Minimum Genetic Mutation

Shortest mutations are BFS levels

Each gene is a node, and an edge changes exactly one character to one of A, C, G, or T, provided the new gene is in the bank. Put the bank in a hash set, enqueue the start gene, and run BFS level by level. For each dequeued gene, try each position and each of the other three bases; when a generated gene is in the bank, remove it from the set as visited and enqueue it. Reaching the end at level d proves d mutations is minimum because BFS explores all shorter sequences first.

Avoid subtle validity mistakes

The start need not be in the bank, but the end generally must be reachable through bank genes; clarify the statement's convention if start equals end. Mark visited when enqueuing, not when dequeuing, to avoid duplicate work. Do not form edges by comparing every pair unless constraints are tiny. With gene length L and B bank entries, generation costs O(BL4) expected time and the set plus queue use O(B) space. Test missing end genes, one-step changes, and paths requiring a temporary mutation away from the target.