Breadth-first search explores an unweighted graph in layers from a starting vertex. A FIFO queue expands vertices discovered earlier before vertices discovered later. The recording traces this behavior and its queue implementation. The layer order supports shortest paths measured in edges when the graph's edges all have equal cost.
Mark when you enqueue
Begin with the start in the queue and visited set. When expanding a vertex, enqueue each neighbor that has not already been discovered and mark it immediately. Marking at enqueue time prevents several vertices from adding the same neighbor before its first queued copy is processed.
For a diamond A to B and C, with both B and C leading to D, D should enter the queue once. Its first discovery has the smallest possible edge distance because all shallower queued layers are processed first. A predecessor map can remember which vertex discovered D and later reconstruct the route back to A.
Use queue and set operations that support the bound
The current Python lesson uses a set for visited membership. Checking a growing list for each neighbor would add a linear search to repeated edge processing. Queue representation matters too: repeatedly shifting the front of a dynamic array can add work that the graph-level analysis does not include. A deque or an advancing head index avoids that particular repeated shift.
With adjacency lists, a traversal that expands each reachable vertex once examines its outgoing edges once, giving O(V + E) work under bounded membership and queue-operation assumptions. The queue, visited set, and predecessor map can each retain linear state. Returning a route adds storage for the route itself.
Keep the objective within the guarantee
A weighted edge may make a two-edge route cheaper than a one-edge route. Ordinary FIFO BFS minimizes edge count, so weighted shortest paths need a different method unless all weights are equal. Starting from one vertex also does not visit disconnected components automatically.
Test a cycle, a converging diamond, an unreachable target, and a start that is already the target. Continue with Pilot a Rocket to reconstruct a route including both endpoints. Use the graph traversal tool to compare BFS with DFS on a graph where the first DFS route is longer.
Trace discovery by layers
For [[1, 2], [3], [3], []] and start 0, the example returns [0, 1, 2, 3]. It marks a neighbor visited when enqueueing it. When vertex 2 also sees vertex 3, that mark prevents a second enqueue, even though 3 has not necessarily been processed yet.
The queue holds vertices discovered earlier ahead of vertices discovered later. Starting from one vertex in an unweighted graph, this processes reachable vertices in nondecreasing edge distance. If you need actual distances or paths, store a distance or parent at that first discovery; the returned order alone is not a path to every vertex.
Test queue order and discovery timing
An invalid start returns an empty result. The graph otherwise assumes valid neighbor identifiers. Include an isolated vertex, a cycle, a shared descendant, self-loops, and disconnected components. Repeated adjacency entries should not repeat a vertex in the output.
Traversal with an on-demand visited set takes O(Vr + Er) time over the reachable subgraph, assuming efficient queue operations. Versions initializing a visited array for every vertex add O(V) time and storage up front. The queue, visited state, and returned order use O(Vr) space. Some versions advance a cursor over a growable array; others use a deque. Repeatedly shifting the front of a contiguous array would add avoidable movement. Replacing this queue with a stack would also change discovery order and remove the shortest-layer argument.
C# example and implementation choices
Queue<int> gives the FIFO order, and HashSet<int>.Add returns true only at first discovery. The code enqueues a neighbor in that successful branch. Count protects Dequeue on the empty boundary. The output is a fresh list, and input adjacency lists are unchanged. Separate distance or parent storage would be needed if callers want more than this visit sequence.
Worked example
using System;
using System.Collections.Generic;
public static class BfsOrderExample {
public static List<int> BfsOrder(List<List<int>> graph, int start) {
var output = new List<int>();
if (start < 0 || start >= graph.Count) return output;
var queue = new Queue<int>();
var visited = new HashSet<int>();
queue.Enqueue(start);
visited.Add(start);
while (queue.Count > 0) {
int node = queue.Dequeue();
output.Add(node);
foreach (int neighbor in graph[node]) {
if (visited.Add(neighbor)) queue.Enqueue(neighbor);
}
}
return output;
}
}
Continue in C#
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