Dijkstra - Family of functions¶
- pgr_dijkstra - Dijkstra’s algorithm for the shortest paths.
- pgr_dijkstraCost - Get the aggregate cost of the shortest paths.
- pgr_dijkstraCostMatrix - Use pgr_dijkstra to create a costs matrix.
- pgr_drivingDistance - Use pgr_dijkstra to calculate catchament information.
- pgr_KSP - Use Yen algorithm with pgr_dijkstra to get the K shortest paths.
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.
- pgr_dijkstraVia - Proposed - Get a route of a seuence of vertices.
- pgr_dijkstraNear - Proposed - Get the route to the nearest vertex.
- pgr_dijkstraNearCost - Proposed - Get the cost to the nearest vertex.
Introduction¶
Dijkstra’s algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1956. It is a graph search algorithm that solves the shortest path problem for a graph with non-negative edge path costs, producing a shortest path from a starting vertex to an ending vertex. This implementation can be used with a directed graph and an undirected graph.
The main characteristics are:
- Process is done only on edges with positive costs.
- A negative value on a cost column is interpreted as the edge does not exist.
- Values are returned when there is a path.
- When there is no path:
- When the starting vertex and ending vertex are the same.
- The aggregate cost of the non included values \((v, v)\) is \(0\)
- When the starting vertex and ending vertex are the different and there is
no path:
- The aggregate cost the non included values \((u, v)\) is \(\infty\)
- When the starting vertex and ending vertex are the same.
- For optimization purposes, any duplicated value in the starting vertices or on the ending vertices are ignored.
- Running time: \(O(| start\ vids | * (V \log V + E))\)
The Dijkstra family functions are based on the Dijkstra algorithm.
Parameters¶
Column | Type | Description |
---|---|---|
Edges SQL | TEXT |
Edges SQL as described below |
Combinations SQL | TEXT |
Combinations SQL as described below |
start vid | BIGINT |
Identifier of the starting vertex of the path. |
start vids | ARRAY[BIGINT] |
Array of identifiers of starting vertices. |
end vid | BIGINT |
Identifier of the ending vertex of the path. |
end vids | ARRAY[BIGINT] |
Array of identifiers of ending vertices. |
Optional parameters¶
Column | Type | Default | Description |
---|---|---|---|
directed |
BOOLEAN |
true |
|
Inner Queries¶
Edges SQL¶
Column | Type | Default | Description |
---|---|---|---|
id |
ANY-INTEGER | Identifier of the edge. | |
source |
ANY-INTEGER | Identifier of the first end point vertex of the edge. | |
target |
ANY-INTEGER | Identifier of the second end point vertex of the edge. | |
cost |
ANY-NUMERICAL | Weight of the edge (source , target ) |
|
reverse_cost |
ANY-NUMERICAL | -1 | Weight of the edge (
|
Where:
ANY-INTEGER: | SMALLINT , INTEGER , BIGINT |
---|---|
ANY-NUMERICAL: | SMALLINT , INTEGER , BIGINT , REAL , FLOAT |
Combinations SQL¶
Parameter | Type | Description |
---|---|---|
source |
ANY-INTEGER | Identifier of the departure vertex. |
target |
ANY-INTEGER | Identifier of the arrival vertex. |
Where:
ANY-INTEGER: | SMALLINT , INTEGER , BIGINT |
---|
Advanced documentation¶
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.