# pgRouting Concepts¶

This is a simple guide that go through some of the steps for getting started with pgRouting. This guide covers:

## Graphs¶

### Graph definition¶

A graph is an ordered pair $$G = (V ,E)$$ where:

• $$V$$ is a set of vertices, also called nodes.

• $$E \subseteq \{( u, v ) \mid u , v \in V \}$$

There are different kinds of graphs:

• Undirected graph

• $$E \subseteq \{( u, v ) \mid u , v \in V\}$$

• Undirected simple graph

• $$E \subseteq \{( u, v ) \mid u , v \in V, u \neq v\}$$

• Directed graph

• $$E \subseteq \{( u, v ) \mid (u , v) \in (V X V) \}$$

• Directed simple graph

• $$E \subseteq \{( u, v ) \mid (u , v) \in (V X V), u \neq v\}$$

Graphs:

• Do not have geometries.

• Some graph theory problems require graphs to have weights, called cost in pgRouting.

In pgRouting there are several ways to represent a graph on the database:

• With cost

• (id, source, target, cost)

• With cost and reverse_cost

• (id, source, target, cost, reverse_cost)

Where:

Column

Description

id

Identifier of the edge. Requirement to use the database in a consistent. manner.

source

Identifier of a vertex.

target

Identifier of a vertex.

cost

Weight of the edge (source, target):

• When negative the edge (source, target) do not exist on the graph.

• cost must exist in the query.

reverse_cost

Weight of the edge (target, source)

• When negative the edge (target, source) do not exist on the graph.

The decision of the graph to be directed or undirected is done when executing a pgRouting algorithm.

### Graph with cost¶

The weighted directed graph, $$G_d(V,E)$$:

• Graph data is obtained with a query

SELECT id, source, target, cost FROM edges

• the set of edges $$E$$

• $$E = \{(source_{id}, target_{id}, cost_{id}) \text{ when } cost_{id} \ge 0 \}$$

• Edges where cost is non negative are part of the graph.

• the set of vertices $$V$$

• $$V = \{source_{id} \cup target_{id}\}$$

• All vertices in source and target are part of the graph.

Directed graph

In a directed graph the edge $$(source_{id}, target_{id}, cost_{id})$$ has directionality: $$source_{id} \rightarrow target_{id}$$

For the following data:

SELECT *
FROM (VALUES (1, 1, 2, 5), (2, 1, 3, -3))
AS t(id, source, target, cost);
id | source | target | cost
----+--------+--------+------
1 |      1 |      2 |    5
2 |      1 |      3 |   -3
(2 rows)



Edge $$2$$ ($$1 \rightarrow 3$$) is not part of the graph.

The data is representing the following graph: Undirected graph

In an undirected graph the edge $$(source_{id}, target_{id}, cost_{id})$$ does not have directionality: $$source_{id} \frac{\;\;\;\;\;}{} target_{id}$$

• In terms of a directed graph is like having two edges: $$source_{id} \leftrightarrow target_{id}$$

For the following data:

SELECT *
FROM (VALUES (1, 1, 2, 5), (2, 1, 3, -3))
AS t(id, source, target, cost);
id | source | target | cost
----+--------+--------+------
1 |      1 |      2 |    5
2 |      1 |      3 |   -3
(2 rows)



Edge $$2$$ ($$1 \frac{\;\;\;\;\;}{} 3$$) is not part of the graph.

The data is representing the following graph: ### Graph with cost and reverse_cost¶

The weighted directed graph, $$G_d(V,E)$$, is defined by:

• Graph data is obtained with a query

SELECT id, source, target, cost, reverse_cost FROM edges

• The set of edges $$E$$:

• E = \begin{split} \begin{align} & {\{(source_{id}, target_{id}, cost_{id}) \text{ when } cost_{id} >=0 \}} \\ & \cup \\ & {\{(target_{id}, source_{id}, reverse\_cost_{id}) \text{ when } reverse\_cost_{id} >=0 \}} \end{align} \end{split}

• Edges $$(source \rightarrow target)$$ where cost is non negative are part of the graph.

• Edges $$(target \rightarrow source)$$ where reverse_cost is non negative are part of the graph.

• The set of vertices $$V$$:

• $$V = \{source_{id} \cup target_{id}\}$$

• All vertices in source and target are part of the graph.

Directed graph

In a directed graph both edges have directionality

• edge $$(source_{id}, target_{id}, cost_{id})$$ has directionality: $$source_{id} \rightarrow target_{id}$$

• edge $$(target_{id}, source_{id}, reverse\_cost_{id})$$ has directionality: $$target_{id} \rightarrow source_{id}$$

For the following data:

SELECT *
FROM (VALUES (1, 1, 2, 5, 2), (2, 1, 3, -3, 4), (3, 2, 3, 7, -1))
AS t(id, source, target, cost, reverse_cost);
id | source | target | cost | reverse_cost
----+--------+--------+------+--------------
1 |      1 |      2 |    5 |            2
2 |      1 |      3 |   -3 |            4
3 |      2 |      3 |    7 |           -1
(3 rows)



Edges not part of the graph:

• $$2$$ ($$1 \rightarrow 3$$)

• $$3$$ ($$3 \rightarrow 2$$)

The data is representing the following graph: Undirected graph

In a directed graph both edges do not have directionality

• Edge $$(source_{id}, target_{id}, cost_{id})$$ is $$source_{id} \frac{\;\;\;\;\;}{} target_{id}$$

• Edge $$(target_{id}, source_{id}, reverse\_cost_{id})$$ is $$target_{id} \frac{\;\;\;\;\;}{} source_{id}$$

• In terms of a directed graph is like having four edges:

• $$source_i \leftrightarrow target_i$$

• $$target_i \leftrightarrow source_i$$

For the following data:

SELECT *
FROM (VALUES (1, 1, 2, 5, 2), (2, 1, 3, -3, 4), (3, 2, 3, 7, -1))
AS t(id, source, target, cost, reverse_cost);
id | source | target | cost | reverse_cost
----+--------+--------+------+--------------
1 |      1 |      2 |    5 |            2
2 |      1 |      3 |   -3 |            4
3 |      2 |      3 |    7 |           -1
(3 rows)



Edges not part of the graph:

• $$2$$ ($$1 \frac{\;\;\;\;\;}{} 3$$)

• $$3$$ ($$3 \frac{\;\;\;\;\;}{} 2$$)

The data is representing the following graph: ## Graphs without geometries¶

Personal relationships, genealogy, file dependency problems can be solved using pgRouting. Those problems, normally, do not come with geometries associated with the graph.

### Wiki example¶

Solve the example problem taken from wikipedia):

Where:

• Problem is to find the shortest path from $$1$$ to $$5$$.

• Is an undirected graph.

• Although visually looks like to have geometries, the drawing is not to scale.

• No geometries associated to the vertices or edges

• Has 6 vertices $$\{1,2,3,4,5,6\}$$

• Has 9 edges:

\begin{split} \begin{align} E = & \{(1,2,7), (1,3,9), (1,6,14), \\ & (2,3,10), (2,4,13), \\ & (3,4,11), (3,6,2), \\ & (4,5,6), \\ & (5,6,9) \} \end{align} \end{split}

• The graph can be represented in many ways for example: #### Prepare the database¶

Create a database for the example, access the database and install pgRouting:

$createdb wiki$ psql wiki
wiki =# CREATE EXTENSION pgRouting CASCADE;


#### Create a table¶

The basic elements needed to perform basic routing on an undirected graph are:

Column

Type

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)

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

Using this table design for this example:

CREATE TABLE wiki (
id SERIAL,
source INTEGER,
target INTEGER,
cost INTEGER);
CREATE TABLE


#### Insert the data¶

INSERT INTO wiki (source, target, cost) VALUES
(1, 2, 7),  (1, 3, 9), (1, 6, 14),
(2, 3, 10), (2, 4, 15),
(3, 6, 2),  (3, 4, 11),
(4, 5, 6),
(5, 6, 9);
INSERT 0 9


#### Find the shortest path¶

To solve this example pgr_dijkstra is used:

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM wiki',
1, 5, false);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         1 |       5 |    1 |    2 |    9 |        0
2 |        2 |         1 |       5 |    3 |    6 |    2 |        9
3 |        3 |         1 |       5 |    6 |    9 |    9 |       11
4 |        4 |         1 |       5 |    5 |   -1 |    0 |       20
(4 rows)



To go from $$1$$ to $$5$$ the path goes thru the following vertices: $$1 \rightarrow 3 \rightarrow 6 \rightarrow 5$$ #### Vertex information¶

To obtain the vertices information, use pgr_extractVertices – Proposed

SELECT id, in_edges, out_edges
FROM pgr_extractVertices('SELECT id, source, target FROM wiki');
id | in_edges | out_edges
----+----------+-----------
3 | {2,4}    | {6,7}
5 | {8}      | {9}
4 | {5,7}    | {8}
2 | {1}      | {4,5}
1 |          | {1,2,3}
6 | {3,6,9}  |
(6 rows)



## Graphs with geometries¶

### Create a routing Database¶

The first step is to create a database and load pgRouting in the database.

Typically create a database for each project.

Once having the database to work in, load your data and build the routing application in that database.

createdb sampledata
psql sampledata -c "CREATE EXTENSION pgrouting CASCADE"


There are various open source tools that can help, like:

shp2pgsql:

ogr2ogr:
• vector data conversion utility

osm2pgsql:
• load OSM data into postgresql

Please note that these tools will not import the data in a structure compatible with pgRouting and when this happens the topology needs to be adjusted.

• Breakup a segments on each segment-segment intersection

• When missing, add columns and assign values to source, target, cost, reverse_cost.

• Connect a disconnected graph.

• Create the complete graph topology

• Create one or more graphs based on the application to be developed.

• Create a contracted graph for the high speed roads

• Create graphs per state/country

In few words:

Prepare the graph

What and how to prepare the graph, will depend on the application and/or on the quality of the data and/or on how close the information is to have a topology usable by pgRouting and/or some other factors not mentioned.

The steps to prepare the graph involve geometry operations using PostGIS and some others involve graph operations like pgr_contraction to contract a graph.

The workshop has a step by step on how to prepare a graph using Open Street Map data, for a small application.

• Have the geometries indexed.

• Have the identifiers columns indexed.

Please consult the PostgreSQL documentation and the PostGIS documentation.

### Build a routing topology¶

The basic information to use the majority of the pgRouting functions id, source, target, cost, [reverse_cost] is what in pgRouting is called the routing topology.

reverse_cost is optional but strongly recommended to have in order to reduce the size of the database due to the size of the geometry columns. Having said that, in this documentation reverse_cost is used in this documentation.

When the data comes with geometries and there is no routing topology, then this step is needed.

All the start and end vertices of the geometries need an identifier that is to be stored in a source and target columns of the table of the data. Likewise, cost and reverse_cost need to have the value of traversing the edge in both directions.

If the columns do not exist they need to be added to the table in question. (see ALTER TABLE)

The function pgr_extractVertices – Proposed is used to create a vertices table based on the edge identifier and the geometry of the edge of the graph.

Finally using the data stored on the vertices tables the source and target are filled up.

See Sample Data for an example for building a topology.

Data coming from OSM and using osm2pgrouting as an import tool, comes with the routing topology. See an example of using osm2pgrouting on the workshop.

For this example the cost and reverse_cost values are going to be the double of the length of the geometry.

#### Update costs to length of geometry¶

Suppose that cost and reverse_cost columns in the sample data represent:

• $$1$$ when the edge exists in the graph

• $$-1$$ when the edge does not exist in the graph

Using that information updating to the length of the geometries:

UPDATE edges SET
cost = sign(cost) * ST_length(geom) * 2,
reverse_cost = sign(reverse_cost) * ST_length(geom) * 2;
UPDATE 18


Which gives the following results:

SELECT id, cost, reverse_cost FROM edges;
id |        cost        |    reverse_cost
----+--------------------+--------------------
6 |                  2 |                  2
7 |                  2 |                  2
4 |                  2 |                  2
5 |                  2 |                 -2
8 |                  2 |                  2
12 |                  2 |                 -2
11 |                  2 |                 -2
10 |                  2 |                  2
17 |     2.999999999998 |     2.999999999998
14 |                  2 |                  2
18 | 3.4000000000000004 | 3.4000000000000004
13 |                  2 |                 -2
15 |                  2 |                  2
16 |                  2 |                  2
9 |                  2 |                  2
3 |                 -2 |                  2
1 |                  2 |                  2
2 |                 -2 |                  2
(18 rows)



Note that to be able to follow the documentation examples, everything is based on the original graph.

Returning to the original data:

UPDATE edges SET
cost = sign(cost),
reverse_cost = sign(reverse_cost);
UPDATE 18


#### Update costs based on codes¶

Other datasets, can have a column with values like

• FT vehicle flow on the direction of the geometry

• TF vehicle flow opposite of the direction of the geometry

• B vehicle flow on both directions

Preparing a code column for the example:

ALTER TABLE edges ADD COLUMN direction TEXT;
ALTER TABLE
UPDATE edges SET
direction = CASE WHEN (cost>0 AND reverse_cost>0) THEN 'B'   /* both ways */
WHEN (cost>0 AND reverse_cost<0) THEN 'FT'  /* direction of the LINESSTRING */
WHEN (cost<0 AND reverse_cost>0) THEN 'TF'  /* reverse direction of the LINESTRING */
ELSE '' END;
UPDATE 18
/* unknown */


Adjusting the costs based on the codes:

UPDATE edges SET
cost = CASE WHEN (direction = 'B' OR direction = 'FT')
THEN ST_length(geom) * 2
ELSE -1 END,
reverse_cost = CASE WHEN (direction = 'B' OR direction = 'TF')
THEN ST_length(geom) * 2
ELSE -1 END;
UPDATE 18


Which gives the following results:

SELECT id, cost, reverse_cost FROM edges;
id |        cost        |    reverse_cost
----+--------------------+--------------------
6 |                  2 |                  2
7 |                  2 |                  2
4 |                  2 |                  2
5 |                  2 |                 -1
8 |                  2 |                  2
12 |                  2 |                 -1
11 |                  2 |                 -1
10 |                  2 |                  2
17 |     2.999999999998 |     2.999999999998
14 |                  2 |                  2
18 | 3.4000000000000004 | 3.4000000000000004
13 |                  2 |                 -1
15 |                  2 |                  2
16 |                  2 |                  2
9 |                  2 |                  2
3 |                 -1 |                  2
1 |                  2 |                  2
2 |                 -1 |                  2
(18 rows)



Returning to the original data:

UPDATE edges SET
cost = sign(cost),
reverse_cost = sign(reverse_cost);
UPDATE 18
ALTER TABLE edges DROP COLUMN direction;
ALTER TABLE


## Check the Routing Topology¶

There are lots of possible problems in a graph.

• The data used may not have been designed with routing in mind.

• A graph has some very specific requirements.

• The graph is disconnected.

• There are unwanted intersections.

• The graph is too large and needs to be contracted.

• A sub graph is needed for the application.

• and many other problems that the pgRouting user, that is the application developer might encounter.

### Crossing edges¶

To get the crossing edges:

SELECT a.id, b.id
FROM edges AS a, edges AS b
WHERE a.id < b.id AND st_crosses(a.geom, b.geom);
id | id
----+----
13 | 18
(1 row)



That information is correct, for example, when in terms of vehicles, is it a tunnel or bride crossing over another road.

It might be incorrect, for example:

1. When it is actually an intersection of roads, where vehicles can make turns.

2. When in terms of electrical lines, the electrical line is able to switch roads even on a tunnel or bridge.

When it is incorrect, it needs fixing:

1. For vehicles and pedestrians

• If the data comes from OSM and was imported to the database using osm2pgrouting, the fix needs to be done in the OSM portal and the data imported again.

• In general when the data comes from a supplier that has the data prepared for routing vehicles, and there is a problem, the data is to be fixed from the supplier

2. For very specific applications

• The data is correct when from the point of view of routing vehicles or pedestrians.

• The data needs a local fix for the specific application.

Once analyzed one by one the crossings, for the ones that need a local fix, the edges need to be split.

SELECT ST_AsText((ST_Dump(ST_Split(a.geom, b.geom))).geom)
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18
UNION
SELECT ST_AsText((ST_Dump(ST_Split(b.geom, a.geom))).geom)
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18;
st_astext
---------------------------
LINESTRING(3.5 2.3,3.5 3)
LINESTRING(3 3,3.5 3)
LINESTRING(3.5 3,4 3)
LINESTRING(3.5 3,3.5 4)
(4 rows)



The new edges need to be added to the edges table, the rest of the attributes need to be updated in the new edges, the old edges need to be removed and the routing topology needs to be updated.

For each pair of crossing edges a process similar to this one must be performed.

The columns inserted and the way are calculated are based on the application. For example, if the edges have a trait name, then that column is to be copied.

For pgRouting calculations

• factor based on the position of the intersection of the edges can be used to adjust the cost and reverse_cost columns.

• Capacity information, used on the Flow - Family of functions functions does not need to change when splitting edges.

WITH
first_edge AS (
SELECT (ST_Dump(ST_Split(a.geom, b.geom))).path,
(ST_Dump(ST_Split(a.geom, b.geom))).geom,
ST_LineLocatePoint(a.geom,ST_Intersection(a.geom,b.geom)) AS factor
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18),
first_segments AS (
SELECT path, first_edge.geom,
capacity, reverse_capacity,
CASE WHEN path=1 THEN factor * cost
ELSE (1 - factor) * cost END AS cost,
CASE WHEN path=1 THEN factor * reverse_cost
ELSE (1 - factor) * reverse_cost END AS reverse_cost
FROM first_edge , edges WHERE id = 13),
second_edge AS (
SELECT (ST_Dump(ST_Split(b.geom, a.geom))).path,
(ST_Dump(ST_Split(b.geom, a.geom))).geom,
ST_LineLocatePoint(b.geom,ST_Intersection(a.geom,b.geom)) AS factor
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18),
second_segments AS (
SELECT path, second_edge.geom,
capacity, reverse_capacity,
CASE WHEN path=1 THEN factor * cost
ELSE (1 - factor) * cost END AS cost,
CASE WHEN path=1 THEN factor * reverse_cost
ELSE (1 - factor) * reverse_cost END AS reverse_cost
FROM second_edge , edges WHERE id = 18),
all_segments AS (
SELECT * FROM first_segments
UNION
SELECT * FROM second_segments)
INSERT INTO edges
(capacity, reverse_capacity,
cost, reverse_cost,
x1, y1, x2, y2,
geom)
(SELECT capacity, reverse_capacity, cost, reverse_cost,
ST_X(ST_StartPoint(geom)), ST_Y(ST_StartPoint(geom)),
ST_X(ST_EndPoint(geom)), ST_Y(ST_EndPoint(geom)),
geom
FROM all_segments);
INSERT 0 4


After adding all the split edges required by the application, the newly created vertices need to be added to the vertices table.

INSERT INTO vertices (in_edges, out_edges, x, y, geom)
(SELECT nv.in_edges, nv.out_edges, nv.x, nv.y, nv.geom
FROM pgr_extractVertices('SELECT id, geom FROM edges') AS nv
LEFT JOIN vertices AS v USING(geom) WHERE v.geom IS NULL);
INSERT 0 1


#### Updating edges topology¶

/* -- set the source information */
UPDATE edges AS e
SET source = v.id
FROM vertices AS v
WHERE source IS NULL AND ST_StartPoint(e.geom) = v.geom;
UPDATE 4
/* -- set the target information */
UPDATE edges AS e
SET target = v.id
FROM vertices AS v
WHERE target IS NULL AND ST_EndPoint(e.geom) = v.geom;
UPDATE 4


#### Removing the surplus edges¶

Once all significant information needed by the application has been transported to the new edges, then the crossing edges can be deleted.

DELETE FROM edges WHERE id IN (13, 18);
DELETE 2


There are other options to do this task, like creating a view, or a materialized view.

#### Updating vertices topology¶

To keep the graph consistent, the vertices topology needs to be updated

UPDATE vertices AS v SET
in_edges = nv.in_edges, out_edges = nv.out_edges
FROM (SELECT * FROM pgr_extractVertices('SELECT id, geom FROM edges')) AS nv
WHERE v.geom = nv.geom;
UPDATE 18


#### Checking for crossing edges¶

There are no crossing edges on the graph.

SELECT a.id, b.id
FROM edges AS a, edges AS b
WHERE a.id < b.id AND st_crosses(a.geom, b.geom);
id | id
----+----
(0 rows)



### Disconnected graphs¶

To get the graph connectivity:

SELECT * FROM pgr_connectedComponents(
'SELECT id, source, target, cost, reverse_cost FROM edges'
);
seq | component | node
-----+-----------+------
1 |         1 |    1
2 |         1 |    3
3 |         1 |    5
4 |         1 |    6
5 |         1 |    7
6 |         1 |    8
7 |         1 |    9
8 |         1 |   10
9 |         1 |   11
10 |         1 |   12
11 |         1 |   13
12 |         1 |   14
13 |         1 |   15
14 |         1 |   16
15 |         1 |   17
16 |         1 |   18
17 |         2 |    2
18 |         2 |    4
(18 rows)



In this example, the component $$2$$ consists of vertices $$\{2, 4\}$$ and both vertices are also part of the dead end result set.

This graph needs to be connected.

Note

With the original graph of this documentation, there would be 3 components as the crossing edge in this graph is a different component.

#### Prepare storage for connection information¶

ALTER TABLE vertices ADD COLUMN component BIGINT;
ALTER TABLE
ALTER TABLE edges ADD COLUMN component BIGINT;
ALTER TABLE


#### Save the vertices connection information¶

UPDATE vertices SET component = c.component
FROM (SELECT * FROM pgr_connectedComponents(
'SELECT id, source, target, cost, reverse_cost FROM edges'
)) AS c
WHERE id = node;
UPDATE 18


#### Save the edges connection information¶

UPDATE edges SET component = v.component
FROM (SELECT id, component FROM vertices) AS v
WHERE source = v.id;
UPDATE 20


#### Get the closest vertex¶

Using pgr_findCloseEdges the closest vertex to component $$1$$ is vertex $$4$$. And the closest edge to vertex $$4$$ is edge $$14$$.

SELECT edge_id, fraction, ST_AsText(edge) AS edge, id AS closest_vertex
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges WHERE component = 1$$,
(SELECT array_agg(geom) FROM vertices WHERE component = 2),
2, partial => false) JOIN vertices USING (geom) ORDER BY distance LIMIT 1;
edge_id | fraction |                 edge                 | closest_vertex
---------+----------+--------------------------------------+----------------
14 |      0.5 | LINESTRING(1.999999999999 3.5,2 3.5) |              4
(1 row)



The edge can be used to connect the components, using the fraction information about the edge $$14$$ to split the connecting edge.

#### Connecting components¶

There are three basic ways to connect the components

• From the vertex to the starting point of the edge

• From the vertex to the ending point of the edge

• From the vertex to the closest vertex on the edge

• This solution requires the edge to be split.

The following query shows the three ways to connect the components:

WITH
info AS (
SELECT
edge_id, fraction, side, distance, ce.geom, edge, v.id AS closest,
source, target, capacity, reverse_capacity, e.geom AS e_geom
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges WHERE component = 1$$,
(SELECT array_agg(geom) FROM vertices WHERE component = 2),
2, partial => false) AS ce
JOIN vertices AS v USING (geom)
JOIN edges AS e ON (edge_id = e.id)
ORDER BY distance LIMIT 1),
three_options AS (
SELECT
closest AS source, target, 0 AS cost, 0 AS reverse_cost,
capacity, reverse_capacity,
ST_X(geom) AS x1, ST_Y(geom) AS y1,
ST_X(ST_EndPoint(e_geom)) AS x2, ST_Y(ST_EndPoint(e_geom)) AS y2,
ST_MakeLine(geom, ST_EndPoint(e_geom)) AS geom
FROM info
UNION
SELECT closest, source, 0, 0, capacity, reverse_capacity,
ST_X(geom) AS x1, ST_Y(geom) AS y1,
ST_X(ST_StartPoint(e_geom)) AS x2, ST_Y(ST_StartPoint(e_geom)) AS y2,
ST_MakeLine(info.geom, ST_StartPoint(e_geom))
FROM info
/*
UNION
-- This option requires splitting the edge
SELECT closest, NULL, 0, 0, capacity, reverse_capacity,
ST_X(geom) AS x1, ST_Y(geom) AS y1,
ST_X(ST_EndPoint(edge)) AS x2, ST_Y(ST_EndPoint(edge)) AS y2,
edge
FROM info */
)
INSERT INTO edges
(source, target,
cost, reverse_cost,
capacity, reverse_capacity,
x1, y1, x2, y2,
geom)
(SELECT
source, target, cost, reverse_cost, capacity, reverse_capacity,
x1, y1, x2, y2, geom
FROM three_options);
INSERT 0 2


#### Checking components¶

Ignoring the edge that requires further work. The graph is now fully connected as there is only one component.

SELECT * FROM pgr_connectedComponents(
'SELECT id, source, target, cost, reverse_cost FROM edges'
);
seq | component | node
-----+-----------+------
1 |         1 |    1
2 |         1 |    2
3 |         1 |    3
4 |         1 |    4
5 |         1 |    5
6 |         1 |    6
7 |         1 |    7
8 |         1 |    8
9 |         1 |    9
10 |         1 |   10
11 |         1 |   11
12 |         1 |   12
13 |         1 |   13
14 |         1 |   14
15 |         1 |   15
16 |         1 |   16
17 |         1 |   17
18 |         1 |   18
(18 rows)



### Contraction of a graph¶

The graph can be reduced in size using Contraction - Family of functions

When to contract will depend on the size of the graph, processing times, correctness of the data, on the final application, or any other factor not mentioned.

A fairly good method of finding out if contraction can be useful is because of the number of dead ends and/or the number of linear edges.

A complete method on how to contract and how to use the contracted graph is described on Contraction - Family of functions

SELECT id FROM vertices
WHERE array_length(in_edges || out_edges, 1) = 1;
id
----
1
5
9
13
14
2
4
(7 rows)



That information is correct, for example, when the dead end is on the limit of the imported graph.

Visually node $$4$$ looks to be as start/ending of 3 edges, but it is not.

Is that correct?

• Is there such a small curb:

• That does not allow a vehicle to use that visual intersection?

• Is the application for pedestrians and therefore the pedestrian can easily walk on the small curb?

• Is the application for the electricity and the electrical lines than can easily be extended on top of the small curb?

• Is there a big cliff and from eagles view look like the dead end is close to the segment?

When there are many dead ends, to speed up, the Contraction - Family of functions functions can be used to divide the problem.

#### Linear edges¶

To get the linear edges:

SELECT id FROM vertices
WHERE array_length(in_edges || out_edges, 1) = 2;
id
----
3
15
17
(3 rows)



This information is correct, for example, when the application is taking into account speed bumps, stop signals.

When there are many linear edges, to speed up, the Contraction - Family of functions functions can be used to divide the problem.

## Function’s structure¶

Once the graph preparation work has been done above, it is time to use a

The general form of a pgRouting function call is:

pgr_<name>(Inner queries, parameters, [ Optional parameters)

Where:

• Inner queries: Are compulsory parameters that are TEXT strings containing SQL queries.

• parameters: Additional compulsory parameters needed by the function.

• Optional parameters: Are non compulsory named parameters that have a default value when omitted.

The compulsory parameters are positional parameters, the optional parameters are named parameters.

For example, for this pgr_dijkstra signature:

pgr_dijkstra(Edges SQL, start vid, end vid [, directed])

• Is the first parameter.

• It is compulsory.

• It is an inner query.

• It has no name, so Edges SQL gives an idea of what kind of inner query needs to be used

• start vid:

• Is the second parameter.

• It is compulsory.

• It has no name, so start vid gives an idea of what the second parameter’s value should contain.

• end vid

• Is the third parameter.

• It is compulsory.

• It has no name, so end vid gives an idea of what the third parameter’s value should contain

• directed

• Is the fourth parameter.

• It is optional.

• It has a name.

The full description of the parameters are found on the Parameters section of each function.

A function might have different overloads. The most common are called:

Depending on the overload the parameters types change.

• One: ANY-INTEGER

• Many: ARRAY [ANY-INTEGER]

Depending of the function the overloads may vary. But the concept of parameter type change remains the same.

### One to One¶

When routing from:

• From one starting vertex

• to one ending vertex

### One to Many¶

When routing from:

• From one starting vertex

• to many ending vertices

### Many to One¶

When routing from:

• From many starting vertices

• to one ending vertex

### Many to Many¶

When routing from:

• From many starting vertices

• to many ending vertices

### Combinations¶

When routing from:

• From many different starting vertices

• to many different ending vertices

• Every tuple specifies a pair of a start vertex and an end vertex

• Users can define the combinations as desired.

• Needs a Combinations SQL

## Inner Queries¶

There are several kinds of valid inner queries and also the columns returned are depending of the function. Which kind of inner query will depend on the function(s) requirements. To simplify variety of types, ANY-INTEGER and ANY-NUMERICAL is used.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

### Edges SQL¶

#### General¶

Edges SQL for

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 (target, source)

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

#### General without id¶

Edges SQL for

Column

Type

Default

Description

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 (target, source)

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

#### General with (X,Y)¶

Edges SQL for

Parameter

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)

• When negative: edge (source, target) does not exist, therefore it’s not part of the graph.

reverse_cost

ANY-NUMERICAL

-1

Weight of the edge (target, source),

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

x1

ANY-NUMERICAL

X coordinate of source vertex.

y1

ANY-NUMERICAL

Y coordinate of source vertex.

x2

ANY-NUMERICAL

X coordinate of target vertex.

y2

ANY-NUMERICAL

Y coordinate of target vertex.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

#### Flow¶

Edges SQL for Flow - Family of functions

Edges SQL for

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.

capacity

ANY-INTEGER

Weight of the edge (source, target)

reverse_capacity

ANY-INTEGER

-1

Weight of the edge (target, source)

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

Edges SQL for the following functions of Flow - Family of functions

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.

capacity

ANY-INTEGER

Capacity of the edge (source, target)

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

reverse_capacity

ANY-INTEGER

-1

Capacity of the edge (target, source)

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

cost

ANY-NUMERICAL

Weight of the edge (source, target) if it exist

reverse_cost

ANY-NUMERICAL

$$-1$$

Weight of the edge (target, source) if it exist

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

### Combinations SQL¶

Used on combination signatures

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

### Restrictions SQL¶

Column

Type

Description

path

ARRAY [ANY-INTEGER]

Sequence of edge identifiers that form a path that is not allowed to be taken. - Empty arrays or NULL arrays are ignored. - Arrays that have a NULL element will raise an exception.

Cost

ANY-NUMERICAL

Cost of taking the forbidden path.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

### Points SQL¶

Points SQL for

Parameter

Type

Default

Description

pid

ANY-INTEGER

value

Identifier of the point.

• Use with positive value, as internally will be converted to negative value

• If column is present, it can not be NULL.

• If column is not present, a sequential negative value will be given automatically.

edge_id

ANY-INTEGER

Identifier of the “closest” edge to the point.

fraction

ANY-NUMERICAL

Value in <0,1> that indicates the relative postition from the first end point of the edge.

side

CHAR

b

Value in [b, r, l, NULL] indicating if the point is:

• In the right r,

• In the left l,

• In both sides b, NULL

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

## Parameters¶

The main parameter of the majority of the pgRouting functions is a query that selects the edges of the graph.

Parameter

Type

Description

Edges SQL

TEXT

Edges SQL as described below.

Depending on the family or category of a function it will have additional parameters, some of them are compulsory and some are optional.

The compulsory parameters are nameless and must be given in the required order. The optional parameters are named parameters and will have a default value.

### Parameters for the Via functions¶

Parameter

Type

Default

Description

Edges SQL

TEXT

SQL query as described.

via vertices

ARRAY [ANY-INTEGER]

Array of ordered vertices identifiers that are going to be visited.

directed

BOOLEAN

true

• When true Graph is considered Directed

• When false the graph is considered as Undirected.

strict

BOOLEAN

false

• When true if a path is missing stops and returns EMPTY SET

• When false ignores missing paths returning all paths found

U_turn_on_edge

BOOLEAN

true

• When true departing from a visited vertex will not try to avoid using the edge used to reach it. In other words, U turn using the edge with same identifier is allowed.

• When false when a departing from a visited vertex tries to avoid using the edge used to reach it. In other words, U turn using the edge with same identifier is used when no other path is found.

### For the TRSP functions¶

Column

Type

Description

Edges SQL

TEXT

SQL query as described.

Restrictions SQL

TEXT

SQL query as described.

Combinations SQL

TEXT

Combinations SQL as described below

start vid

ANY-INTEGER

Identifier of the departure vertex.

start vids

ARRAY [ANY-INTEGER]

Array of identifiers of destination vertices.

end vid

ANY-INTEGER

Identifier of the departure vertex.

end vids

ARRAY [ANY-INTEGER]

Array of identifiers of destination vertices.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

## Return columns¶

There are several kinds of columns returned are depending of the function.

### Return columns for a path¶

Used on functions that return one path solution

Returns set of (seq, path_seq [, start_vid] [, end_vid], node, edge, cost, agg_cost)

Column

Type

Description

seq

INTEGER

Sequential value starting from 1.

path_seq

INTEGER

Relative position in the path. Has value 1 for the beginning of a path.

start_vid

BIGINT

Identifier of the starting vertex. Returned when multiple starting vetrices are in the query.

end_vid

BIGINT

Identifier of the ending vertex. Returned when multiple ending vertices are in the query.

node

BIGINT

Identifier of the node in the path from start_vid to end_vid.

edge

BIGINT

Identifier of the edge used to go from node to the next node in the path sequence. -1 for the last node of the path.

cost

FLOAT

Cost to traverse from node using edge to the next node in the path sequence.

agg_cost

FLOAT

Aggregate cost from start_vid to node.

Used on functions the following:

Returns set of (seq, path_seq [, start_pid] [, end_pid], node, edge, cost, agg_cost)

Column

Type

Description

seq

INTEGER

Sequential value starting from 1.

path_seq

INTEGER

Relative position in the path.

• 1 For the first row of the path.

start_pid

BIGINT

Identifier of a starting vertex/point of the path.

• When positive is the identifier of the starting vertex.

• When negative is the identifier of the starting point.

• Returned on Many to One and Many to Many

end_pid

BIGINT

Identifier of an ending vertex/point of the path.

• When positive is the identifier of the ending vertex.

• When negative is the identifier of the ending point.

• Returned on One to Many and Many to Many

node

BIGINT

Identifier of the node in the path from start_pid to end_pid.

• When positive is the identifier of the a vertex.

• When negative is the identifier of the a point.

edge

BIGINT

Identifier of the edge used to go from node to the next node in the path sequence.

• -1 for the last row of the path.

cost

FLOAT

Cost to traverse from node using edge to the next node in the path sequence.

• 0 For the first row of the path.

agg_cost

FLOAT

Aggregate cost from start_vid to node.

• 0 For the first row of the path.

Used on functions the following:

Returns (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)

Column

Type

Description

seq

INTEGER

Sequential value starting from 1.

path_seq

INTEGER

Relative position in the path. Has value 1 for the beginning of a path.

start_vid

BIGINT

Identifier of the starting vertex of the current path.

end_vid

BIGINT

Identifier of the ending vertex of the current path.

node

BIGINT

Identifier of the node in the path from start_vid to end_vid.

edge

BIGINT

Identifier of the edge used to go from node to the next node in the path sequence. -1 for the last node of the path.

cost

FLOAT

Cost to traverse from node using edge to the next node in the path sequence.

agg_cost

FLOAT

Aggregate cost from start_vid to node.

### Multiple paths¶

#### Selective for multiple paths.¶

The columns depend on the function call.

Set of (seq, path_id, path_seq [, start_vid] [, end_vid], node, edge, cost, agg_cost)

Column

Type

Description

seq

INTEGER

Sequential value starting from 1.

path_id

INTEGER

Path identifier.

• Has value 1 for the first of a path from start_vid to end_vid.

path_seq

INTEGER

Relative position in the path. Has value 1 for the beginning of a path.

start_vid

BIGINT

Identifier of the starting vertex. Returned when multiple starting vetrices are in the query.

end_vid

BIGINT

Identifier of the ending vertex. Returned when multiple ending vertices are in the query.

node

BIGINT

Identifier of the node in the path from start_vid to end_vid.

edge

BIGINT

Identifier of the edge used to go from node to the next node in the path sequence. -1 for the last node of the path.

cost

FLOAT

Cost to traverse from node using edge to the next node in the path sequence.

agg_cost

FLOAT

Aggregate cost from start_vid to node.

#### Non selective for multiple paths¶

Regardless of the call, al the columns are returned.

Returns set of (seq, path_id, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)

Column

Type

Description

seq

INTEGER

Sequential value starting from 1.

path_id

INTEGER

Path identifier.

• Has value 1 for the first of a path from start_vid to end_vid.

path_seq

INTEGER

Relative position in the path. Has value 1 for the beginning of a path.

start_vid

BIGINT

Identifier of the starting vertex.

end_vid

BIGINT

Identifier of the ending vertex.

node

BIGINT

Identifier of the node in the path from start_vid to end_vid.

edge

BIGINT

Identifier of the edge used to go from node to the next node in the path sequence. -1 for the last node of the path.

cost

FLOAT

Cost to traverse from node using edge to the next node in the path sequence.

agg_cost

FLOAT

Aggregate cost from start_vid to node.

### Return columns for cost functions¶

Used in the following

Set of (start_vid, end_vid, agg_cost)

Column

Type

Description

start_vid

BIGINT

Identifier of the starting vertex.

end_vid

BIGINT

Identifier of the ending vertex.

agg_cost

FLOAT

Aggregate cost from start_vid to end_vid.

Note

When start_vid or end_vid columns have negative values, the identifier is for a Point.

### Return columns for flow functions¶

Edges SQL for the following

Column

Type

Description

seq

INT

Sequential value starting from 1.

edge

BIGINT

Identifier of the edge in the original query (edges_sql).

start_vid

BIGINT

Identifier of the first end point vertex of the edge.

end_vid

BIGINT

Identifier of the second end point vertex of the edge.

flow

BIGINT

Flow through the edge in the direction (start_vid, end_vid).

residual_capacity

BIGINT

Residual capacity of the edge in the direction (start_vid, end_vid).

Edges SQL for the following functions of Flow - Family of functions

Column

Type

Description

seq

INT

Sequential value starting from 1.

edge

BIGINT

Identifier of the edge in the original query (edges_sql).

source

BIGINT

Identifier of the first end point vertex of the edge.

target

BIGINT

Identifier of the second end point vertex of the edge.

flow

BIGINT

Flow through the edge in the direction (source, target).

residual_capacity

BIGINT

Residual capacity of the edge in the direction (source, target).

cost

FLOAT

The cost of sending this flow through the edge in the direction (source, target).

agg_cost

FLOAT

The aggregate cost.

### Return columns for spanning tree functions¶

Edges SQL for the following

Returns SET OF (edge, cost)

Column

Type

Description

edge

BIGINT

Identifier of the edge.

cost

FLOAT

Cost to traverse the edge.

## Performance Tips¶

### For the Routing functions¶

To get faster results bound the queries to an area of interest of routing.

In this example Use an inner query SQL that does not include some edges in the routing function and is within the area of the results.

SELECT * FROM pgr_dijkstra($$SELECT id, source, target, cost, reverse_cost from edges WHERE geom && (SELECT st_buffer(geom, 1) AS myarea FROM edges WHERE id = 2)$$,
1, 2);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
(0 rows)



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