How to Solve Longest Unequal Adjacent Groups Subsequence II

To solve this problem, use dynamic programming to track the longest subsequence ending at each word that meets the constraints. The primary condition involves the Hamming distance between selected strings, and the group numbers of adjacent strings in the subsequence must be unequal. This problem relies on efficiently managing these conditions using state transitions.

Problem Statement

Given two arrays, words and groups, both of length n, you need to find the longest subsequence of indices such that the strings at these indices meet two conditions: the Hamming distance between consecutive strings in the subsequence must be at least one, and the group numbers of adjacent strings in the subsequence must differ.

The Hamming distance is defined as the number of positions at which the corresponding characters of two strings are different. Your goal is to select the longest subsequence of indices such that these conditions are satisfied. A valid answer could involve multiple subsequences with the same length.

Examples

Example 1

Input: words = ["bab","dab","cab"], groups = [1,2,2]

Output: ["bab","cab"]

A subsequence that can be selected is [0,2] . So, a valid answer is [words[0],words[2]] = ["bab","cab"] . Another subsequence that can be selected is [0,1] . So, another valid answer is [words[0],words[1]] = ["bab","dab"] . It can be shown that the length of the longest subsequence of indices that satisfies the conditions is 2 .

Example 2

Input: words = ["a","b","c","d"], groups = [1,2,3,4]

Output: ["a","b","c","d"]

We can select the subsequence [0,1,2,3] . It satisfies both conditions. Hence, the answer is [words[0],words[1],words[2],words[3]] = ["a","b","c","d"] . It has the longest length among all subsequences of indices that satisfy the conditions. Hence, it is the only answer.

Constraints

• 1 <= n == words.length == groups.length <= 1000
• 1 <= words[i].length <= 10
• 1 <= groups[i] <= n
• words consists of distinct strings.
• words[i] consists of lowercase English letters.

Solution Approach

Dynamic Programming Approach

Define a dynamic programming array dp[] where dp[i] represents the longest subsequence ending at index i that satisfies the problem conditions. The solution involves iterating through each word and checking every prior word to determine if they can be part of the subsequence by comparing their Hamming distances and ensuring their group numbers differ.

State Transition

For each i, check all previous indices j < i. If groups[i] != groups[j] and the Hamming distance between words[i] and words[j] is greater than zero, then consider updating dp[i] = max(dp[i], dp[j] + 1).

Final Solution

The final solution will be the maximum value found in the dp[] array, as it represents the length of the longest subsequence that satisfies the given conditions.

Complexity Analysis

The time complexity is O(n^2L) because for each pair of words (i, j), we check their Hamming distance, which takes O(L) time where L is the average word length. The space complexity is O(n) for storing the dynamic programming array dp[].

What Interviewers Usually Probe

• Candidate understands dynamic programming and state transitions.
• Able to manage group constraints and Hamming distance in the context of subsequences.
• Efficient in applying the dynamic programming approach to solve sequence problems.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to check the group numbers for adjacent subsequence elements.
• Not handling edge cases with words that are identical or have zero Hamming distance.
• Not updating dp[i] correctly when a valid subsequence is found.

Follow-up variants

• Allow subsequences where group numbers can be equal.
• Change the minimum required Hamming distance between consecutive words.
• Modify the problem to find the shortest subsequence instead of the longest.

How GhostInterview Helps

• GhostInterview helps by guiding you through the dynamic programming approach step-by-step, ensuring you understand state transitions.
• We provide optimizations for handling group constraints and Hamming distance checks to avoid performance bottlenecks.
• By breaking down the problem with clear examples, GhostInterview ensures you can avoid common pitfalls and optimize your approach.

Topic Pages

• Array
• String
• Dynamic Programming

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FAQ

What is the primary approach to solving the Longest Unequal Adjacent Groups Subsequence II problem?

The primary approach involves dynamic programming, where you track the longest subsequence ending at each word while ensuring the Hamming distance and group constraints are met.

How does the Hamming distance affect the subsequences in this problem?

The Hamming distance ensures that consecutive words in the subsequence are different at least at one position. This is a crucial condition for forming valid subsequences.

What does the dynamic programming state transition involve?

For each word, check all previous words to see if they can form a valid subsequence based on group differences and the Hamming distance.

How do group constraints impact the subsequence?

Group constraints ensure that no two consecutive words in the subsequence belong to the same group, adding an extra layer of restriction to the subsequences.

What is the time complexity of solving the Longest Unequal Adjacent Groups Subsequence II?

The time complexity is O(n^2L), where n is the number of words, and L is the average length of the words.