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.sh
Replay: real traced execution (multi-file project)
#!/usr/bin/env bash
set -euo pipefail
declare -A adj
adj[1]="2 3"
adj[2]="1 4"
adj[3]="1 4"
adj[4]="2 3 5"
adj[5]="4 6"
adj[6]="5"
src=1
dst=6
declare -A dist
declare -A parent
dist[$src]=0
parent[$src]=0
queue=("$src")
head=0
while [ "$head" -lt "${#queue[@]}" ]; do
	v=${queue[head]}
	head=$((head + 1))
	read -ra nbrs <<< "${adj[$v]}"
	for nb in "${nbrs[@]}"; do
		if [ -z "${dist[$nb]+_}" ]; then
			dist[$nb]=$((dist[$v] + 1))
			parent[$nb]=$v
			queue+=("$nb")
		fi
	done
done
path=()
node=$dst
while [ "$node" -ne 0 ]; do
	path+=("$node")
	node=${parent[$node]}
done
printf '['
sep=''
for ((k = ${#path[@]} - 1; k >= 0; k--)); do
	printf '%s%d' "$sep" "${path[k]}"
	sep=', '
done
printf ']\n'
printf '%d\n' "${dist[$dst]}"

  1. dist ← {1: 0}

    13declare -A parent14dist[$src]=015parent[$src]=0
    values this step{1: 0}dist

  2. parent ← {1: null}

    14dist[$src]=015parent[$src]=016queue=("$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]

    18while [ "$head" -lt "${#queue[@]}" ]; do19	v=${queue[head]}20	head=$((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}

    18while [ "$head" -lt "${#queue[@]}" ]; do19	v=${queue[head]}20	head=$((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}

    18while [ "$head" -lt "${#queue[@]}" ]; do19	v=${queue[head]}20	head=$((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}

    18while [ "$head" -lt "${#queue[@]}" ]; do19	v=${queue[head]}20	head=$((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}

    18while [ "$head" -lt "${#queue[@]}" ]; do19	v=${queue[head]}20	head=$((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}

    18while [ "$head" -lt "${#queue[@]}" ]; do19	v=${queue[head]}20	head=$((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]

    35done36printf '['37sep=''
    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]

    41done42printf ']\n'43printf '%d\n' "${dist[$dst]}"
    values this step[1, 2, 4, 5, 6]stdout[1, 2, 4, 5, 6]path

  11. stdout ← 4

    42printf ']\n'43printf '%d\n' "${dist[$dst]}"
    values this step4stdout4dist[6]

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

    42printf ']\n'43printf '%d\n' "${dist[$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

  • Bash: a dist associative array doubles as the visited check, parent records predecessors (0 marks the source), and a head index walks the queue array.
  • 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.