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:

• the set of vertices \(V\)
  • \(V = source \cup target \cup {start_{vid}} \cup {end_{vid}}\)
• the set of edges \(E\)
  • \(E = \begin{cases} &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} &\quad \text{ if } reverse\_cost = \varnothing \\ \\ &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} &\quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

Undirected graph

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

• the set of vertices \(V\)
  • \(V = source \cup target \cup {start_v{vid}} \cup {end_{vid}}\)
• the set of edges \(E\)
  • \(E = \begin{cases} &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, cost_i) \text{ when } cost >=0 \} &\quad \text{ if } reverse\_cost = \varnothing \\ \\ &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} \\ \cup &\{(source_i, target_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} &\quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

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:

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

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.