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.lua
Replay: real traced execution (multi-file project)
local 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}
local src = 1
local dst = 6
local dist = {}
local parent = {}
dist[src] = 0
parent[src] = 0
local queue = {src}
local head = 1
while head <= #queue do
	local v = queue[head]
	head = head + 1
	for i = 1, #adj[v] do
		local nb = adj[v][i]
		if dist[nb] == nil then
			dist[nb] = dist[v] + 1
			parent[nb] = v
			queue[#queue + 1] = nb
		end
	end
end
local path = {}
local node = dst
while node ~= 0 do
	path[#path + 1] = node
	node = parent[node]
end
io.write("[")
for k = #path, 1, -1 do
	if k < #path then io.write(", ") end
	io.write(tostring(path[k]))
end
io.write("]\n")
io.write(tostring(dist[dst]) .. "\n")

  1. dist ← {1: 0}

    11local parent = {}12dist[src] = 013parent[src] = 0
    values this step{1: 0}dist

  2. parent ← {1: null}

    12dist[src] = 013parent[src] = 014local 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 <= #queue do17	local 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 <= #queue do17	local 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 <= #queue do17	local 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 <= #queue do17	local 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 <= #queue do17	local 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 <= #queue do17	local 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]

    33end34io.write("[")35for k = #path, 1, -1 do
    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]

    38end39io.write("]\n")40io.write(tostring(dist[dst]) .. "\n")
    values this step[1, 2, 4, 5, 6]stdout[1, 2, 4, 5, 6]path

  11. stdout ← 4

    39io.write("]\n")40io.write(tostring(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)

    39io.write("]\n")40io.write(tostring(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

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