pgr_aStar

pgr_aStar — Shortest path using the A* algorithm.

_images/boost-inside.jpeg

Boost Graph Inside

Availability

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
           [, directed] [, heuristic] [, factor] [, epsilon])
pgr_aStar(Edges SQL, start vid, end vids
           [, directed] [, heuristic] [, factor] [, epsilon])
pgr_aStar(Edges SQL, start vids, end vid
           [, directed] [, heuristic] [, factor] [, epsilon])
pgr_aStar(Edges SQL, start vids, end 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.

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)
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
           [, 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
           [, 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
           [, directed] [, heuristic] [, factor] [, epsilon])
RETURNS (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
           [, directed] [, heuristic] [, factor] [, epsilon])
RETURNS (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.

Additional Examples

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)