Sum of Beauty of All Substrings

Problem Statement❯

Given a string s, your task is to find the sum of beauty for all its substrings.

The beauty of a substring is defined as the difference between the highest frequency character and the lowest frequency character (ignoring characters that do not appear).

For example, in substring "aabcb", the frequencies are: a → 2, b → 2, c → 1. So the beauty is 2 - 1 = 1.

You need to compute this value for every possible substring of s and return the total sum of all these beauties.

Examples❯

Input String	Output	Description
"aabcb"	5	Each substring is checked for character frequency difference. Total beauty = 5
"abcab"	5	Beauty is computed for substrings like "abc", "bca", etc.
"aaaa"	0	All characters are the same in every substring, so beauty = 0
"ab"	0	Each substring has unique characters or equal frequencies
"a"	0	Only one character. No frequency difference
""	0	Empty string has no substrings. Answer is 0

Solution❯

To solve this problem, we need to understand what beauty means for a substring. The beauty is the difference between how often the most frequent character appears and how often the least frequent (but present) character appears in that substring.

We consider every possible substring of the input string. For each substring, we count how many times each character appears. We ignore characters that have a count of 0, and then find the difference between the highest and lowest frequency among the characters that do appear.

Let’s understand this through different types of input:

Case 1 – Normal string (e.g., "aabcb"):
There are substrings where characters appear with different frequencies. For example, in "aab", 'a' appears twice, 'b' once → beauty = 1. We do this for every substring and keep adding up their beauty values.
Case 2 – All same characters (e.g., "aaaa"):
Every substring has all characters the same, so the highest and lowest frequencies are equal. Beauty = 0 for all of them.
Case 3 – Short strings like "ab" or "a":
In "ab", both characters appear once, so no frequency difference → beauty = 0. In "a", there's only one character → beauty = 0.
Case 4 – Empty string ""
There are no substrings to evaluate, so the total beauty is 0.

This problem involves many substrings, but we can optimize the solution by using a frequency array (of size 26 for each character) and reusing it while expanding the substring window. In each iteration, we quickly find the max and min (non-zero) frequencies and calculate the beauty.

By combining smart counting with substring expansion, we can solve this efficiently even for longer strings. The time complexity is roughly O(n²) but works well with optimization.

Algorithm Steps❯

Initialize a variable totalBeauty = 0.
Loop over each start index i of the string.
Initialize a frequency array count[26] for characters 'a' to 'z'.
For each end index j ≥ i, do:
→ Increment the count of character s[j].
→ Find maxFreq and minFreq from the count array (ignoring 0s).
→ Add (maxFreq - minFreq) to totalBeauty.
Return totalBeauty.

Code

Python

Java

JavaScript

C++

Kotlin

Swift

Php

def beautySum(s):
    total_beauty = 0
    n = len(s)

    for i in range(n):
        count = [0] * 26
        for j in range(i, n):
            idx = ord(s[j]) - ord('a')
            count[idx] += 1

            freq_list = [c for c in count if c > 0]
            max_freq = max(freq_list)
            min_freq = min(freq_list)

            total_beauty += (max_freq - min_freq)

    return total_beauty

# Example
s = "aabcb"
print("Sum of Beauty:", beautySum(s))

Time Complexity❯

Case	Time Complexity	Explanation
Best Case	O(n^2)	We iterate through all possible substrings (n^2 combinations), and each frequency check takes constant time.
Average Case	O(n^2)	The main loop processes all substrings, and frequency difference is computed in O(26) = O(1).
Worst Case	O(n^2)	Even if all characters are distinct, the double loop covers all substrings of the input string.

Space Complexity❯

O(1)

Explanation: Only a fixed-size array of 26 elements is used for frequency counting.

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