# pgr_dijkstra¶

pgr_dijkstra — Shortest path(s) using Dijkstra algorithm.

Availability

• Version 3.1.0

• New Proposed functions:

• Version 3.0.0

• Official functions

• Version 2.2.0

• Version 2.1.0

• Signature change on pgr_dijkstra (One to One)

• Version 2.0.0

• Official pgr_dijkstra (One to One)

## Description¶

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.

• 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 $$\infty$$

• For optimization purposes, any duplicated value in the starting vertices or on the ending vertices are ignored.

• Running time: $$O(| start\ vids | * (V \log V + E))$$

## Signatures¶

Summary

pgr_dijkstra(Edges SQL, start vid, end vid , [directed])
pgr_dijkstra(Edges SQL, start vid, end vids , [directed])
pgr_dijkstra(Edges SQL, start vids, end vid , [directed])
pgr_dijkstra(Edges SQL, start vids, end vids , [directed])
pgr_dijkstra(Edges SQL, Combinations SQL , [directed])
RETURNS SET OF (seq, path_seq, [start_vid], [end_vid], node, edge, cost, agg_cost)
OR EMPTY SET

### One to One¶

pgr_dijkstra(Edges SQL, start vid, end vid , [directed])
RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost)
OR EMPTY SET
Example:

From vertex $$6$$ to vertex $$10$$ on a directed graph

SELECT * FROM pgr_Dijkstra(
'select id, source, target, cost, reverse_cost from edges',
6, 10, true);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |    8 |    1 |        1
3 |        3 |   11 |    9 |    1 |        2
4 |        4 |   16 |   16 |    1 |        3
5 |        5 |   15 |    3 |    1 |        4
6 |        6 |   10 |   -1 |    0 |        5
(6 rows)



### One to Many¶

pgr_dijkstra(Edges SQL, start vid, end vids , [directed])
RETURNS SET OF (seq, path_seq, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET
Example:

From vertex $$6$$ to vertices $$\{10, 17\}$$ on a directed

SELECT * FROM pgr_Dijkstra(
'select id, source, target, cost, reverse_cost from edges',
6, ARRAY[10, 17]);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 |        1 |      10 |    6 |    4 |    1 |        0
2 |        2 |      10 |    7 |    8 |    1 |        1
3 |        3 |      10 |   11 |    9 |    1 |        2
4 |        4 |      10 |   16 |   16 |    1 |        3
5 |        5 |      10 |   15 |    3 |    1 |        4
6 |        6 |      10 |   10 |   -1 |    0 |        5
7 |        1 |      17 |    6 |    4 |    1 |        0
8 |        2 |      17 |    7 |    8 |    1 |        1
9 |        3 |      17 |   11 |    9 |    1 |        2
10 |        4 |      17 |   16 |   15 |    1 |        3
11 |        5 |      17 |   17 |   -1 |    0 |        4
(11 rows)



### Many to One¶

pgr_dijkstra(Edges SQL, start vids, end vid , [directed])
RETURNS SET OF (seq, path_seq, start_vid, node, edge, cost, agg_cost)
OR EMPTY SET
Example:

From vertices $$\{6, 1\}$$ to vertex $$17$$ on a directed graph

SELECT * FROM pgr_Dijkstra(
'select id, source, target, cost, reverse_cost from edges',
ARRAY[6, 1], 17);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 |        1 |         1 |    1 |    6 |    1 |        0
2 |        2 |         1 |    3 |    7 |    1 |        1
3 |        3 |         1 |    7 |    8 |    1 |        2
4 |        4 |         1 |   11 |   11 |    1 |        3
5 |        5 |         1 |   12 |   13 |    1 |        4
6 |        6 |         1 |   17 |   -1 |    0 |        5
7 |        1 |         6 |    6 |    4 |    1 |        0
8 |        2 |         6 |    7 |    8 |    1 |        1
9 |        3 |         6 |   11 |   11 |    1 |        2
10 |        4 |         6 |   12 |   13 |    1 |        3
11 |        5 |         6 |   17 |   -1 |    0 |        4
(11 rows)



### Many to Many¶

pgr_dijkstra(Edges SQL, start vids, end vids , [directed])
RETURNS SET OF (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET
Example:

From vertices $$\{6, 1\}$$ to vertices $$\{10, 17\}$$ on an undirected graph

SELECT * FROM pgr_Dijkstra(
'select id, source, target, cost, reverse_cost from edges',
ARRAY[6, 1], ARRAY[10, 17],
directed => false);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         1 |      10 |    1 |    6 |    1 |        0
2 |        2 |         1 |      10 |    3 |    7 |    1 |        1
3 |        3 |         1 |      10 |    7 |    4 |    1 |        2
4 |        4 |         1 |      10 |    6 |    2 |    1 |        3
5 |        5 |         1 |      10 |   10 |   -1 |    0 |        4
6 |        1 |         1 |      17 |    1 |    6 |    1 |        0
7 |        2 |         1 |      17 |    3 |    7 |    1 |        1
8 |        3 |         1 |      17 |    7 |    8 |    1 |        2
9 |        4 |         1 |      17 |   11 |    9 |    1 |        3
10 |        5 |         1 |      17 |   16 |   15 |    1 |        4
11 |        6 |         1 |      17 |   17 |   -1 |    0 |        5
12 |        1 |         6 |      10 |    6 |    2 |    1 |        0
13 |        2 |         6 |      10 |   10 |   -1 |    0 |        1
14 |        1 |         6 |      17 |    6 |    4 |    1 |        0
15 |        2 |         6 |      17 |    7 |    8 |    1 |        1
16 |        3 |         6 |      17 |   11 |   11 |    1 |        2
17 |        4 |         6 |      17 |   12 |   13 |    1 |        3
18 |        5 |         6 |      17 |   17 |   -1 |    0 |        4
(18 rows)



### Combinations¶

pgr_dijkstra(Edges SQL, Combinations SQL , [directed])
RETURNS SET OF (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET
Example:

Using a combinations table on an undirected graph

The combinations table:

SELECT source, target FROM combinations;
source | target
--------+--------
5 |      6
5 |     10
6 |      5
6 |     15
6 |     14
(5 rows)



The query:

SELECT * FROM pgr_Dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
'SELECT source, target FROM combinations',
false);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         5 |       6 |    5 |    1 |    1 |        0
2 |        2 |         5 |       6 |    6 |   -1 |    0 |        1
3 |        1 |         5 |      10 |    5 |    1 |    1 |        0
4 |        2 |         5 |      10 |    6 |    2 |    1 |        1
5 |        3 |         5 |      10 |   10 |   -1 |    0 |        2
6 |        1 |         6 |       5 |    6 |    1 |    1 |        0
7 |        2 |         6 |       5 |    5 |   -1 |    0 |        1
8 |        1 |         6 |      15 |    6 |    2 |    1 |        0
9 |        2 |         6 |      15 |   10 |    3 |    1 |        1
10 |        3 |         6 |      15 |   15 |   -1 |    0 |        2
(10 rows)



## 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

• When true the graph is considered Directed

• When false the graph is considered as Undirected.

## 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 (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

### 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

## Result Columns¶

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.

Example:

Demonstration of repeated values are ignored, and result is sorted.

SELECT * FROM pgr_Dijkstra(
'select id, source, target, cost, reverse_cost from edges',
ARRAY[7, 10, 15, 10, 10, 15], ARRAY[10, 7, 10, 15]);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         7 |      10 |    7 |    8 |    1 |        0
2 |        2 |         7 |      10 |   11 |    9 |    1 |        1
3 |        3 |         7 |      10 |   16 |   16 |    1 |        2
4 |        4 |         7 |      10 |   15 |    3 |    1 |        3
5 |        5 |         7 |      10 |   10 |   -1 |    0 |        4
6 |        1 |         7 |      15 |    7 |    8 |    1 |        0
7 |        2 |         7 |      15 |   11 |    9 |    1 |        1
8 |        3 |         7 |      15 |   16 |   16 |    1 |        2
9 |        4 |         7 |      15 |   15 |   -1 |    0 |        3
10 |        1 |        10 |       7 |   10 |    5 |    1 |        0
11 |        2 |        10 |       7 |   11 |    8 |    1 |        1
12 |        3 |        10 |       7 |    7 |   -1 |    0 |        2
13 |        1 |        10 |      15 |   10 |    5 |    1 |        0
14 |        2 |        10 |      15 |   11 |    9 |    1 |        1
15 |        3 |        10 |      15 |   16 |   16 |    1 |        2
16 |        4 |        10 |      15 |   15 |   -1 |    0 |        3
17 |        1 |        15 |       7 |   15 |   16 |    1 |        0
18 |        2 |        15 |       7 |   16 |    9 |    1 |        1
19 |        3 |        15 |       7 |   11 |    8 |    1 |        2
20 |        4 |        15 |       7 |    7 |   -1 |    0 |        3
21 |        1 |        15 |      10 |   15 |    3 |    1 |        0
22 |        2 |        15 |      10 |   10 |   -1 |    0 |        1
(22 rows)


Example 2:

Making start_vids the same as end_vids

SELECT * FROM pgr_Dijkstra(
'select id, source, target, cost, reverse_cost from edges',
ARRAY[7, 10, 15], ARRAY[7, 10, 15]);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         7 |      10 |    7 |    8 |    1 |        0
2 |        2 |         7 |      10 |   11 |    9 |    1 |        1
3 |        3 |         7 |      10 |   16 |   16 |    1 |        2
4 |        4 |         7 |      10 |   15 |    3 |    1 |        3
5 |        5 |         7 |      10 |   10 |   -1 |    0 |        4
6 |        1 |         7 |      15 |    7 |    8 |    1 |        0
7 |        2 |         7 |      15 |   11 |    9 |    1 |        1
8 |        3 |         7 |      15 |   16 |   16 |    1 |        2
9 |        4 |         7 |      15 |   15 |   -1 |    0 |        3
10 |        1 |        10 |       7 |   10 |    5 |    1 |        0
11 |        2 |        10 |       7 |   11 |    8 |    1 |        1
12 |        3 |        10 |       7 |    7 |   -1 |    0 |        2
13 |        1 |        10 |      15 |   10 |    5 |    1 |        0
14 |        2 |        10 |      15 |   11 |    9 |    1 |        1
15 |        3 |        10 |      15 |   16 |   16 |    1 |        2
16 |        4 |        10 |      15 |   15 |   -1 |    0 |        3
17 |        1 |        15 |       7 |   15 |   16 |    1 |        0
18 |        2 |        15 |       7 |   16 |    9 |    1 |        1
19 |        3 |        15 |       7 |   11 |    8 |    1 |        2
20 |        4 |        15 |       7 |    7 |   -1 |    0 |        3
21 |        1 |        15 |      10 |   15 |    3 |    1 |        0
22 |        2 |        15 |      10 |   10 |   -1 |    0 |        1
(22 rows)


Example:

Manually assigned vertex combinations.

SELECT * FROM pgr_Dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
'SELECT * FROM (VALUES (6, 10), (6, 7), (12, 10)) AS combinations (source, target)');
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
3 |        1 |         6 |      10 |    6 |    4 |    1 |        0
4 |        2 |         6 |      10 |    7 |    8 |    1 |        1
5 |        3 |         6 |      10 |   11 |    9 |    1 |        2
6 |        4 |         6 |      10 |   16 |   16 |    1 |        3
7 |        5 |         6 |      10 |   15 |    3 |    1 |        4
8 |        6 |         6 |      10 |   10 |   -1 |    0 |        5
9 |        1 |        12 |      10 |   12 |   13 |    1 |        0
10 |        2 |        12 |      10 |   17 |   15 |    1 |        1
11 |        3 |        12 |      10 |   16 |   16 |    1 |        2
12 |        4 |        12 |      10 |   15 |    3 |    1 |        3
13 |        5 |        12 |      10 |   10 |   -1 |    0 |        4
(13 rows)



The examples of this section are based on the Sample Data network.

### For directed graphs with cost and reverse_cost columns¶

#### 1) Path from $$6$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, 10
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |    8 |    1 |        1
3 |        3 |   11 |    9 |    1 |        2
4 |        4 |   16 |   16 |    1 |        3
5 |        5 |   15 |    3 |    1 |        4
6 |        6 |   10 |   -1 |    0 |        5
(6 rows)



#### 2) Path from $$6$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, 7
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |   -1 |    0 |        1
(2 rows)



#### 3) Path from $$12$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
12, 10
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |   12 |   13 |    1 |        0
2 |        2 |   17 |   15 |    1 |        1
3 |        3 |   16 |   16 |    1 |        2
4 |        4 |   15 |    3 |    1 |        3
5 |        5 |   10 |   -1 |    0 |        4
(5 rows)



#### 4) Path from $$12$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
12, 7
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |   12 |   13 |    1 |        0
2 |        2 |   17 |   15 |    1 |        1
3 |        3 |   16 |    9 |    1 |        2
4 |        4 |   11 |    8 |    1 |        3
5 |        5 |    7 |   -1 |    0 |        4
(5 rows)



#### 5) Using One to Many to get the solution of examples 1 and 2¶

Paths $$\{6\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, ARRAY[10, 7]
);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 |        1 |       7 |    6 |    4 |    1 |        0
2 |        2 |       7 |    7 |   -1 |    0 |        1
3 |        1 |      10 |    6 |    4 |    1 |        0
4 |        2 |      10 |    7 |    8 |    1 |        1
5 |        3 |      10 |   11 |    9 |    1 |        2
6 |        4 |      10 |   16 |   16 |    1 |        3
7 |        5 |      10 |   15 |    3 |    1 |        4
8 |        6 |      10 |   10 |   -1 |    0 |        5
(8 rows)



#### 6) Using Many to One to get the solution of examples 2 and 4¶

Paths $$\{6, 12\}\rightarrow\{7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6, 12], 7
);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 |        1 |         6 |    6 |    4 |    1 |        0
2 |        2 |         6 |    7 |   -1 |    0 |        1
3 |        1 |        12 |   12 |   13 |    1 |        0
4 |        2 |        12 |   17 |   15 |    1 |        1
5 |        3 |        12 |   16 |    9 |    1 |        2
6 |        4 |        12 |   11 |    8 |    1 |        3
7 |        5 |        12 |    7 |   -1 |    0 |        4
(7 rows)



#### 7) Using Many to Many to get the solution of examples 1 to 4¶

Paths $$\{6, 12\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6, 12], ARRAY[10,7]
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
3 |        1 |         6 |      10 |    6 |    4 |    1 |        0
4 |        2 |         6 |      10 |    7 |    8 |    1 |        1
5 |        3 |         6 |      10 |   11 |    9 |    1 |        2
6 |        4 |         6 |      10 |   16 |   16 |    1 |        3
7 |        5 |         6 |      10 |   15 |    3 |    1 |        4
8 |        6 |         6 |      10 |   10 |   -1 |    0 |        5
9 |        1 |        12 |       7 |   12 |   13 |    1 |        0
10 |        2 |        12 |       7 |   17 |   15 |    1 |        1
11 |        3 |        12 |       7 |   16 |    9 |    1 |        2
12 |        4 |        12 |       7 |   11 |    8 |    1 |        3
13 |        5 |        12 |       7 |    7 |   -1 |    0 |        4
14 |        1 |        12 |      10 |   12 |   13 |    1 |        0
15 |        2 |        12 |      10 |   17 |   15 |    1 |        1
16 |        3 |        12 |      10 |   16 |   16 |    1 |        2
17 |        4 |        12 |      10 |   15 |    3 |    1 |        3
18 |        5 |        12 |      10 |   10 |   -1 |    0 |        4
(18 rows)



#### 8) Using Combinations to get the solution of examples 1 to 3¶

Paths $$\{6\}\rightarrow\{10, 7\}\cup\{12\}\rightarrow\{10\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
'SELECT * FROM (VALUES (6, 10), (6, 7), (12, 10)) AS combinations (source, target)'
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
3 |        1 |         6 |      10 |    6 |    4 |    1 |        0
4 |        2 |         6 |      10 |    7 |    8 |    1 |        1
5 |        3 |         6 |      10 |   11 |    9 |    1 |        2
6 |        4 |         6 |      10 |   16 |   16 |    1 |        3
7 |        5 |         6 |      10 |   15 |    3 |    1 |        4
8 |        6 |         6 |      10 |   10 |   -1 |    0 |        5
9 |        1 |        12 |      10 |   12 |   13 |    1 |        0
10 |        2 |        12 |      10 |   17 |   15 |    1 |        1
11 |        3 |        12 |      10 |   16 |   16 |    1 |        2
12 |        4 |        12 |      10 |   15 |    3 |    1 |        3
13 |        5 |        12 |      10 |   10 |   -1 |    0 |        4
(13 rows)



### For undirected graphs with cost and reverse_cost columns¶

#### 9) Path from $$6$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, 10,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    2 |    1 |        0
2 |        2 |   10 |   -1 |    0 |        1
(2 rows)



#### 10) Path from $$6$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, 7,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |   -1 |    0 |        1
(2 rows)



#### 11) Path from $$12$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
12, 10,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |   12 |   11 |    1 |        0
2 |        2 |   11 |    5 |    1 |        1
3 |        3 |   10 |   -1 |    0 |        2
(3 rows)



#### 12) Path from $$12$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
12, 7,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |   12 |   12 |    1 |        0
2 |        2 |    8 |   10 |    1 |        1
3 |        3 |    7 |   -1 |    0 |        2
(3 rows)



#### 13) Using One to Many to get the solution of examples 9 and 10¶

Paths $$\{6\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, ARRAY[10,7],
false
);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 |        1 |       7 |    6 |    4 |    1 |        0
2 |        2 |       7 |    7 |   -1 |    0 |        1
3 |        1 |      10 |    6 |    2 |    1 |        0
4 |        2 |      10 |   10 |   -1 |    0 |        1
(4 rows)



#### 14) Using Many to One to get the solution of examples 10 and 12¶

Paths $$\{6, 12\}\rightarrow\{7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6,12], 7,
false
);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 |        1 |         6 |    6 |    4 |    1 |        0
2 |        2 |         6 |    7 |   -1 |    0 |        1
3 |        1 |        12 |   12 |   12 |    1 |        0
4 |        2 |        12 |    8 |   10 |    1 |        1
5 |        3 |        12 |    7 |   -1 |    0 |        2
(5 rows)



#### 15) Using Many to Many to get the solution of examples 9 to 12¶

Paths $$\{6, 12\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6, 12], ARRAY[10,7],
false
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
3 |        1 |         6 |      10 |    6 |    2 |    1 |        0
4 |        2 |         6 |      10 |   10 |   -1 |    0 |        1
5 |        1 |        12 |       7 |   12 |   12 |    1 |        0
6 |        2 |        12 |       7 |    8 |   10 |    1 |        1
7 |        3 |        12 |       7 |    7 |   -1 |    0 |        2
8 |        1 |        12 |      10 |   12 |   11 |    1 |        0
9 |        2 |        12 |      10 |   11 |    5 |    1 |        1
10 |        3 |        12 |      10 |   10 |   -1 |    0 |        2
(10 rows)



#### 16) Using Combinations to get the solution of examples 9 to 11¶

Paths $$\{6\}\rightarrow\{10, 7\}\cup\{12\}\rightarrow\{10\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
'SELECT * FROM (VALUES (6, 10), (6, 7), (12, 10)) AS combinations (source, target)',
false
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
3 |        1 |         6 |      10 |    6 |    2 |    1 |        0
4 |        2 |         6 |      10 |   10 |   -1 |    0 |        1
5 |        1 |        12 |      10 |   12 |   11 |    1 |        0
6 |        2 |        12 |      10 |   11 |    5 |    1 |        1
7 |        3 |        12 |      10 |   10 |   -1 |    0 |        2
(7 rows)



### For directed graphs only with cost column¶

#### 17) Path from $$6$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
6, 10
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
(0 rows)



#### 18) Path from $$6$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
6, 7
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |   -1 |    0 |        1
(2 rows)



#### 19) Path from $$12$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
12, 10
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
(0 rows)



#### 20) Path from $$12$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
12, 7
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
(0 rows)



#### 21) Using One to Many to get the solution of examples 17 and 18¶

Paths $$\{6\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
6, ARRAY[10,7]
);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 |        1 |       7 |    6 |    4 |    1 |        0
2 |        2 |       7 |    7 |   -1 |    0 |        1
(2 rows)



#### 22) Using Many to One to get the solution of examples 18 and 20¶

Paths $$\{6, 12\}\rightarrow\{7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
ARRAY[6,12], 7
);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 |        1 |         6 |    6 |    4 |    1 |        0
2 |        2 |         6 |    7 |   -1 |    0 |        1
(2 rows)



#### 23) Using Many to Many to get the solution of examples 17 to 20¶

Paths $$\{6, 12\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
ARRAY[6, 12], ARRAY[10,7]
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
(2 rows)



#### 24) Using Combinations to get the solution of examples 17 to 19¶

Paths $$\{6\}\rightarrow\{10, 7\}\cup\{12\}\rightarrow\{10\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
'SELECT * FROM (VALUES (6, 10), (6, 7), (12, 10)) AS combinations (source, target)'
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
(2 rows)



### For undirected graphs only with cost column¶

#### 25) Path from $$6$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
6, 10,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |    8 |    1 |        1
3 |        3 |   11 |    5 |    1 |        2
4 |        4 |   10 |   -1 |    0 |        3
(4 rows)



#### 26) Path from $$6$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
6, 7,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |   -1 |    0 |        1
(2 rows)



#### 27) Path from $$12$$ to $$10$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
12, 10,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |   12 |   11 |    1 |        0
2 |        2 |   11 |    5 |    1 |        1
3 |        3 |   10 |   -1 |    0 |        2
(3 rows)



#### 28) Path from $$12$$ to $$7$$¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
12, 7,
false
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |   12 |   12 |    1 |        0
2 |        2 |    8 |   10 |    1 |        1
3 |        3 |    7 |   -1 |    0 |        2
(3 rows)



#### 29) Using One to Many to get the solution of examples 25 and 26¶

Paths $$\{6\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
6, ARRAY[10,7],
false
);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 |        1 |       7 |    6 |    4 |    1 |        0
2 |        2 |       7 |    7 |   -1 |    0 |        1
3 |        1 |      10 |    6 |    4 |    1 |        0
4 |        2 |      10 |    7 |    8 |    1 |        1
5 |        3 |      10 |   11 |    5 |    1 |        2
6 |        4 |      10 |   10 |   -1 |    0 |        3
(6 rows)



#### 30) Using Many to One to get the solution of examples 26 and 28¶

Paths $$\{6, 12\}\rightarrow\{7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
ARRAY[6,12], 7,
false
);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 |        1 |         6 |    6 |    4 |    1 |        0
2 |        2 |         6 |    7 |   -1 |    0 |        1
3 |        1 |        12 |   12 |   12 |    1 |        0
4 |        2 |        12 |    8 |   10 |    1 |        1
5 |        3 |        12 |    7 |   -1 |    0 |        2
(5 rows)



#### 31) Using Many to Many to get the solution of examples 25 to 28¶

Paths $$\{6, 12\}\rightarrow\{10, 7\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
ARRAY[6, 12], ARRAY[10,7],
false
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
3 |        1 |         6 |      10 |    6 |    4 |    1 |        0
4 |        2 |         6 |      10 |    7 |    8 |    1 |        1
5 |        3 |         6 |      10 |   11 |    5 |    1 |        2
6 |        4 |         6 |      10 |   10 |   -1 |    0 |        3
7 |        1 |        12 |       7 |   12 |   12 |    1 |        0
8 |        2 |        12 |       7 |    8 |   10 |    1 |        1
9 |        3 |        12 |       7 |    7 |   -1 |    0 |        2
10 |        1 |        12 |      10 |   12 |   11 |    1 |        0
11 |        2 |        12 |      10 |   11 |    5 |    1 |        1
12 |        3 |        12 |      10 |   10 |   -1 |    0 |        2
(12 rows)



#### 32) Using Combinations to get the solution of examples 25 to 27¶

Paths $$\{6\}\rightarrow\{10, 7\}\cup\{12\}\rightarrow\{10\}$$

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM edges',
'SELECT * FROM (VALUES (6, 10), (6, 7), (12, 10)) AS combinations (source, target)',
false
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |       7 |    6 |    4 |    1 |        0
2 |        2 |         6 |       7 |    7 |   -1 |    0 |        1
3 |        1 |         6 |      10 |    6 |    4 |    1 |        0
4 |        2 |         6 |      10 |    7 |    8 |    1 |        1
5 |        3 |         6 |      10 |   11 |    5 |    1 |        2
6 |        4 |         6 |      10 |   10 |   -1 |    0 |        3
7 |        1 |        12 |      10 |   12 |   11 |    1 |        0
8 |        2 |        12 |      10 |   11 |    5 |    1 |        1
9 |        3 |        12 |      10 |   10 |   -1 |    0 |        2
(9 rows)



### Equvalences between signatures¶

The following examples find the path for $$\{6\}\rightarrow\{10\}$$

#### 33) Using One to One¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, 10
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |    8 |    1 |        1
3 |        3 |   11 |    9 |    1 |        2
4 |        4 |   16 |   16 |    1 |        3
5 |        5 |   15 |    3 |    1 |        4
6 |        6 |   10 |   -1 |    0 |        5
(6 rows)



#### 34) Using One to Many¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, ARRAY[10]
);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 |        1 |      10 |    6 |    4 |    1 |        0
2 |        2 |      10 |    7 |    8 |    1 |        1
3 |        3 |      10 |   11 |    9 |    1 |        2
4 |        4 |      10 |   16 |   16 |    1 |        3
5 |        5 |      10 |   15 |    3 |    1 |        4
6 |        6 |      10 |   10 |   -1 |    0 |        5
(6 rows)



#### 35) Using Many to One¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6], 10
);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 |        1 |         6 |    6 |    4 |    1 |        0
2 |        2 |         6 |    7 |    8 |    1 |        1
3 |        3 |         6 |   11 |    9 |    1 |        2
4 |        4 |         6 |   16 |   16 |    1 |        3
5 |        5 |         6 |   15 |    3 |    1 |        4
6 |        6 |         6 |   10 |   -1 |    0 |        5
(6 rows)



#### 36) Using Many to Many¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6], ARRAY[10]
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |      10 |    6 |    4 |    1 |        0
2 |        2 |         6 |      10 |    7 |    8 |    1 |        1
3 |        3 |         6 |      10 |   11 |    9 |    1 |        2
4 |        4 |         6 |      10 |   16 |   16 |    1 |        3
5 |        5 |         6 |      10 |   15 |    3 |    1 |        4
6 |        6 |         6 |      10 |   10 |   -1 |    0 |        5
(6 rows)



#### 37) Using Combinations¶

SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost, reverse_cost FROM edges',
'SELECT * FROM (VALUES(6, 10)) AS combinations (source, target)'
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         6 |      10 |    6 |    4 |    1 |        0
2 |        2 |         6 |      10 |    7 |    8 |    1 |        1
3 |        3 |         6 |      10 |   11 |    9 |    1 |        2
4 |        4 |         6 |      10 |   16 |   16 |    1 |        3
5 |        5 |         6 |      10 |   15 |    3 |    1 |        4
6 |        6 |         6 |      10 |   10 |   -1 |    0 |        5
(6 rows)