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: \(O((E + V) * \log V)\)

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 .

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)

See Also

Indices and tables