Substring With Largest Variance - Practice Coding Problems

Substring With Largest Variance - Problem

The Variance Challenge: Finding Maximum Character Frequency Differences

You're given a string containing only lowercase English letters, and your mission is to find the largest possible variance among all its substrings.

What is variance? The variance of a string is defined as the largest difference between the number of occurrences of any 2 characters present in the string. These two characters can be the same or different.

For example, in the string "aaba", character 'a' appears 3 times and 'b' appears 1 time, so the variance is 3 - 1 = 2.

Your Goal: Among all possible contiguous substrings of the input string, find the one with the maximum variance and return that maximum variance value.

Input: A string s consisting of lowercase English letters only
Output: An integer representing the largest variance possible among all substrings

Input & Output

example_1.py — Basic case

$ Input: s = "aababbb"

› Output: 3

💡 Note: The substring "abbb" has the largest variance where 'b' appears 3 times and 'a' appears 1 time, giving variance = 3 - 1 = 2. However, the substring "aababbb" has 'b' appearing 4 times and 'a' appearing 3 times, giving variance = 4 - 3 = 1. Actually, "bbb" has 'b' appearing 3 times with no other characters, but we need at least 2 different characters. The optimal is "abbb" or "bbba" with variance 2. Wait, let me recalculate: "ababbb" has a=2, b=4, variance=2. "aababbb" has a=3, b=4, variance=1. The answer should be 3 from substring "bbb" if we consider implicit characters, but since we need 2 different chars, it's "abbb" with variance 2. Actually, "aababbb" gives us variance of 4-3=1, "ababbb" gives 4-2=2, "babbb" gives 4-1=3. So the answer is 3.

example_2.py — Single character

$ Input: s = "abcde"

› Output: 0

💡 Note: Every character appears exactly once, so in any substring containing at least 2 different characters, the maximum frequency is 1 and minimum frequency is 1, giving variance = 1 - 1 = 0.

example_3.py — Two characters

$ Input: s = "aaaa"

› Output: 0

💡 Note: The string contains only one unique character 'a', so we cannot form any substring with at least 2 different characters. Therefore, the variance is 0.

Constraints

1 ≤ s.length ≤ 10⁴
s consists of lowercase English letters only
The variance is calculated only for substrings with at least 2 different characters

Visualization

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Understanding the Visualization

📈 Choose Stock Pair

Select two stocks (characters) to compare - one as 'winner' (+1) and one as 'loser' (-1)

📊 Transform Timeline

Convert the timeline: winner stock days = +1, loser stock days = -1, others = 0

🔍 Find Best Period

Use Kadane's algorithm to find the time period with maximum net gain

🏆 Compare All Pairs

Repeat for all possible stock pairs and take the maximum variance

Key Takeaway

🎯 Key Insight: By transforming the variance problem into a maximum subarray problem using Kadane's algorithm, we can efficiently find the optimal substring in linear time for each character pair!

Asked in

G Google 45 f Meta 32 a Amazon 28 ⊞ Microsoft 15

The optimal solution uses a brilliant insight: transform the variance problem into Kadane's maximum subarray algorithm. For each pair of characters (major, minor), treat major as +1 and minor as -1, then find the maximum subarray sum while ensuring both characters appear. This gives us O(n) time complexity with the power of dynamic programming.

Common Approaches

Approach	Time	Space	Notes
✓ Brute Force (Check All Substrings)	O(n³)	O(1)	Generate all possible substrings and calculate variance for each
Kadane's Algorithm Variation (Optimal)	O(26² × n) = O(n)	O(1)	Transform the problem into maximum subarray using Kadane's algorithm for each character pair

Brute Force (Check All Substrings) — Algorithm Steps

Generate all possible substrings (O(n²) combinations)
For each substring, count frequency of each character
Find the maximum and minimum frequencies in the substring
Calculate variance as max_freq - min_freq
Keep track of the maximum variance seen so far

Visualization

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Step-by-Step Walkthrough

Generate Substrings

Start with substring of length 1, then 2, etc.

Count Frequencies

For each substring, count how many times each character appears

Calculate Variance

Find max frequency - min frequency for current substring

Update Maximum

Keep track of the largest variance seen so far

Code -

solution.c — C

#include <stdio.h>
#include <string.h>
#include <stdlib.h>

int largestVariance(char* s) {
    int n = strlen(s);
    int maxVariance = 0;
    
    // Try all possible substrings
    for (int i = 0; i < n; i++) {
        for (int j = i; j < n; j++) {
            // Count frequency of each character (a-z)
            int freq[26] = {0};
            int uniqueChars = 0;
            
            for (int k = i; k <= j; k++) {
                if (freq[s[k] - 'a'] == 0) {
                    uniqueChars++;
                }
                freq[s[k] - 'a']++;
            }
            
            // Find max and min frequencies
            if (uniqueChars >= 2) {
                int maxFreq = 0, minFreq = 1000000;
                for (int c = 0; c < 26; c++) {
                    if (freq[c] > 0) {
                        if (freq[c] > maxFreq) maxFreq = freq[c];
                        if (freq[c] < minFreq) minFreq = freq[c];
                    }
                }
                if (maxFreq - minFreq > maxVariance) {
                    maxVariance = maxFreq - minFreq;
                }
            }
        }
    }
    
    return maxVariance;
}

int main() {
    char s[1000];
    fgets(s, sizeof(s), stdin);
    // Remove newline and quotes
    s[strcspn(s, "\n")] = 0;
    if (s[0] == '"') {
        memmove(s, s+1, strlen(s));
        s[strlen(s)-1] = 0;
    }
    printf("%d\n", largestVariance(s));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n³)

O(n²) to generate all substrings, O(n) to count frequencies for each substring

⚠ Quadratic Growth

Space Complexity

O(1)

Only using constant extra space for frequency counting (at most 26 characters)

✓ Linear Space

38.4K Views

Medium-High Frequency

~25 min Avg. Time

1.2K Likes

Ln 1, Col 1

Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force (Check All Substrings) — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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