Queue 0, then 1, 2, then 3: distances 0, 1, 1, 2. Marking at enqueue keeps every vertex queued once.
For the worked walkthrough, continue to the topic article and execution trace.
Stage-by-stage execution walkthrough
This request flow decomposes Breadth-First Search in Queue Order into 5 sequential stages so you can trace where latency accumulates, which state transitions are irreversible, and where production failure modes occur.
Stage 1: Store neighbors
Keep adjacency lists 0 to 1 and 2, 1 to 3, 2 to 3 in linear space. Sparse graphs stay compact; matrices would store sixteen cells for the same facts. During this transition, the runtime enforces strict state invariants before handing control to the next stage: inputs are validated at the boundary, state mutations are ordered deterministically, and latency/resource costs are bounded so downstream consumers never observe a torn or partially initialized state.
Stage 2: Seed the queue
Enqueue source 0 at distance 0 and mark it visited immediately, so no later edge can queue it again. During this transition, the runtime enforces strict state invariants before handing control to the next stage: inputs are validated at the boundary, state mutations are ordered deterministically, and latency/resource costs are bounded so downstream consumers never observe a torn or partially initialized state.
Stage 3: Expand in rings
Dequeue 0 to discover 1 and 2; dequeue 1 to discover 3; dequeue 2 to skip visited 3; dequeue 3 to finish. Visits leave in nondecreasing distance order. During this transition, the runtime enforces strict state invariants before handing control to the next stage: inputs are validated at the boundary, state mutations are ordered deterministically, and latency/resource costs are bounded so downstream consumers never observe a torn or partially initialized state.
Stage 4: Respect edge costs
Ring order counts edges, not cost. Non-negative weights switch to Dijkstra with a priority queue; negative edges need Bellman-Ford. During this transition, the runtime enforces strict state invariants before handing control to the next stage: inputs are validated at the boundary, state mutations are ordered deterministically, and latency/resource costs are bounded so downstream consumers never observe a torn or partially initialized state.
Stage 5: Cover every component
One launch reaches only its reachable set. Directed or disconnected graphs need a fresh launch from each unvisited vertex to count components. During this transition, the runtime enforces strict state invariants before handing control to the next stage: inputs are validated at the boundary, state mutations are ordered deterministically, and latency/resource costs are bounded so downstream consumers never observe a torn or partially initialized state.
Production invariants and failure modes
BFS finds fewest-edge paths in linear time on unweighted graphs; weights and unreachable components change the algorithm or the launch loop. When debugging or capacity-planning this flow in production, instrument each stage boundary separately (measuring both p50 median and p99.9 tail latency), verify what happens when an intermediate stage times out or drops packets, and ensure retry or fallback paths preserve end-to-end idempotency.