# pgRouting Manual (3.0-rc1)

## pgr_aStar

`pgr_aStar` — Shortest path using A* algorithm.

Boost Graph Inside

Availability

- Version 3.0.0
  - Official function
- Version 2.4.0
  - New Proposed functions:
    - 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)

Support

- Supported versions: current(3.0) 2.6
- Unsupported versions: 2.5 2.4 2.3 2.2 2.1 2.0

## 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:
  - pgr_aStar( One to Many )
  - pgr_aStar( Many to One )
  - pgr_aStar( Many to Many )
- `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:

- One to One
- One to Many

from_vids

`ARRAY[ANY-INTEGER]`

Array of starting vertices identifiers. Parameter in:

- Many to One
- Many to Many

to_vid

`ANY-INTEGER`

Ending vertex identifier. Parameter in:

- One to One
- Many to One

to_vids

`ARRAY[ANY-INTEGER]`

Array of ending vertices identifiers. Parameter in:

- One to Many
- Many to Many

### 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.

- 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_v` to `node`.

## See Also

- aStar - 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