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Longest Unequal Adjacent Groups Subsequence II

Difficulty: Medium

Problem Description

You are given a string array words, and an array groups, both arrays having length n. You need to select the longest subsequence from an array of indices [0, 1, ..., n - 1], such that for the subsequence having length k, the following holds:

  • For adjacent indices in the subsequence, their corresponding groups are unequal, i.e., groups[i_j] != groups[i_j+1].
  • words[i_j] and words[i_j+1] are equal in length, and the Hamming distance between them is 1.

Return a string array containing the words corresponding to the indices in the selected subsequence. If there are multiple answers, return any of them.

Key Insights

  • The problem requires ensuring that adjacent words in the subsequence belong to different groups.
  • The Hamming distance of 1 implies that only one character differs between two words of the same length.
  • We can use a graph-like approach where each word is a node, and edges connect words that meet the criteria.

Space and Time Complexity

Time Complexity: O(n^2) - We need to compare each pair of words, leading to a quadratic complexity in the worst case. Space Complexity: O(n) - We may use additional space to store the subsequence and adjacency information.

Solution

To solve this problem, we can use a depth-first search (DFS) or backtracking approach:

  1. Create a graph where nodes are words indexed by their position, and edges represent valid transitions based on group inequality and Hamming distance of 1.
  2. Use a DFS to explore all possible subsequences, keeping track of the last selected word to ensure group inequality and Hamming distance constraints.
  3. Store and return the longest valid subsequence found.

Code Solutions

def hamming_distance(word1, word2):
    return sum(c1 != c2 for c1, c2 in zip(word1, word2))

def longest_unequal_adjacent_groups(words, groups):
    n = len(words)
    graph = [[] for _ in range(n)]
    
    # Build the graph
    for i in range(n):
        for j in range(n):
            if i != j and groups[i] != groups[j] and len(words[i]) == len(words[j]) and hamming_distance(words[i], words[j]) == 1:
                graph[i].append(j)

    longest = []
    
    def dfs(index, path):
        nonlocal longest
        if len(path) > len(longest):
            longest = path[:]
        
        for neighbor in graph[index]:
            dfs(neighbor, path + [words[neighbor]])

    for i in range(n):
        dfs(i, [words[i]])

    return longest

# Example usage
words = ["bab", "dab", "cab"]
groups = [1, 2, 2]
print(longest_unequal_adjacent_groups(words, groups))
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