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.swift
Replay: real traced execution (multi-file project)
var adj: [Int: [Int]] = [:]
adj[1] = [2, 3]
adj[2] = [1, 4]
adj[3] = [1, 4]
adj[4] = [2, 3, 5]
adj[5] = [4, 6]
adj[6] = [5]
let src = 1
let dst = 6
var dist: [Int: Int] = [src: 0]
var parent: [Int: Int] = [src: 0]
var queue: [Int] = [src]
while !queue.isEmpty {
	let v = queue.removeFirst()
	for nb in adj[v]! {
		if dist[nb] == nil {
			dist[nb] = dist[v]! + 1
			parent[nb] = v
			queue.append(nb)
		}
	}
}
var path: [Int] = []
var node = dst
while node != 0 {
	path.append(node)
	node = parent[node]!
}
print(Array(path.reversed()))
print(dist[dst]!)

  1. dist ← {1: 0}

    9let dst = 610var dist: [Int: Int] = [src: 0]11var parent: [Int: Int] = [src: 0]
    values this step{1: 0}dist

  2. parent ← {1: null}

    10var dist: [Int: Int] = [src: 0]11var parent: [Int: Int] = [src: 0]12var queue: [Int] = [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]

    13while !queue.isEmpty {14	let v = queue.removeFirst()15	for nb in adj[v]! {
    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}

    13while !queue.isEmpty {14	let v = queue.removeFirst()15	for nb in adj[v]! {
    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}

    13while !queue.isEmpty {14	let v = queue.removeFirst()15	for nb in adj[v]! {
    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}

    13while !queue.isEmpty {14	let v = queue.removeFirst()15	for nb in adj[v]! {
    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}

    13while !queue.isEmpty {14	let v = queue.removeFirst()15	for nb in adj[v]! {
    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}

    13while !queue.isEmpty {14	let v = queue.removeFirst()15	for nb in adj[v]! {
    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]

    23var path: [Int] = []24var node = dst25while node != 0 {
    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]

    28}29print(Array(path.reversed()))30print(dist[dst]!)
    values this step[1, 2, 4, 5, 6]stdout[1, 2, 4, 5, 6]path

  11. stdout ← 4

    29print(Array(path.reversed()))30print(dist[dst]!)
    values this step4stdout4dist[6]

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

    29print(Array(path.reversed()))30print(dist[dst]!)
    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

  • Swift: a dist dictionary doubles as the visited check, parent records predecessors (0 marks the source), and an array 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.