#2272

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Substring With Largest Variance

Find the largest variance possible in any substring of a given string using dynamic programming.

Problem Statement

Given a string of lowercase English letters, you need to find the largest variance that can be achieved in any of its substrings. The variance is the largest difference in the frequency of any two distinct characters present in the substring, which may or may not be the same character.

For example, the string 'aababbb' has different substrings with varying variances. By calculating the variances across all possible substrings, the solution determines that the largest variance is 3. This problem requires an efficient approach to check and maximize this difference using dynamic programming techniques.

Examples

Example 1

Input: s = "aababbb"

Output: 3

All possible variances along with their respective substrings are listed below:

Variance 0 for substrings "a", "aa", "ab", "abab", "aababb", "ba", "b", "bb", and "bbb".
Variance 1 for substrings "aab", "aba", "abb", "aabab", "ababb", "aababbb", and "bab".
Variance 2 for substrings "aaba", "ababbb", "abbb", and "babb".
Variance 3 for substring "babbb". Since the largest possible variance is 3, we return it.

Example 2

Input: s = "abcde"

Output: 0

No letter occurs more than once in s, so the variance of every substring is 0.

Constraints

1 <= s.length <= 104
s consists of lowercase English letters.

Solution Approach

State Transition Dynamic Programming

The core idea is to use dynamic programming where the state represents the frequency difference of characters. By iterating through possible substrings and updating the frequency differences based on transitions, we can track the largest variance. This approach efficiently reduces the problem complexity.

Two Character Consideration

Initially, consider only substrings with two distinct characters. This simplification reduces the complexity and helps in understanding how variance changes when adding or removing characters. Once the general solution for two characters is clear, extend it to handle all characters.

Iterative Frequency Update

For each new substring, the character frequencies are updated incrementally, avoiding recalculation from scratch. This optimization reduces unnecessary operations and speeds up the process of finding the largest variance.

Complexity Analysis

Metric	Value
Time	O(n \cdot k^2)
Space	O(1)

The time complexity of the solution is O(n * k^2), where n is the length of the string and k is the number of distinct characters considered. Space complexity is O(1) as we only maintain a few variables for state transitions without using additional space proportional to the input size.

What Interviewers Usually Probe

Look for understanding of dynamic programming and state transitions.
Check if the candidate can optimize by considering only two characters initially.
Assess whether the candidate uses iterative updates to minimize redundant calculations.

Common Pitfalls or Variants

Common pitfalls

Forgetting to optimize by limiting the number of distinct characters considered initially.
Overcomplicating the solution by recalculating character frequencies for every substring.
Neglecting the importance of updating states incrementally in dynamic programming.

Follow-up variants

What if the string contains more than two characters? Extend the solution to account for all characters while maintaining efficiency.
Consider solving the problem without using dynamic programming and assess the trade-offs in terms of time complexity.
Explore edge cases where the string is very short or where all characters are identical.

FAQ

What is the largest variance in a substring?

The largest variance is the greatest difference in the frequency of any two distinct characters in a substring. For example, in 'aababbb', the largest variance is 3, which occurs in 'babbb'.

How can dynamic programming help solve this problem?

Dynamic programming can be used to track the frequency differences between characters in substrings and update them efficiently, reducing unnecessary recalculations.

What is the primary pattern used in 'Substring With Largest Variance'?

The primary pattern used is state transition dynamic programming, where we track the differences in character frequencies as we move through the string.

How do you handle substrings with more than two characters?

Start by considering substrings with two distinct characters to simplify the problem, then extend the solution to handle all characters without losing efficiency.

What are the time and space complexities of this solution?

The time complexity is O(n * k^2), where n is the length of the string and k is the number of distinct characters. The space complexity is O(1), as no additional space proportional to the input is used.

terminal

Solution

Solution 1: Enumeration + Dynamic Programming

Since the character set only contains lowercase letters, we can consider enumerating the most frequent character $a$ and the least frequent character $b$. For a substring, the difference in the number of occurrences of these two characters is the variance of the substring.

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class Solution:
    def largestVariance(self, s: str) -> int:
        ans = 0
        for a, b in permutations(ascii_lowercase, 2):
            if a == b:
                continue
            f = [0, -inf]
            for c in s:
                if c == a:
                    f[0], f[1] = f[0] + 1, f[1] + 1
                elif c == b:
                    f[1] = max(f[1] - 1, f[0] - 1)
                    f[0] = 0
                if ans < f[1]:
                    ans = f[1]
        return ans

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