# pgr_aStar¶

pgr_aStar — Shortest path using A* algorithm.

Availability

• Version 3.2.0

• New proposed function:

• pgr_aStar(Combinations)

• Version 3.0.0

• Official function

• Version 2.4.0

• New Proposed functions:

• pgr_aStar(One to Many)

• pgr_aStar(Many to One)

• pgr_aStar(Many to Many)

• Version 2.3.0

• Signature change on pgr_astar(One to One)

• Old signature no longer supported

• Version 2.0.0

• Official pgr_aStar(One to One)

## Description¶

The main characteristics are:

• Default kind of graph is directed when

• directed flag is missing.

• directed flag is set to true

• Unless specified otherwise, ordering is:

• first by start_vid (if exists)

• then by end_vid

• Values are returned when there is a path

• Let $$v$$ and $$u$$ be nodes on the graph:

• If there is no path from $$v$$ to $$u$$:

• no corresponding row is returned

• agg_cost from $$v$$ to $$u$$ is $$\infty$$

• There is no path when $$v = u$$ therefore

• no corresponding row is returned

• agg_cost from v to u is $$0$$

• Edges with negative costs are not included in the graph.

• When (x,y) coordinates for the same vertex identifier differ:

• A random selection of the vertex’s (x,y) coordinates is used.

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

• The results are equivalent to the union of the results of the pgr_aStar( One to One ) on the:

• start_vid and end_vid in the result is used to distinguish to which path it belongs.

## Signatures¶

Summary

pgr_aStar(Edges SQL, from_vid,  to_vid  [, directed] [, heuristic] [, factor] [, epsilon])
pgr_aStar(Edges SQL, from_vid,  to_vids [, directed] [, heuristic] [, factor] [, epsilon])
pgr_aStar(Edges SQL, from_vids, to_vid  [, directed] [, heuristic] [, factor] [, epsilon])
pgr_aStar(Edges SQL, from_vids, to_vids [, directed] [, heuristic] [, factor] [, epsilon])
pgr_aStar(Edges SQL, Combinations SQL  [, directed] [, heuristic] [, factor] [, epsilon])

RETURNS SET OF (seq, path_seq [, start_vid] [, end_vid], node, edge, cost, agg_cost)
OR EMPTY SET


Optional parameters are named parameters and have a default value.

Using defaults

pgr_aStar(Edges SQL, from_vid, to_vid)
RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost)
OR EMPTY SET

Example

From vertex $$2$$ to vertex $$12$$ on a directed graph

SELECT * FROM pgr_astar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2 FROM edge_table',
2, 12);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    2 |    4 |    1 |        0
2 |        2 |    5 |    8 |    1 |        1
3 |        3 |    6 |   11 |    1 |        2
4 |        4 |   11 |   13 |    1 |        3
5 |        5 |   12 |   -1 |    0 |        4
(5 rows)



### One to One¶

pgr_aStar(Edges SQL, from_vid,  to_vid  [, directed] [, heuristic] [, factor] [, epsilon])

RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost)
OR EMPTY SET

Example

From vertex $$2$$ to vertex $$12$$ on an undirected graph using heuristic $$2$$

SELECT * FROM pgr_astar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2 FROM edge_table',
2, 12,
directed := false, heuristic := 2);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    2 |    2 |    1 |        0
2 |        2 |    3 |    3 |    1 |        1
3 |        3 |    4 |   16 |    1 |        2
4 |        4 |    9 |   15 |    1 |        3
5 |        5 |   12 |   -1 |    0 |        4
(5 rows)



### One to many¶

pgr_aStar(Edges SQL, from_vid,  to_vids [, directed] [, heuristic] [, factor] [, epsilon])
RETURNS SET OF (seq, path_seq, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET

Example

From vertex $$2$$ to vertices $$\{3, 12\}$$ on a directed graph using heuristic $$2$$

SELECT * FROM pgr_astar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2 FROM edge_table',
2, ARRAY[3, 12], heuristic := 2);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 |        1 |       3 |    2 |    4 |    1 |        0
2 |        2 |       3 |    5 |    8 |    1 |        1
3 |        3 |       3 |    6 |    9 |    1 |        2
4 |        4 |       3 |    9 |   16 |    1 |        3
5 |        5 |       3 |    4 |    3 |    1 |        4
6 |        6 |       3 |    3 |   -1 |    0 |        5
7 |        1 |      12 |    2 |    4 |    1 |        0
8 |        2 |      12 |    5 |   10 |    1 |        1
9 |        3 |      12 |   10 |   12 |    1 |        2
10 |        4 |      12 |   11 |   13 |    1 |        3
11 |        5 |      12 |   12 |   -1 |    0 |        4
(11 rows)



### Many to One¶

pgr_aStar(Edges SQL, from_vids, to_vid  [, directed] [, heuristic] [, factor] [, epsilon])
RETURNS SET OF (seq, path_seq, start_vid, node, edge, cost, agg_cost)
OR EMPTY SET

Example

From vertices $$\{7, 2\}$$ to vertex $$12$$ on a directed graph using heuristic $$0$$

SELECT * FROM pgr_astar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2 FROM edge_table',
ARRAY[7, 2], 12, heuristic := 0);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 |        1 |         2 |    2 |    4 |    1 |        0
2 |        2 |         2 |    5 |   10 |    1 |        1
3 |        3 |         2 |   10 |   12 |    1 |        2
4 |        4 |         2 |   11 |   13 |    1 |        3
5 |        5 |         2 |   12 |   -1 |    0 |        4
6 |        1 |         7 |    7 |    6 |    1 |        0
7 |        2 |         7 |    8 |    7 |    1 |        1
8 |        3 |         7 |    5 |   10 |    1 |        2
9 |        4 |         7 |   10 |   12 |    1 |        3
10 |        5 |         7 |   11 |   13 |    1 |        4
11 |        6 |         7 |   12 |   -1 |    0 |        5
(11 rows)



### Many to Many¶

pgr_aStar(Edges SQL, from_vids, to_vids [, directed] [, heuristic] [, factor] [, epsilon])
RETURNS SET OF (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET

Example

From vertices $$\{7, 2\}$$ to vertices $$\{3, 12\}$$ on a directed graph using heuristic $$2$$

SELECT * FROM pgr_astar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2 FROM edge_table',
ARRAY[7, 2], ARRAY[3, 12], heuristic := 2);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         2 |       3 |    2 |    4 |    1 |        0
2 |        2 |         2 |       3 |    5 |    8 |    1 |        1
3 |        3 |         2 |       3 |    6 |    9 |    1 |        2
4 |        4 |         2 |       3 |    9 |   16 |    1 |        3
5 |        5 |         2 |       3 |    4 |    3 |    1 |        4
6 |        6 |         2 |       3 |    3 |   -1 |    0 |        5
7 |        1 |         2 |      12 |    2 |    4 |    1 |        0
8 |        2 |         2 |      12 |    5 |   10 |    1 |        1
9 |        3 |         2 |      12 |   10 |   12 |    1 |        2
10 |        4 |         2 |      12 |   11 |   13 |    1 |        3
11 |        5 |         2 |      12 |   12 |   -1 |    0 |        4
12 |        1 |         7 |       3 |    7 |    6 |    1 |        0
13 |        2 |         7 |       3 |    8 |    7 |    1 |        1
14 |        3 |         7 |       3 |    5 |    8 |    1 |        2
15 |        4 |         7 |       3 |    6 |    9 |    1 |        3
16 |        5 |         7 |       3 |    9 |   16 |    1 |        4
17 |        6 |         7 |       3 |    4 |    3 |    1 |        5
18 |        7 |         7 |       3 |    3 |   -1 |    0 |        6
19 |        1 |         7 |      12 |    7 |    6 |    1 |        0
20 |        2 |         7 |      12 |    8 |    7 |    1 |        1
21 |        3 |         7 |      12 |    5 |   10 |    1 |        2
22 |        4 |         7 |      12 |   10 |   12 |    1 |        3
23 |        5 |         7 |      12 |   11 |   13 |    1 |        4
24 |        6 |         7 |      12 |   12 |   -1 |    0 |        5
(24 rows)



### Combinations¶

pgr_aStar(Edges SQL, Combinations SQL  [, directed] [, heuristic] [, factor] [, epsilon])
RETURNS SET OF (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET

Example

Using a combinations table on a directed graph using heuristic $$2$$.

SELECT * FROM pgr_astar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2 FROM edge_table',
'SELECT * FROM ( VALUES (7, 3), (2, 12) ) AS t(source, target)',
heuristic := 2);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 |        1 |         2 |      12 |    2 |    4 |    1 |        0
2 |        2 |         2 |      12 |    5 |   10 |    1 |        1
3 |        3 |         2 |      12 |   10 |   12 |    1 |        2
4 |        4 |         2 |      12 |   11 |   13 |    1 |        3
5 |        5 |         2 |      12 |   12 |   -1 |    0 |        4
6 |        1 |         7 |       3 |    7 |    6 |    1 |        0
7 |        2 |         7 |       3 |    8 |    7 |    1 |        1
8 |        3 |         7 |       3 |    5 |    8 |    1 |        2
9 |        4 |         7 |       3 |    6 |    9 |    1 |        3
10 |        5 |         7 |       3 |    9 |   16 |    1 |        4
11 |        6 |         7 |       3 |    4 |    3 |    1 |        5
12 |        7 |         7 |       3 |    3 |   -1 |    0 |        6
(12 rows)



## Parameters¶

Parameter

Type

Description

Edges SQL

TEXT

Edges query as described below.

Combinations SQL

TEXT

Combinations query as described below.

from_vid

ANY-INTEGER

Starting vertex identifier. Parameter in:

from_vids

ARRAY[ANY-INTEGER]

Array of starting vertices identifiers. Parameter in:

to_vid

ANY-INTEGER

Ending vertex identifier. Parameter in:

to_vids

ARRAY[ANY-INTEGER]

Array of ending vertices identifiers. Parameter in:

### Optional Parameters¶

Parameter

Type

Default

Description

directed

BOOLEAN

true

• When true the graph is considered as Directed.

• When false the graph is considered as Undirected.

heuristic

INTEGER

5

Heuristic number. Current valid values 0~5. Default 5

• 0: h(v) = 0 (Use this value to compare with pgr_dijkstra)

• 1: h(v) abs(max(dx, dy))

• 2: h(v) abs(min(dx, dy))

• 3: h(v) = dx * dx + dy * dy

• 4: h(v) = sqrt(dx * dx + dy * dy)

• 5: h(v) = abs(dx) + abs(dy)

factor

FLOAT

1

For units manipulation. $$factor > 0$$. See Factor

epsilon

FLOAT

1

For less restricted results. $$epsilon >= 1$$.

## Inner queries¶

### Edges query¶

edges_sql

an SQL query, which should return a set of rows with the following columns:

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)

• 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

### Combinations query¶

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.

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

INT

Sequential value starting from 1.

path_seq

INT

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_v to node.