Vowels of All Substrings - Problem
Imagine you're a linguist analyzing the vowel density of every possible phrase within a word. Your task is to calculate the total sum of vowels across all possible substrings of a given word.
A substring is any contiguous sequence of characters within the string. For example, the word "abc" has substrings: "a", "b", "c", "ab", "bc", and "abc".
The vowels we're counting are: 'a', 'e', 'i', 'o', 'u' (case-sensitive).
Goal: Return the sum of vowel counts in every substring of the input word.
Note: Due to large constraints, use 64-bit integers to avoid overflow.
Input & Output
example_1.py — Basic Case
$
Input:
word = "aba"
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Output:
6
💡 Note:
The substrings are "a", "b", "a", "ab", "ba", "aba". Vowel counts: 1+0+1+1+1+2 = 6
example_2.py — All Vowels
$
Input:
word = "abc"
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Output:
3
💡 Note:
The substrings are "a", "b", "c", "ab", "bc", "abc". Vowel counts: 1+0+0+1+0+1 = 3
example_3.py — Single Character
$
Input:
word = "a"
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Output:
1
💡 Note:
Only one substring "a" which contains 1 vowel
Visualization
Tap to expand
Understanding the Visualization
1
Position Analysis
For each vowel, determine how many substrings can include it
2
Prefix Count
Count how many starting positions come before or at the vowel
3
Suffix Count
Count how many ending positions come at or after the vowel
4
Multiply
Total substrings containing the vowel = prefix × suffix
Key Takeaway
🎯 Key Insight: Instead of generating all substrings, we calculate how many times each vowel 'echoes' across all possible substrings using the mathematical formula (i+1) × (n-i) for position i.
Time & Space Complexity
Time Complexity
O(n)
Single pass through the string, constant time calculation per character
✓ Linear Growth
Space Complexity
O(1)
Only using a few variables, no extra data structures needed
✓ Linear Space
Constraints
- 1 ≤ word.length ≤ 105
- word consists of lowercase English letters only
- Answer may exceed 32-bit integer range - use 64-bit integers
💡
Explanation
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