Minimum Number of String Groups Through Transformations - Solution & Explanation

Problem Overview: You need to partition strings into the minimum number of groups where strings inside the same group can be connected through valid transformations. The core challenge is efficiently checking whether two strings belong to the same connected component after one or more transformation operations.

Approach 1: Brute Force Pair Comparison with DFS/BFS (O(n² * m) time, O(n²) space)

Build a graph where each node represents a string. Iterate through every pair of strings and check whether one can transform into the other in O(m) time, where m is the string length. If two strings satisfy the transformation rule, add an undirected edge. After constructing the graph, run DFS or BFS to count connected components. This approach is straightforward and useful when constraints are small because the graph construction logic mirrors the problem statement directly.

This method usually relies on graph traversal techniques and adjacency lists. The downside is the quadratic pair comparison cost. For large input sizes, repeated transformation checks become the bottleneck.

Approach 2: Union-Find / DSU Optimization (O(n² * m) time, O(n) space)

Instead of explicitly storing the graph, use a Disjoint Set Union structure to merge indices whenever two strings are transformable. Iterate through all pairs, perform the same transformation validation, and call union(u, v) when the condition matches. At the end, count distinct parents to get the number of groups.

The key insight is that the problem only needs the number of connected components, not the actual traversal order. DSU removes adjacency storage overhead and simplifies connectivity management with path compression and union by rank. This is the standard interview solution because it combines clean implementation with strong amortized performance characteristics. Problems involving grouping, merging, or equivalence classes often use Union-Find.

Approach 3: Hash-Based Neighbor Generation (Near O(n * m²) average time, O(n * m) space)

If transformations are based on limited edits such as swaps, replacements, or character modifications, you can generate canonical intermediate states and store them in a hash map. Instead of comparing every pair directly, generate all possible transformation signatures for a string and look up matching candidates in constant average time.

This reduces unnecessary comparisons and performs well when strings are short but the number of strings is large. The approach typically combines hashing with DSU merges. You trade additional memory for faster candidate discovery. This optimization is common in hash table and string-grouping problems.

Recommended for interviews: Start with the brute force graph interpretation to show you understand the connectivity model. Then move to the Union-Find solution because interviewers usually expect component counting without explicit graph traversal overhead. If constraints are aggressive, discuss hash-based neighbor generation as the scalability optimization.