Dijkstra - Family of functions

proposed

Warning

Proposed functions for next mayor release.

  • They are not officially in the current release.

  • They will likely officially be part of the next mayor release:

    • The functions make use of ANY-INTEGER and ANY-NUMERICAL

    • Name might not change. (But still can)

    • Signature might not change. (But still can)

    • Functionality might not change. (But still can)

    • pgTap tests have being done. But might need more.

    • Documentation might need refinement.

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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

Indices and tables