Solve unique triplets summing to target

Implement a function that, given an integer array nums and an integer target T, returns all unique triplets [x, y, z] from nums such that x + y + z = T. Requirements: avoid duplicate triplets (enforce x <= y <= z within each triplet), return results in lexicographic order, and aim for O(n^ 2) time using sorting and a two-pointer strategy with O(1) extra space beyond the output. Discuss edge cases (duplicates, all zeros, no solution, mixed signs, large n) and provide a concise set of test cases. Analyze time and space complexity.

What a Strong Answer Covers

• A clear algorithm with the right data structures and enough pseudocode or code-level detail to implement it.
• A correctness argument that explains why the algorithm covers all required cases.
• Time and space complexity, plus at least one alternative approach when relevant.
• Focused tests for normal cases, edge cases, and failure modes.

Follow-up Questions

• How would the approach change if the input were streaming or too large for memory?
• What invariants would you assert in production code?
• Which tests would catch off-by-one, duplicate, or tie-breaking bugs?

Solution

Solution Alignment

The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests.

Problem Restatement

Implement a function that, given an integer array nums and an integer target T, returns all unique triplets [x, y, z] from nums such that x + y + z = T. Requirements: avoid duplicate triplets (enforce x <= y <= z within each triplet), return results in lexicographic order, and aim for O(n^ 2) time using sorting and a two-pointer strategy with O(1) extra space beyond the output. Discuss edge cases (duplicates, all zeros, no solution, mixed signs, large n) and provide a concise set of test cases. Analyze time and space complexity.

Recommended Approach

Model the states explicitly and use BFS for unweighted shortest paths, Dijkstra for weighted non-negative paths, or topological DP for DAGs. Track visited states at the right granularity so cycles do not cause repeated work.

Correctness

The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call.

Complexity

BFS is O(V + E) time and O(V) space for a standard graph. Expanded-state problems multiply those bounds by the number of state dimensions.

Edge Cases and Tests

Disconnected graph, source equals target, cycles, duplicate edges, unreachable target, and whether the answer counts nodes, edges, moves, or transfers.