Multi-source Dijkstra

Finds the nearest of several sources for every vertex, and the distance from it, in one search.

Time complexity O((V+E) log V), gograph function path.DijkstraMultiSource.

How it works

Multi-source Dijkstra answers which source is nearest to each vertex, such as which warehouse can reach each address soonest. It puts every source in the queue at distance 0 and then runs Dijkstra's algorithm once. Each vertex is settled by whichever source gets there first, so the areas around the sources grow together until they meet.

One search settles each vertex once, so it costs O((V+E) log V) however many sources there are. Running path.Dijkstra from each source instead settles every vertex once per source. The performance card on this page times both.

path.DijkstraMultiSource returns a NearestSources value with Source, Distance and PathTo for any vertex. When two sources are equally near, the one listed first wins, so the same sources give the same answer on every run. Distances run along the edges, from a source to the vertex. On a map with one-way streets, that is the drive from the warehouse to the address, and the drive back can differ.

Metro region

A hundred towns, T1 to T100, where T stands for town, numbered row by row from the top left with ten to a row. Two fast motorways cross at T45: one runs from T41 to T50, the other through T5, T15 and on down to T95. Local roads join neighboring towns, a few of them one way, and each edge is the travel time in minutes.

A directed, weighted graph with 100 vertices and 346 edges.

With three warehouses in the region, which one should deliver to each town, and what is the longest drive?

Warehouse 1 at T6 serves 37, 2 at T52 serves 30 and 3 at T68 serves 33. The longest drive is 29, from warehouse 1 to T1.

Step by step

Put every warehouse in the queue at 0: 1 at T6, 2 at T52 and 3 at T68. Each place goes to the warehouse that can reach it soonest, so one search serves them all.
Take T6 from the queue. It is warehouse 1, at 0 from itself.
T6 to T5: 0 + 4 = 4, so T5 goes to warehouse 1 at 4.
T6 to T7: 0 + 4 = 4, so T7 goes to warehouse 1 at 4.
T6 to T16: 0 + 5 = 5, so T16 goes to warehouse 1 at 5.
Take T52 from the queue. It is warehouse 2, at 0 from itself.

268 more steps lead to the last one:

Done. Every place has its nearest warehouse. One search settled 100 places. path.Dijkstra from each warehouse in turn would settle 300 between them.

Use it in Go

nearest, err := path.DijkstraMultiSource(g, "T6", "T52", "T68")
if err != nil {
	return err
}
for _, v := range g.GetAllVertices() {
	from, ok := nearest.Source(v.Label())
	if !ok {
		continue
	}
	minutes, _ := nearest.Distance(v.Label())
	fmt.Println(v.Label(), from.Label(), minutes)
}

Example graphs

New York streets, 295 vertices and 669 edges
London streets, 274 vertices and 598 edges
San Francisco streets, 248 vertices and 548 edges
Rasht streets, 298 vertices and 568 edges
Delivery area, 8 vertices and 12 edges
Downtown grid, 20 vertices and 60 edges
One-way city, 45 vertices and 88 edges
Metro region, 100 vertices and 346 edges