Step Breakdown

Let's trace through the algorithm with "xxyzy":

from collections import Counter

def solve_with_trace(s):
    res = 0
    print(f"Analyzing string: '{s}'")
    print("-" * 40)
    
    for i in range(len(s)):
        for j in range(i + 2, len(s) + 1):
            substring = s[i:j]
            c = Counter(substring)
            v = list(c.values())
            beauty = max(v) - min(v) if v else 0
            
            if beauty > 0:
                print(f"Substring: '{substring}' ? Beauty: {beauty}")
                res += beauty
    
    print("-" * 40)
    print(f"Total beauty sum: {res}")
    return res

s = "xxyzy"
solve_with_trace(s)

Analyzing string: 'xxyzy'
----------------------------------------
Substring: 'xxy' ? Beauty: 1
Substring: 'xxyz' ? Beauty: 1
Substring: 'xxyzy' ? Beauty: 1
Substring: 'xyzy' ? Beauty: 1
Substring: 'yzy' ? Beauty: 1
----------------------------------------
Total beauty sum: 5

Time Complexity

The algorithm has O(n³) time complexity:

• Two nested loops: O(n²) to generate all substrings
• Counter operation: O(n) for each substring
• Overall: O(n³)

Conclusion

This solution efficiently calculates the sum of beauty for all substrings by using nested loops and the Counter class. The beauty is determined by the difference between maximum and minimum character frequencies in each substring.