Implement core graph algorithms for graphics

Given a scene or dependency graph, implement topological sort, BFS/DFS, and shortest path (Dijkstra). Discuss time/space complexity, memory layouts (CSR vs adjacency lists), and how you would parallelize traversals on CPU vs GPU while avoiding contention and ensuring determinism in a test framework.

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

Given a scene or dependency graph, implement topological sort, BFS/DFS, and shortest path (Dijkstra). Discuss time/space complexity, memory layouts (CSR vs adjacency lists), and how you would parallelize traversals on CPU vs GPU while avoiding contention and ensuring determinism in a test framework.

Recommended Approach

Model each reachable configuration as a graph state and choose the traversal by edge cost: BFS for unweighted shortest paths, Dijkstra for non-negative weighted paths, or topological DP for DAGs. Track visited states at the correct 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.