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 distance and predecessor while exploring, then walk predecessors 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 the predecessor of each vertex: 6 -> 5 -> 4 -> 2 -> 1, reversed.

Basic Implementation

basic.R

adj <- list("1" = c(2, 3), "2" = c(1, 4), "3" = c(1, 4), "4" = c(2, 3, 5), "5" = c(4, 6), "6" = c(5))
src <- 1
dst <- 6
dist <- list()
parent <- list()
dist[[as.character(src)]] <- 0
parent[[as.character(src)]] <- 0
queue <- c(src)
head <- 1
while (head <= length(queue)) {
	v <- queue[head]
	head <- head + 1
	for (nb in adj[[as.character(v)]]) {
		if (is.null(dist[[as.character(nb)]])) {
			dist[[as.character(nb)]] <- dist[[as.character(v)]] + 1
			parent[[as.character(nb)]] <- v
			queue[length(queue) + 1] <- nb
		}
	}
}
path <- integer(0)
node <- dst
while (node != 0) {
	path[length(path) + 1] <- node
	node <- parent[[as.character(node)]]
}
path <- rev(path)
cat("[", paste(path, collapse = ", "), "]\n", sep = "")
cat(dist[[as.character(dst)]], "\n", sep = "")

Complexity

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

Implementation notes

R: a dist list doubles as the visited check, parent records predecessors (0 marks the source), and a head index walks the queue vector.
The replay shows the distances and predecessors filling in, then the reconstructed path, matching the lesson spec.

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