# pgr_aStar¶

## Name¶

pgr_aStar — Returns the shortest path using A* algorithm.

Availability:

• pgr_astar(one to one) 2.0.0, Signature changed 2.3.0
• pgr_astar(other signatures) 2.4.0

## Characteristics¶

The main Characteristics are:

• Process is done only on edges with positive costs.
• Vertices of the graph are:
• positive when it belongs to the edges_sql
• Values are returned when there is a path.
• When the starting vertex and ending vertex are the same, there is no path.
• The agg_cost the non included values (v, v) is 0
• When the starting vertex and ending vertex are the different and there is no path:
• The agg_cost the non included values (u, v) is ∞
• When (x,y) coordinates for the same vertex identifier differ:
• A random selection of the vertex’s (x,y) coordinates is used.
• Running time: $$O((E + V) * \log V)$$

## Signature Summary¶

pgr_aStar(edges_sql, start_vid, end_vid)
pgr_aStar(edges_sql, start_vid, end_vid, directed, heuristic, factor, epsilon)


Warning

Proposed functions for next mayor release.

• They are not officially in the current release.
• They will likely officially be part of the next mayor release:
• The functions make use of ANY-INTEGER and ANY-NUMERICAL
• Name might not change. (But still can)
• Signature might not change. (But still can)
• Functionality might not change. (But still can)
• pgTap tests have being done. But might need more.
• Documentation might need refinement.
pgr_aStar(edges_sql, start_vid, end_vids, directed, heuristic, factor, epsilon) -- proposed
pgr_aStar(edges_sql, starts_vid, end_vid, directed, heuristic, factor, epsilon) -- proposed
pgr_aStar(edges_sql, starts_vid, end_vids, directed, heuristic, factor, epsilon) -- proposed
RETURNS SET OF (seq, path_seq [, start_vid] [, end_vid], node, edge, cost, agg_cost)
OR EMPTY SET


## Signatures¶

### Minimal Signature¶

pgr_aStar(edges_sql, start_vid, end_vid)
RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost)

Example: Using the defaults
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 |   10 |    1 |        1
3 |        3 |   10 |   12 |    1 |        2
4 |        4 |   11 |   13 |    1 |        3
5 |        5 |   12 |   -1 |    0 |        4
(5 rows)



### One to One¶

pgr_aStar(edges_sql, start_vid, end_vid, directed, heuristic, factor, epsilon)
RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost)

Example: Undirected 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, start_vid, end_vids, directed, heuristic, factor, epsilon) -- Proposed
RETURNS SET OF (seq, path_seq, end_vid, node, edge, cost, agg_cost) or EMPTY SET

This signature finds the shortest path from one start_vid to each end_vid in end_vids:
• on a directed graph when directed flag is missing or is set to true.
• on an undirected graph when directed flag is set to false.

Using this signature, will load once the graph and perform a one to one pgr_astar where the starting vertex is fixed, and stop when all end_vids are reached.

• The result is equivalent to the union of the results of the one to one pgr_astar.
• The extra end_vid in the result is used to distinguish to which path it belongs.
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, starts_vid, end_vid, directed, heuristic, factor, epsilon) -- Proposed
RETURNS SET OF (seq, path_seq, start_vid, node, edge, cost, agg_cost) or EMPTY SET

This signature finds the shortest path from each start_vid in start_vids to one end_vid:
• on a directed graph when directed flag is missing or is set to true.
• on an undirected graph when directed flag is set to false.

Using this signature, will load once the graph and perform several one to one pgr_aStar where the ending vertex is fixed.

• The result is the union of the results of the one to one pgr_aStar.
• The extra start_vid in the result is used to distinguish to which path it belongs.
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, starts_vid, end_vids, directed, heuristic, factor, epsilon) -- Proposed
RETURNS SET OF (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost) or EMPTY SET

This signature finds the shortest path from each start_vid in start_vids to each end_vid in end_vids:
• on a directed graph when directed flag is missing or is set to true.
• on an undirected graph when directed flag is set to false.

Using this signature, will load once the graph and perform several one to Many pgr_dijkstra for all start_vids.

• The result is the union of the results of the one to one pgr_dijkstra.
• The extra start_vid in the result is used to distinguish to which path it belongs.

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

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 |         7 |       3 |    7 |    6 |    1 |        0
8 |        2 |         7 |       3 |    8 |    7 |    1 |        1
9 |        3 |         7 |       3 |    5 |    8 |    1 |        2
10 |        4 |         7 |       3 |    6 |    9 |    1 |        3
11 |        5 |         7 |       3 |    9 |   16 |    1 |        4
12 |        6 |         7 |       3 |    4 |    3 |    1 |        5
13 |        7 |         7 |       3 |    3 |   -1 |    0 |        6
14 |        1 |         2 |      12 |    2 |    4 |    1 |        0
15 |        2 |         2 |      12 |    5 |   10 |    1 |        1
16 |        3 |         2 |      12 |   10 |   12 |    1 |        2
17 |        4 |         2 |      12 |   11 |   13 |    1 |        3
18 |        5 |         2 |      12 |   12 |   -1 |    0 |        4
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)



## Description of the Signatures¶

### Description of the edges_sql query for astar like functions¶

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 SMALLINT, INTEGER, BIGINT, REAL, FLOAT

### Description of the parameters of the signatures¶

Parameter Type Description
edges_sql TEXT Edges SQL query as described above.
start_vid ANY-INTEGER Starting vertex identifier.
end_vid ANY-INTEGER Ending vertex identifier.
directed BOOLEAN
• Optional.
• When false the graph is considered as Undirected.
• Default is true which considers the graph as Directed.
heuristic INTEGER

(optional). 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 (optional). For units manipulation. $$factor > 0$$. Default 1. see Factor
epsilon FLOAT (optional). For less restricted results. $$epsilon >= 1$$. Default 1.

### Description of the return values for a path¶

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_id INT Path identifier. Has value 1 for the first of a path. Used when there are multiple paths for the same start_vid to end_vid combination.
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. Used when multiple starting vetrices are in the query.
end_vid BIGINT Identifier of the ending vertex. Used 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.