# pgr_aStar¶

pgr_aStar — Shortest path using the A* algorithm.

Availability

• Version 3.2.0

• New proposed signature:

• Version 3.0.0

• Official function

• Version 2.4.0

• Version 2.3.0

• Signature change on pgr_astar (One to One)

• Old signature no longer supported

• Version 2.0.0

## Description¶

The main characteristics are:

• Process works for directed and undirected graphs.

• 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 $$0$$

• 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, start vid, end vid, [options])
pgr_aStar(Edges SQL, start vid, end vids, [options])
pgr_aStar(Edges SQL, start vids, end vid, [options])
pgr_aStar(Edges SQL, start vids, end vids, [options])
pgr_aStar(Edges SQL, Combinations SQL, [options])
options: [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.

### One to One¶

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

From vertex $$6$$ to vertex $$12$$ on a directed graph with heuristic $$2$$

SELECT * FROM pgr_aStar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2
FROM edges',
6, 12,
directed => true, heuristic => 2);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 |        1 |    6 |    4 |    1 |        0
2 |        2 |    7 |   10 |    1 |        1
3 |        3 |    8 |   12 |    1 |        2
4 |        4 |   12 |   -1 |    0 |        3
(4 rows)



### One to Many¶

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

From vertex $$6$$ to vertices $$\{10, 12\}$$ on a directed graph with heuristic $$3$$ and factor $$3.5$$

SELECT * FROM pgr_aStar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2
FROM edges',
6, ARRAY[10, 12],
heuristic => 3, factor := 3.5);
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 |      12 |    6 |    4 |    1 |        0
8 |        2 |      12 |    7 |    8 |    1 |        1
9 |        3 |      12 |   11 |   11 |    1 |        2
10 |        4 |      12 |   12 |   -1 |    0 |        3
(10 rows)



### Many to One¶

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

From vertices $$\{6, 8\}$$ to vertex $$10$$ on an undirected graph with heuristic $$4$$

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



### Many to Many¶

pgr_aStar(Edges SQL, start vids, end vids, [options])
options: [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 $$\{6, 8\}$$ to vertices $$\{10, 12\}$$ on a directed graph with factor $$0.5$$

SELECT * FROM pgr_aStar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2
FROM edges',
ARRAY[6, 8], ARRAY[10, 12],
factor => 0.5);
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
7 |        1 |         6 |      12 |    6 |    4 |    1 |        0
8 |        2 |         6 |      12 |    7 |   10 |    1 |        1
9 |        3 |         6 |      12 |    8 |   12 |    1 |        2
10 |        4 |         6 |      12 |   12 |   -1 |    0 |        3
11 |        1 |         8 |      10 |    8 |   10 |    1 |        0
12 |        2 |         8 |      10 |    7 |    8 |    1 |        1
13 |        3 |         8 |      10 |   11 |    9 |    1 |        2
14 |        4 |         8 |      10 |   16 |   16 |    1 |        3
15 |        5 |         8 |      10 |   15 |    3 |    1 |        4
16 |        6 |         8 |      10 |   10 |   -1 |    0 |        5
17 |        1 |         8 |      12 |    8 |   12 |    1 |        0
18 |        2 |         8 |      12 |   12 |   -1 |    0 |        1
(18 rows)



### Combinations¶

pgr_aStar(Edges SQL, Combinations SQL, [options])
options: [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 with factor $$0.5$$.

The combinations table:

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



The query:

SELECT * FROM pgr_aStar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2
FROM edges',
'SELECT * FROM combinations',
factor => 0.5);
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 |    4 |    1 |        1
5 |        3 |         5 |      10 |    7 |    8 |    1 |        2
6 |        4 |         5 |      10 |   11 |    9 |    1 |        3
7 |        5 |         5 |      10 |   16 |   16 |    1 |        4
8 |        6 |         5 |      10 |   15 |    3 |    1 |        5
9 |        7 |         5 |      10 |   10 |   -1 |    0 |        6
10 |        1 |         6 |       5 |    6 |    1 |    1 |        0
11 |        2 |         6 |       5 |    5 |   -1 |    0 |        1
12 |        1 |         6 |      15 |    6 |    4 |    1 |        0
13 |        2 |         6 |      15 |    7 |    8 |    1 |        1
14 |        3 |         6 |      15 |   11 |    9 |    1 |        2
15 |        4 |         6 |      15 |   16 |   16 |    1 |        3
16 |        5 |         6 |      15 |   15 |   -1 |    0 |        4
(16 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.

### aStar optional Parameters¶

Parameter

Type

Default

Description

heuristic

INTEGER

5

Heuristic number. Current valid values 0~5.

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

• 1: $$h(v) = abs(max(\Delta x, \Delta y))$$

• 2: $$h(v) = abs(min(\Delta x, \Delta y))$$

• 3: $$h(v) = \Delta x * \Delta x + \Delta y * \Delta y$$

• 4: $$h(v) = sqrt(\Delta x * \Delta x + \Delta y * \Delta y)$$

• 5: $$h(v) = abs(\Delta x) + abs(\Delta y)$$

factor

FLOAT

1

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

epsilon

FLOAT

1

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

See heuristics available and factor handling.

## Inner Queries¶

### Edges SQL¶

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

### 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 1:

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

 SELECT * FROM pgr_aStar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2
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 |    3 |    1 |        0
18 |        2 |        15 |       7 |   10 |    2 |    1 |        1
19 |        3 |        15 |       7 |    6 |    4 |    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_aStar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2
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 |    3 |    1 |        0
18 |        2 |        15 |       7 |   10 |    2 |    1 |        1
19 |        3 |        15 |       7 |    6 |    4 |    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 3:

Manually assigned vertex combinations.

SELECT * FROM pgr_aStar(
'SELECT id, source, target, cost, reverse_cost, x1, y1, x2, y2
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)