#828

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Count Unique Characters of All Substrings of a Given String

Calculate the sum of unique characters in all substrings of a string using state transition dynamic programming.

Problem Statement

Given a string s, you are tasked with finding the sum of unique characters in all of its substrings. A substring is defined as a contiguous segment of the string. To solve this problem, you must compute the sum of the number of unique characters in each possible substring of s.

For example, if s = 'ABC', the substrings are 'A', 'B', 'C', 'AB', 'BC', and 'ABC'. Each substring contains only unique characters, and their sum is calculated as the total length of these substrings. Your goal is to compute this sum for any given string, considering repeated substrings where they occur.

Examples

Example 1

Input: s = "ABC"

Output: 10

All possible substrings are: "A","B","C","AB","BC" and "ABC". Every substring is composed with only unique letters. Sum of lengths of all substring is 1 + 1 + 1 + 2 + 2 + 3 = 10

Example 2

Input: s = "ABA"

Output: 8

The same as example 1, except countUniqueChars("ABA") = 1.

Example 3

Input: s = "LEETCODE"

Output: 92

Example details omitted.

Constraints

1 <= s.length <= 105
s consists of uppercase English letters only.

Solution Approach

Dynamic Programming with Hash Map

To efficiently solve this problem, maintain a dynamic programming table where each entry represents the sum of unique characters for substrings ending at each index. Use a hash map to track the last occurrence of each character, allowing us to compute the unique characters in each substring as we process the string.

State Transitions for Optimization

By considering the transition of substring states as we move through the string, we avoid redundant calculations. Specifically, when a character repeats, adjust the starting point of the substring to exclude previous occurrences of the character, ensuring only unique characters are counted for each substring.

Efficient Calculation with Sliding Window

A sliding window approach can be used to maintain the current valid substring of unique characters. This helps to efficiently calculate the unique character count for each substring while avoiding recalculating for overlapping parts of substrings.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is typically O(n) where n is the length of the string, as each character is processed once. The space complexity is O(n) due to the hash map used to store character positions, allowing for efficient state transitions and substring calculations.

What Interviewers Usually Probe

Look for understanding of dynamic programming and state transitions in string problems.
Assess how the candidate uses hash maps or sliding windows to manage repeated characters.
Check if the candidate can optimize the solution to avoid recalculating substrings unnecessarily.

Common Pitfalls or Variants

Common pitfalls

Not accounting for repeated substrings and recalculating unique characters multiple times.
Failing to efficiently track character occurrences with a hash map, leading to higher time complexity.
Overcomplicating the solution by neglecting the sliding window or dynamic programming optimization.

Follow-up variants

Consider extending the problem to handle strings with lowercase letters or special characters.
Modify the problem to count the total number of unique characters across substrings that have specific lengths.
Explore how the solution can be adapted to substrings that need to be palindromes or contain a specific number of characters.

FAQ

How do I apply state transition dynamic programming to solve this problem?

State transition dynamic programming involves storing the results of previously computed states and using them to transition to the next state. In this problem, the states represent the sum of unique characters in substrings, and the transition is managed by tracking character occurrences.

What is the most efficient way to handle repeated substrings?

The most efficient approach is to use a hash map to track the last occurrence of each character and adjust the starting index of the substring accordingly to ensure only unique characters are counted.

How can I optimize my solution for larger input sizes?

To optimize the solution, use dynamic programming to avoid redundant calculations and a sliding window to efficiently track the valid substrings. These techniques help reduce both time and space complexity.

What is the time complexity of this solution?

The time complexity is O(n), where n is the length of the string. This is because each character is processed once with efficient state transitions using dynamic programming and a hash map.

What if the string contains repeated characters or is large?

For strings with repeated characters or large input sizes, it's essential to optimize using dynamic programming and sliding windows. This ensures that we only compute the unique characters once for each substring.

terminal

Solution

Solution 1: Calculate the Contribution of Each Character

For each character $c_i$ in the string $s$, when it appears only once in a substring, it contributes to the count of unique characters in that substring.

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class Solution:
    def uniqueLetterString(self, s: str) -> int:
        d = defaultdict(list)
        for i, c in enumerate(s):
            d[c].append(i)
        ans = 0
        for v in d.values():
            v = [-1] + v + [len(s)]
            for i in range(1, len(v) - 1):
                ans += (v[i] - v[i - 1]) * (v[i + 1] - v[i])
        return ans

Continue Practicing

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