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pgr_dijkstra - Shortest Path Dijkstra¶
- pgr_dijkstra - Dijkstra’s algorithm for the shortest paths.
- pgr_dijkstraCost -Get the aggregate cost of the shortest paths.
The problem definition¶
Given the following query:
pgr_dijkstra(sql,startvid,endvid,directed)
where sql={(idi,sourcei,targeti,costi,reverse_costi)}
and
- source=⋃sourcei,
- target=⋃targeti,
The graphs are defined as follows:
Directed graph
The weighted directed graph, Gd(V,E), is definied by:
- the set of vertices V
- V=source∪target∪startvid∪endvid
- the set of edges E
- E={{(sourcei,targeti,costi) when cost>=0} if reverse_cost=∅{(sourcei,targeti,costi) when cost>=0}∪{(targeti,sourcei,reverse_costi) when reverse_costi>=0)} if reverse_cost≠∅
Undirected graph
The weighted undirected graph, Gu(V,E), is definied by:
- the set of vertices V
- V=source∪target∪startvvid∪endvid
- the set of edges E
- E={{(sourcei,targeti,costi) when cost>=0}∪{(targeti,sourcei,costi) when cost>=0} if reverse_cost=∅{(sourcei,targeti,costi) when cost>=0}∪{(targeti,sourcei,costi) when cost>=0}∪{(targeti,sourcei,reverse_costi) when reverse_costi>=0)}∪{(sourcei,targeti,reverse_costi) when reverse_costi>=0)} if reverse_cost≠∅
The problem
Given:
- startvid∈V a starting vertex
- endvid∈V an ending vertex
- G(V,E)={Gd(V,E) if directed=trueGu(V,E) if directed=false
Then:
pgr_dijkstra(sql,startvid,endvid,directed)={shortest path π between startvidand endvidif ∃π∅otherwise
\boldsymbol{\pi} = \{(path_\seq_i, node_i, edge_i, cost_i, agg\_cost_i)\}
- where:
- path_\seq_i = i
- path_\seq_{| \pi |} = | \pi |
- nodei∈V
- node1=startvid
- node|π|=endvid
- ∀i≠|π|,(nodei,nodei+1,costi)∈E
- edgei={id(nodei,nodei+1,costi)when i≠|π|−1when i=|π|
- costi=cost(nodei,nodei+1)
- agg_costi={0when i=1i∑k=1cost(nodek−1,nodek)when i≠1
- In other words: The algorithm returns a the shortest path between startvid and endvid , if it exists, in terms of a sequence of nodes and of edges,
- path\seq indicates the relative position in the path of the node or edge.
- cost is the cost of the edge to be used to go to the next node.
- agg_cost is the cost from the startvid up to the node.
If there is no path, the resulting set is empty.