Vowels of All Substrings

Vowels of All Substrings - Problem

Given a string word, return the sum of the number of vowels ('a', 'e', 'i', 'o', and 'u') in every substring of word.

A substring is a contiguous (non-empty) sequence of characters within a string.

Note: Due to the large constraints, the answer may not fit in a signed 32-bit integer. Please be careful during the calculations.

Input & Output

Example 1 — Basic Case

$ Input: word = "aba"

› Output: 6

💡 Note: Substrings are: "a" (1 vowel), "ab" (1 vowel), "aba" (2 vowels), "b" (0 vowels), "ba" (1 vowel), "a" (1 vowel). Total: 1+1+2+0+1+1 = 6

Example 2 — No Vowels

$ Input: word = "abc"

› Output: 3

💡 Note: Substrings are: "a" (1), "ab" (1), "abc" (1), "b" (0), "bc" (0), "c" (0). Total: 1+1+1+0+0+0 = 3

Example 3 — All Vowels

$ Input: word = "aei"

› Output: 10

💡 Note: Each position contributes: a at pos 0: (0+1)*(3-0)=3, e at pos 1: (1+1)*(3-1)=4, i at pos 2: (2+1)*(3-2)=3. Total: 3+4+3 = 10

Constraints

1 ≤ word.length ≤ 10⁵
word consists of lowercase English letters only

Visualization

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Asked in

G Google 35 f Facebook 28 a Amazon 22 M Microsoft 18

The key insight is that instead of generating all substrings, we can calculate how many times each vowel contributes to the final sum. Each character at position i appears in exactly (i+1) × (n-i) substrings. Best approach uses mathematical formula with Time: O(n), Space: O(1).

Common Approaches

✓ Brute Force

⏱️ Time: O(n³) Space: O(1)

Generate every possible substring of the input word, then count the vowels in each substring and sum them all up. This involves nested loops to create all possible substrings.

Mathematical Formula

⏱️ Time: O(n) Space: O(1)

Instead of generating substrings, calculate how many times each vowel appears across all possible substrings. A character at position i appears in (i+1) * (n-i) substrings, where n is the string length.

Brute Force — Algorithm Steps

Generate all possible substrings using nested loops
For each substring, count the number of vowels
Sum up all vowel counts from all substrings

Visualization

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Step-by-Step Walkthrough

Generate Substrings

Create all possible substrings using nested loops

Count Vowels

For each substring, count vowels individually

Sum Results

Add all vowel counts together

Code -

solution.c — C

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

bool isVowel(char c) {
    return c == 'a' || c == 'e' || c == 'i' || c == 'o' || c == 'u';
}

long long solution(char* word) {
    long long total = 0;
    int n = strlen(word);
    
    for (int i = 0; i < n; i++) {
        for (int j = i; j < n; j++) {
            for (int k = i; k <= j; k++) {
                if (isVowel(word[k])) {
                    total++;
                }
            }
        }
    }
    
    return total;
}

int main() {
    char word[100001];
    fgets(word, sizeof(word), stdin);
    word[strcspn(word, "\n")] = 0;
    
    long long result = solution(word);
    printf("%lld\n", result);
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n³)

Two nested loops to generate n² substrings, each taking O(n) time to count vowels

✓ Linear Growth

Space Complexity

O(1)

Only using variables to track counts, no extra data structures

✓ Linear Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Brute Force — Algorithm Steps

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Code -

Time & Space Complexity

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