Shortest Path (Unweighted, via BFS)

BFS explores a graph layer by layer, so the first time it reaches a vertex is along a shortest path. Track dist[v] and parent[v] while exploring, then walk parents back from the target to reconstruct the route.

Algorithm

On the canonical graph from graph-adjacency-list, the shortest path from 1 to 6 is [1, 2, 4, 5, 6] with distance 4. The path is rebuilt from parent: 6 -> 5 -> 4 -> 2 -> 1, reversed.

Basic Implementation

basic.rb

adj = {1 => [2, 3], 2 => [1, 4], 3 => [1, 4], 4 => [2, 3, 5], 5 => [4, 6], 6 => [5]}
src = 1
dst = 6
dist = {src => 0}
parent = {src => nil}
queue = [src]
head = 0
while head < queue.length
	v = queue[head]
	head += 1
	adj[v].each do |nb|
		unless dist.key?(nb)
			dist[nb] = dist[v] + 1
			parent[nb] = v
			queue << nb
		end
	end
end
path = []
node = dst
while !node.nil?
	path << node
	node = parent[node]
end
path.reverse!
puts path.inspect
puts dist[dst]

Complexity

Time: O(V + E)
Space: O(V)

Implementation notes

Ruby: a dist Hash doubles as the visited check, parent records predecessors (nil at the source), and a head index walks the queue Array.
The replay shows dist, parent, and the queue filling in, then the reconstructed path, matching the lesson spec.

layers equal distance BFS order equals distance in an unweighted graph.