Graphs

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.

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

Basic Implementation

basic.php
Replay: real traced execution (multi-file project)
<?php
$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 => null];
$queue = [$src];
$head = 0;
while ($head < count($queue)) {
	$v = $queue[$head];
	$head = $head + 1;
	foreach ($adj[$v] as $nb) {
		if (!array_key_exists($nb, $dist)) {
			$dist[$nb] = $dist[$v] + 1;
			$parent[$nb] = $v;
			$queue[] = $nb;
		}
	}
}
$path = [];
$node = $dst;
while ($node !== null) {
	$path[] = $node;
	$node = $parent[$node];
}
$path = array_reverse($path);
echo "[" . implode(", ", $path) . "]\n";
echo $dist[$dst] . "\n";

  1. dist ← {1: 0}

    4$dst = 6;5$dist = [$src => 0];6$parent = [$src => null];
    values this step{1: 0}dist

  2. parent ← {1: null}

    5$dist = [$src => 0];6$parent = [$src => null];7$queue = [$src];
    values this step{1: null}parent

  3. dist ← {1: 0, 2: 1, 3: 1}, parent ← {1: null, 2: 1, 3: 1}, queue ← [2, 3]

    9while ($head < count($queue)) {10	$v = $queue[$head];11	$head = $head + 1;
    values this step{1: 0, 2: 1, 3: 1}dist{1: null, 2: 1, 3: 1}parent[2, 3]queue1dequeue

  4. dist ← {1: 0, 2: 1, 3: 1, 4: 2}, parent ← {1: null, 2: 1, 3: 1, 4: 2}

    9while ($head < count($queue)) {10	$v = $queue[$head];11	$head = $head + 1;
    values this step{1: 0, 2: 1, 3: 1, 4: 2}dist{1: null, 2: 1, 3: 1, 4: 2}parent[3, 4]queue2dequeue

  5. dist ← {1: 0, 2: 1, 3: 1, 4: 2}, parent ← {1: null, 2: 1, 3: 1, 4: 2}

    9while ($head < count($queue)) {10	$v = $queue[$head];11	$head = $head + 1;
    values this step{1: 0, 2: 1, 3: 1, 4: 2}dist{1: null, 2: 1, 3: 1, 4: 2}parent[4]queue3dequeue

  6. dist ← {1: 0, 2: 1, 3: 1, 4: 2, 5: 3}, parent ← {1: null, 2: 1, 3: 1, 4: 2, 5: 4}

    9while ($head < count($queue)) {10	$v = $queue[$head];11	$head = $head + 1;
    values this step{1: 0, 2: 1, 3: 1, 4: 2, 5: 3}dist{1: null, 2: 1, 3: 1, 4: 2, 5: 4}parent[5]queue4dequeue

  7. dist ← {1: 0, 2: 1, 3: 1, 4: 2, 5: 3, 6: 4}, parent ← {1: null, 2: 1, 3: 1, 4: 2, 5: 4, 6: 5}

    9while ($head < count($queue)) {10	$v = $queue[$head];11	$head = $head + 1;
    values this step{1: 0, 2: 1, 3: 1, 4: 2, 5: 3, 6: 4}dist{1: null, 2: 1, 3: 1, 4: 2, 5: 4, 6: 5}parent[6]queue5dequeue

  8. dist ← {1: 0, 2: 1, 3: 1, 4: 2, 5: 3, 6: 4}, parent ← {1: null, 2: 1, 3: 1, 4: 2, 5: 4, 6: 5}

    9while ($head < count($queue)) {10	$v = $queue[$head];11	$head = $head + 1;
    values this step{1: 0, 2: 1, 3: 1, 4: 2, 5: 3, 6: 4}dist{1: null, 2: 1, 3: 1, 4: 2, 5: 4, 6: 5}parent[]queue6dequeue

  9. path ← [1, 2, 4, 5, 6]

    25}26$path = array_reverse($path);27echo "[" . implode(", ", $path) . "]\n";
    values this step[1, 2, 4, 5, 6]path{1: null, 2: 1, 3: 1, 4: 2, 5: 4, 6: 5}parent

  10. stdout ← [1, 2, 4, 5, 6]

    26$path = array_reverse($path);27echo "[" . implode(", ", $path) . "]\n";28echo $dist[$dst] . "\n";
    values this step[1, 2, 4, 5, 6]stdout[1, 2, 4, 5, 6]path

  11. stdout ← 4

    27echo "[" . implode(", ", $path) . "]\n";28echo $dist[$dst] . "\n";
    values this step4stdout4dist[6]

  12. BFS path ← 1 -> 2 (1 edge, cost 10), cheaper weighted path ← 1 -> 3 -> 2 (2 edges, cost 2)

    27echo "[" . implode(", ", $path) . "]\n";28echo $dist[$dst] . "\n";
    values this step1 -> 2 (1 edge, cost 10)BFS path1 -> 3 -> 2 (2 edges, cost 2)cheaper weighted pathuse Dijkstra with a priority queueweighted algorithm1->2 weight 10, 1->3 weight 1, 3->2 weight 1edge weights

Complexity

  • Time: O(V + E)
  • Space: O(V)

Implementation notes

  • PHP: $dist doubles as the visited check, $parent records predecessors (null at the source), and a head index walks the queue.
  • The replay shows dist, parent, and the queue filling in, then the reconstructed path. It also contrasts that unweighted result with a weighted graph where Dijkstra with a priority queue is required.