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.
