# pgRouting Manual (3.6)

## pgr_aStar

# `pgr_aStar`

`pgr_aStar` — Shortest path using the A* algorithm.

Boost Graph Inside

Availability

- Version 3.6.0

  - Standarizing output columns to `(seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)`

    - `pgr_aStar` (One to One) added `start_vid` and `end_vid` columns.

    - `pgr_aStar` (One to Many) added `end_vid` column.

    - `pgr_aStar` (Many to One) added `start_vid` column.

- Version 3.2.0

  - New proposed signature:

    - `pgr_aStar` (Combinations)

- Version 3.0.0

  - Official function

- Version 2.4.0

  - New Proposed signatures:

    - `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:

- 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:

  - `pgr_aStar` (One to Many)

  - `pgr_aStar` (Many to One)

  - `pgr_aStar` (Many to Many)

  - `pgr_aStar` (Combinations)

## 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, start_vid, end_vid, 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 | start_vid | end_vid | node | edge | cost | agg_cost
    -----+----------+-----------+---------+------+------+------+----------
       1 |        1 |         6 |      12 |    6 |    4 |    1 |        0
       2 |        2 |         6 |      12 |    7 |   10 |    1 |        1
       3 |        3 |         6 |      12 |    8 |   12 |    1 |        2
       4 |        4 |         6 |      12 |   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, start_vid, 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 | 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 |    8 |    1 |        1
       9 |        3 |         6 |      12 |   11 |   11 |    1 |        2
      10 |        4 |         6 |      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, end_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 | end_vid | node | edge | cost | agg_cost
    -----+----------+-----------+---------+------+------+------+----------
       1 |        1 |         6 |      10 |    6 |    2 |    1 |        0
       2 |        2 |         6 |      10 |   10 |   -1 |    0 |        1
       3 |        1 |         8 |      10 |    8 |   12 |    1 |        0
       4 |        2 |         8 |      10 |   12 |   11 |    1 |        1
       5 |        3 |         8 |      10 |   11 |    5 |    1 |        2
       6 |        4 |         8 |      10 |   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.

- Many to One

- Many to Many

`end_vid`

`BIGINT`

Identifier of the ending vertex. Returned when multiple ending vertices are in the query.

- One to Many

- Many to Many

`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

- A* - Family of functions

- Bidirectional A* - Family of functions

- Sample Data

- https://www.boost.org/libs/graph/doc/astar_search.html

- https://en.wikipedia.org/wiki/A*_search_algorithm

Indices and tables

- Index

- Search Page