#1593

Medium

auto_awesomeBacktracking search with pruning

LeetCode Problem Workspace

Split a String Into the Max Number of Unique Substrings

Maximize unique substrings in a string using backtracking with pruning and hash tables to track substrings.

Problem Statement

Given a string s, you need to split it into the maximum number of unique substrings. Each substring must be a contiguous part of the original string, and no two substrings can repeat.

You can split the string s into any list of non-empty substrings, but they must be distinct. Your task is to determine the maximum number of unique substrings that the string can be divided into, using backtracking search with pruning.

Examples

Example 1

Input: s = "ababccc"

Output: 5

One way to split maximally is ['a', 'b', 'ab', 'c', 'cc']. Splitting like ['a', 'b', 'a', 'b', 'c', 'cc'] is not valid as you have 'a' and 'b' multiple times.

Example 2

Input: s = "aba"

Output: 2

One way to split maximally is ['a', 'ba'].

Example 3

Input: s = "aa"

Output: 1

It is impossible to split the string any further.

Constraints

1 <= s.length <= 16
s contains only lower case English letters.

Solution Approach

Backtracking Search with Pruning

Start by trying to split the string into substrings using a backtracking approach. For each split, verify that the substrings are unique by checking against a set that stores the already used substrings. If a substring has been used before, prune that branch of the search tree.

Use Hash Set for Uniqueness Tracking

Utilize a hash set to track which substrings have been used. This allows for O(1) checks on whether a substring has already been included, enabling efficient pruning of duplicate substrings during the backtracking process.

Recursive Backtracking with State Management

Implement the backtracking algorithm recursively. For each character in the string, attempt to add a new substring to the set. If the substring is unique, proceed to explore the next possible substrings, tracking the number of unique substrings found. Return the maximum count after exploring all possibilities.

Complexity Analysis

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

The time complexity of this approach is O(n^2 * 2^n), where n is the length of the string. This arises from the need to explore all possible substring combinations and check for uniqueness. The space complexity is O(n) due to the hash set storing substrings and the recursion stack during the backtracking process.

What Interviewers Usually Probe

Look for understanding of recursive backtracking and pruning techniques.
Check the candidate's ability to optimize space and time complexity using a set for uniqueness tracking.
Ensure familiarity with managing recursive states and backtracking tree exploration.

Common Pitfalls or Variants

Common pitfalls

Forgetting to prune duplicates properly, leading to redundant computations.
Not managing the recursion state efficiently, causing unnecessary recomputation of results.
Misunderstanding the problem requirements, such as assuming substrings can repeat.

Follow-up variants

Limit the length of the substrings (e.g., substrings must be at least 2 characters long).
Extend the problem to work with strings containing uppercase letters or non-alphabetical characters.
Change the problem to return the substrings instead of their count, requiring a different approach.

FAQ

How do I optimize the backtracking search for unique substrings?

You optimize the search by using a hash set to track which substrings have been used, enabling O(1) checks and pruning of duplicate branches during backtracking.

What is the time complexity of this problem?

The time complexity is O(n^2 * 2^n), as we explore all possible substring combinations and check each for uniqueness.

What happens if I don't use pruning during backtracking?

Without pruning, the algorithm would explore duplicate substrings, leading to redundant computations and higher time complexity.

What is the space complexity of this problem?

The space complexity is O(n) due to the hash set storing substrings and the recursion stack during backtracking.

Can I extend this problem to work with non-alphabet characters?

Yes, the problem can be extended to work with strings containing any characters, as long as the backtracking approach and hash set management are adapted to handle them.

terminal

Solution

Solution 1: Backtracking + Pruning

We define a hash table $\textit{st}$ to store the currently split substrings. Then we use a depth-first search approach to try to split the string $\textit{s}$ into several unique substrings.

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class Solution:
    def maxUniqueSplit(self, s: str) -> int:
        def dfs(i: int):
            nonlocal ans
            if len(st) + len(s) - i <= ans:
                return
            if i >= len(s):
                ans = max(ans, len(st))
                return
            for j in range(i + 1, len(s) + 1):
                if s[i:j] not in st:
                    st.add(s[i:j])
                    dfs(j)
                    st.remove(s[i:j])

        ans = 0
        st = set()
        dfs(0)
        return ans

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