pgRouting Manual (2.2.0)

pgr_dijkstra - Shortest Path Dijkstra

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pgr_dijkstra - Shortest Path Dijkstra

The problem definition

Given the following query:

pgr_dijkstra(\(sql, start_{vid}, end_{vid}, directed\))

where \(sql = \{(id_i, source_i, target_i, cost_i, reverse\_cost_i)\}\)

and

  • \(source = \bigcup source_i\),
  • \(target = \bigcup target_i\),

The graphs are defined as follows:

Directed graph

The weighted directed graph, \(G_d(V,E)\), is definied by:

Undirected graph

The weighted undirected graph, \(G_u(V,E)\), is definied by:

The problem

Given:

  • \(start_{vid} \in V\) a starting vertex
  • \(end_{vid} \in V\) an ending vertex
  • \(G(V,E) = \begin{cases} G_d(V,E) &\quad \text{ if } directed = true \\ G_u(V,E) &\quad \text{ if } directed = false \\ \end{cases}\)

Then:

\[\begin{split}\text{pgr_dijkstra}(sql, start_{vid}, end_{vid}, directed) = \begin{cases} \text{shortest path } \boldsymbol{\pi} \text{ between } start_{vid} \text{and } end_{vid} &\quad \text{if } \exists \boldsymbol{\pi} \\ \varnothing &\quad \text{otherwise} \\ \end{cases}\end{split}\]

\(\boldsymbol{\pi} = \{(path_\seq_i, node_i, edge_i, cost_i, agg\_cost_i)\}\)

where:
  • \(path_\seq_i = i\)
  • \(path_\seq_{| \pi |} = | \pi |\)
  • \(node_i \in V\)
  • \(node_1 = start_{vid}\)
  • \(node_{| \pi |} = end_{vid}\)
  • \(\forall i \neq | \pi |, \quad (node_i, node_{i+1}, cost_i) \in E\)
  • \(edge_i = \begin{cases} id_{(node_i, node_{i+1},cost_i)} &\quad \text{when } i \neq | \pi | \\ -1 &\quad \text{when } i = | \pi | \\ \end{cases}\)
  • \(cost_i = cost_{(node_i, node_{i+1})}\)
  • \(agg\_cost_i = \begin{cases} 0 &\quad \text{when } i = 1 \\ \displaystyle\sum_{k=1}^{i} cost_{(node_{k-1}, node_k)} &\quad \text{when } i \neq 1 \\ \end{cases}\)
In other words: The algorithm returns a the shortest path between \(start_{vid}\) and \(end_{vid}\) , 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 \(start_{vid}\) up to the node.

If there is no path, the resulting set is empty.

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