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.

New York streets

The main streets of Lower Manhattan and Downtown Brooklyn, from OpenStreetMap, with the Brooklyn and Manhattan Bridges between them. Each intersection is named after its streets, one-way streets go one way, and each edge is the minutes to drive it at the speed limit, or at 24 to 40 km/h by road type where the map has no limit.

A directed, weighted graph with 295 vertices and 669 edges.

Map data © OpenStreetMap contributors.

With warehouses in Lower Manhattan, Williamsburg and Downtown Brooklyn, which one should deliver to each intersection, and what is the longest drive any of them has?

Warehouse 1 at Canal St & Walker St serves 139, 2 at Roebling St & Division Ave serves 53 and 3 at Tillary St & Gold St serves 103. The longest drive is 6.04, from warehouse 1 to Church St & Vesey St.

Step by step

Put every warehouse in the queue at 0: 1 at Canal St & Walker St, 2 at Roebling St & Division Ave and 3 at Tillary St & Gold St. Each place goes to the warehouse that can reach it soonest, so one search serves them all.
Take Canal St & Walker St from the queue. It is warehouse 1, at 0 from itself.
Canal St & Walker St to Bowery & Canal St: 0 + 0.5 = 0.5, so Bowery & Canal St goes to warehouse 1 at 0.5.
Canal St & Walker St to Centre St & Canal St: 0 + 0.23 = 0.23, so Centre St & Canal St goes to warehouse 1 at 0.23.
Take Roebling St & Division Ave from the queue. It is warehouse 2, at 0 from itself.
Roebling St & Division Ave to Broadway & Roebling St: 0 + 0.32 = 0.32, so Broadway & Roebling St goes to warehouse 2 at 0.32.

654 more steps lead to the last one:

Done. Every place has its nearest warehouse. One search settled 295 places. path.Dijkstra from each warehouse in turn would settle 885 between them.

Use it in Go

nearest, err := path.DijkstraMultiSource(g, "Canal St & Walker St", "Roebling St & Division Ave", "Tillary St & Gold St")
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