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.pl
Replay: real traced execution (multi-file project)
use strict; use warnings;
my %adj = (
	1 => [2, 3],
	2 => [1, 4],
	3 => [1, 4],
	4 => [2, 3, 5],
	5 => [4, 6],
	6 => [5],
);
my $src = 1;
my $dst = 6;
my %dist = ($src => 0);
my %parent = ($src => 0);
my @queue = ($src);
my $head = 0;
while ($head < scalar @queue) {
	my $v = $queue[$head];
	$head = $head + 1;
	for my $nb (@{$adj{$v}}) {
		if (!exists $dist{$nb}) {
			$dist{$nb} = $dist{$v} + 1;
			$parent{$nb} = $v;
			push @queue, $nb;
		}
	}
}
my @path = ();
my $node = $dst;
while ($node != 0) {
	push @path, $node;
	$node = $parent{$node};
}
@path = reverse @path;
print "[" . join(", ", @path) . "]\n";
print "$dist{$dst}\n";

  1. dist ← {1: 0}

    11my $dst = 6;12my %dist = ($src => 0);13my %parent = ($src => 0);
    values this step{1: 0}dist

  2. parent ← {1: null}

    12my %dist = ($src => 0);13my %parent = ($src => 0);14my @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]

    16while ($head < scalar @queue) {17	my $v = $queue[$head];18	$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}

    16while ($head < scalar @queue) {17	my $v = $queue[$head];18	$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}

    16while ($head < scalar @queue) {17	my $v = $queue[$head];18	$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}

    16while ($head < scalar @queue) {17	my $v = $queue[$head];18	$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}

    16while ($head < scalar @queue) {17	my $v = $queue[$head];18	$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}

    16while ($head < scalar @queue) {17	my $v = $queue[$head];18	$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]

    32}33@path = reverse @path;34print "[" . join(", ", @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]

    33@path = reverse @path;34print "[" . join(", ", @path) . "]\n";35print "$dist{$dst}\n";
    values this step[1, 2, 4, 5, 6]stdout[1, 2, 4, 5, 6]path

  11. stdout ← 4

    34print "[" . join(", ", @path) . "]\n";35print "$dist{$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)

    34print "[" . join(", ", @path) . "]\n";35print "$dist{$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

  • Perl: %dist doubles as the visited check, %parent records predecessors (0 marks 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.