LeetCode #1781 — MEDIUM

Sum of Beauty of All Substrings

Move from brute-force thinking to an efficient approach using hash map strategy.

Solve on LeetCode
The Problem

Problem Statement

The beauty of a string is the difference in frequencies between the most frequent and least frequent characters.

  • For example, the beauty of "abaacc" is 3 - 1 = 2.

Given a string s, return the sum of beauty of all of its substrings.

Example 1:

Input: s = "aabcb"
Output: 5
Explanation: The substrings with non-zero beauty are ["aab","aabc","aabcb","abcb","bcb"], each with beauty equal to 1.

Example 2:

Input: s = "aabcbaa"
Output: 17

Constraints:

  • 1 <= s.length <= 500
  • s consists of only lowercase English letters.

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: The beauty of a string is the difference in frequencies between the most frequent and least frequent characters. For example, the beauty of "abaacc" is 3 - 1 = 2. Given a string s, return the sum of beauty of all of its substrings.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Hash Map

Example 1

"aabcb"

Example 2

"aabcbaa"

Related Problems

  • Substrings That Begin and End With the Same Letter (substrings-that-begin-and-end-with-the-same-letter)
Step 02

Core Insight

What unlocks the optimal approach

  • Maintain a prefix sum for the frequencies of characters.
  • You can iterate over all substring then iterate over the alphabet and find which character appears most and which appears least using the prefix sum array
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. Continue until all input is consumed.
Use the first example testcase as your mental trace to verify each transition.
Step 04

Edge Cases

Minimum Input
Single element / shortest valid input
Validate boundary behavior before entering the main loop or recursion.
Duplicates & Repeats
Repeated values / repeated states
Decide whether duplicates should be merged, skipped, or counted explicitly.
Extreme Constraints
Upper-end input sizes
Re-check complexity target against constraints to avoid time-limit issues.
Invalid / Corner Shape
Empty collections, zeros, or disconnected structures
Handle special-case structure before the core algorithm path.
Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #1781: Sum of Beauty of All Substrings
class Solution {
    public int beautySum(String s) {
        int ans = 0;
        int n = s.length();
        for (int i = 0; i < n; ++i) {
            int[] cnt = new int[26];
            for (int j = i; j < n; ++j) {
                ++cnt[s.charAt(j) - 'a'];
                int mi = 1000, mx = 0;
                for (int v : cnt) {
                    if (v > 0) {
                        mi = Math.min(mi, v);
                        mx = Math.max(mx, v);
                    }
                }
                ans += mx - mi;
            }
        }
        return ans;
    }
}
Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(n^2 × C)
Space
O(C)

Approach Breakdown

BRUTE FORCE
O(n²) time
O(1) space

For each element, scan the rest of the array looking for a match. Two nested loops give n × (n−1)/2 comparisons = O(n²). No extra space since we only use loop indices.

HASH MAP
O(n) time
O(n) space

One pass through the input, performing O(1) hash map lookups and insertions at each step. The hash map may store up to n entries in the worst case. This is the classic space-for-time tradeoff: O(n) extra memory eliminates an inner loop.

Shortcut: Need to check “have I seen X before?” → hash map → O(n) time, O(n) space.
Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.