# pgRouting Manual (2.5)

## pgr_aStar

## Name

`pgr_aStar` — Returns the shortest path using A* algorithm.

Boost Graph Inside

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.

Example: |

    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.

Example: |

    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.

Example: |

    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
ANY-NUMERICAL: | 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`.

## See Also

- aStar - Family of functions
- Sample Data
- http://www.boost.org/libs/graph/doc/astar_search.html
- http://en.wikipedia.org/wiki/A*_search_algorithm

Indices and tables

- Index
- Search Page