## 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  .