pgr_aStar

pgr_aStar — Returns the shortest path using A* algorithm.

_images/boost-inside.jpeg

Boost Graph Inside

Availability

  • Official on v3.0.0
  • Proposed on v2.4.0:
    • pgr_astar(One to Many)
    • pgr_astar(Many to One)
    • pgr_astar(Many to Many)
  • Signature change on v2.3.0:
    • pgr_astar(One to One)
  • New on v2.0.0:
    • 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])

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)

Parameters

Parameter Type Description
edges_sql TEXT edges_sql inner query.
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 query

edges_sql

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

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.