Maximum Length Of A Concatenated String With Unique Characters [Python]

In this blog post, we're going to tackle the Maximum Length Of A Concatenated String With Unique Characters problem from LeetCode.

This problem falls under the category of backtracking and is rated as a medium difficulty challenge.

We'll explore the problem, outline the constraints, provide a brute-force approach, and then dive into an efficient solution along with its Python code.

Problem Overview

Question Link: Maximum Length of a Concatenated String With Unique Characters

Problem Description

You are given an array of strings, arr.

A string s is formed by the concatenation of a subsequence of arr that has unique characters.

Your task is to return the maximum possible length of s.

A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.

Example 1:

Input: arr = ["un","iq","ue"]
Output: 4
Explanation: All the valid concatenations are:
– ""
– "un"
– "iq"
– "ue"
– "uniq" ("un" + "iq")
– "ique" ("iq" + "ue")
Maximum length is 4.

Example 2:

Input: arr = ["cha","r","act","ers"]
Output: 6
Explanation: Possible longest valid concatenations are "chaers" ("cha" + "ers") and "acters" ("act" + "ers").

Example 3:

Input: arr = ["abcdefghijklmnopqrstuvwxyz"]
Output: 26
Explanation: The only string in arr has all 26 characters.

Constraints

1 <= arr.length <= 16
1 <= arr[i].length <= 26
arr[i] contains only lowercase English letters.

Now, let's take a closer look at the problem's constraints and discuss potential solutions.

Understanding the Constraints

Before diving into the solutions, it's crucial to understand the constraints provided in the problem.

This understanding will guide us in selecting the right approach to solve the problem.

1 <= arr.length <= 16: This constraint tells us that the array arr can contain a maximum of 16 strings.

We should keep this limit in mind when designing our solution, as it impacts time complexity.

1 <= arr[i].length <= 26: Each string in the array can have a length between 1 and 26 characters.

This constraint informs us about the potential size of the strings in arr.

arr[i] contains only lowercase English letters: All characters in the strings are lowercase English letters.

This constraint simplifies the problem, as we don't need to consider other character sets.

Now that we have a clear understanding of the problem and its constraints, let's explore the approaches to solving it.

Brute-Force Approach

When dealing with backtracking problems, the brute-force approach often involves considering all possible combinations or sequences.

In this case, we can iterate through all subsequences of the input array arr and check if each subsequence forms a valid concatenation with unique characters.

We'll keep track of the maximum length found during this process.

Pseudocode for Brute-Force Approach

Here's a high-level pseudocode for the brute-force approach:

Initialize a variable max_length to 0.
Iterate through all possible subsequences of arr:
- For each subsequence, check if it forms a valid concatenation with unique characters:
  - If it does, update max_length if the length of the concatenation is greater than the current max_length.
Return max_length as the result.

While the brute-force approach is straightforward, it's not the most efficient solution, especially when the array contains a large number of strings.

The time complexity of this approach can be exponential, with a worst-case time complexity of O(2^n), where n is the number of strings in arr.

Therefore, we should consider a more efficient approach.

Efficient Approach

To optimize the solution, we can use a backtracking approach.

We'll recursively explore all possible combinations of strings in the array while maintaining a character set to track the unique characters.

This approach ensures that we only consider valid concatenations with unique characters, and it avoids unnecessary calculations.

Efficient Python Code Solution

Now, let's dive into the Python code that implements this efficient approach:

View on GitHub ➹

from collections import Counter

class Solution:
    def maxLength(self, arr):
        # Initialize a character set to track unique characters
        charSet = set()

        def overlap(charSet, s):
            # Check if there are overlapping characters
            c = Counter(charSet) + Counter(s)
            return max(c.values()) &gt; 1

        def backtrack(i):
            if i == len(arr):
                return len(charSet)
            res = 0
            if not overlap(charSet, arr[i]):
                for c in arr[i]:
                    charSet.add(c)
                res = backtrack(i + 1)
                for c in arr[i]:
                    charSet.remove(c)
            return max(res, backtrack(i + 1))  # Don&#39;t concatenate arr[i]

        return backtrack(0)

In this Python code solution, we use a class called Solution, as it is common for LeetCode problems.

The maxLength function takes the input array arr and returns the maximum possible length of a valid concatenation.

Let's break down the key components of the code:

We start by initializing a character set charSet to keep track of unique characters.

This set will be used throughout the backtracking process.

We define a helper function overlap to check if there are overlapping characters between the current character set and a new string s.

This function ensures that we only include strings that do not introduce duplicate characters.

The core logic of the backtracking solution is implemented in the backtrack function.

This function takes an index i as a parameter, which represents the current position in the array arr.

If i reaches the length of the array arr, it means we have explored all possible combinations, and we return the length of the current character set.

This represents the length of the valid concatenation.

Within the backtrack function, we have two choices:
- If the current string at index i can be included without introducing duplicate characters (checked using the overlap function), we add its characters to the character set, call the backtrack function recursively for the next index (i + 1), and then remove the added characters.

This allows us to explore the possibility of including the current string.
– We also call the backtrack function for the next index without including the current string, effectively skipping it.

We return the maximum of the results obtained from both choices.

This ensures that we consider all possible combinations and select the one with the maximum length.

The use of backtracking allows us to

explore all valid concatenations efficiently while avoiding unnecessary calculations.

This approach significantly reduces the time complexity compared to the brute-force approach.

Time and Space Complexity

Let's analyze the time and space complexity of the efficient backtracking solution.

Time Complexity

The time complexity of the backtracking solution is determined by the number of recursive calls made.

In the worst case, the algorithm explores all possible combinations of strings in arr.

Since there are 2^n possible subsequences of arr, where n is the length of arr, the worst-case time complexity is O(2^n).

Space Complexity

The space complexity is determined by the space used for the character set and the recursive call stack.

The character set charSet can store at most 26 lowercase letters, which is constant.

The space used by the recursive call stack depends on the depth of recursion, which is at most n (the length of arr).

Therefore, the space complexity is O(n).

Reasoning Behind Our Approach

The efficient backtracking approach takes advantage of the properties of valid concatenations with unique characters.

By carefully considering each string in the array and tracking unique characters, we can ensure that the resulting concatenation is valid.

The backtracking process explores all possible combinations while avoiding duplicates.

This approach is designed to handle the constraints of the problem efficiently and provides a solution with a reasonable runtime.

In summary, the Maximum Length Of A Concatenated String With Unique Characters problem can be solved using a backtracking approach that explores all valid concatenations while maintaining a character set.

This approach allows us to efficiently find the maximum length of a valid concatenation.

While the problem may have exponential time complexity in the worst case, the efficient implementation significantly reduces the number of calculations, making it a suitable solution for the given constraints.

Conclusion

In this blog post, we've tackled the Maximum Length Of A Concatenated String With Unique Characters problem from LeetCode.

We've explored the problem, considered the constraints, and provided both a brute-force and an efficient backtracking approach along with Python code for the latter.

The efficient approach ensures that we find the maximum length of a valid concatenation with unique characters while efficiently handling the constraints of the problem.

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