Dijkstra - Family of functions - pgRouting Manual (2.5)

The problem definition (Advanced documentation)

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 \(E\)
  • \(E = \begin{cases} \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{if } reverse\_cost = \varnothing \\ \text{ } \text{ } & \quad \text{ } \\ \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ } \\ \cup \{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i>=0 \} & \quad \text{if } reverse\_cost \neq \varnothing \\ \end{cases}\)

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 \(E\)
  • \(E = \begin{cases} \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ if } reverse\_cost = \varnothing \\ \text{ } \text{ } & \text{ } \\ \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \text{ } \\ \cup \{(source_i, target_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

The problem

Given:

• \(start_{vid} \in V\) a starting vertex
• \(end_{vid} \in V\) an ending vertex
• \(G(V,E) = \begin{cases} G_d(V,E) & \quad \text{ if6 } directed = true \\ G_u(V,E) & \quad \text{ if5 } directed = false \\ \end{cases}\)

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.

See Also

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