# aStar - Family of functions¶

The A* (pronounced “A Star”) algorithm is based on Dijkstra’s algorithm with a heuristic that allow it to solve most shortest path problems by evaluation only a sub-set of the overall graph.

pgr_aStar - A* algorithm for the shortest path.

pgr_aStarCost - Get the aggregate cost of the shortest paths.

pgr_aStarCostMatrix - Get the cost matrix of the shortest paths.

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## General Information¶

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

## Advanced documentation¶

The A* (pronounced “A Star”) algorithm is based on Dijkstra’s algorithm with a heuristic, that is an estimation of the remaining cost from the vertex to the goal, that allows to solve most shortest path problems by evaluation only a sub-set of the overall graph. Running time: \(O((E + V) * \log V)\)

### Heuristic¶

Currently the heuristic functions available are:

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: \(h(v) = abs(\Delta x) + abs(\Delta y)\)

where \(\Delta x = x_1 - x_0\) and \(\Delta y = y_1 - y_0\)

## Factor¶

Analysis 1

Working with cost/reverse_cost as length in degrees, x/y in lat/lon: Factor = 1 (no need to change units)

Analysis 2

Working with cost/reverse_cost as length in meters, x/y in lat/lon: Factor = would depend on the location of the points:

Latitude |
Conversion |
Factor |
---|---|---|

45 |
1 longitude degree is 78846.81 m |
78846 |

0 |
1 longitude degree is 111319.46 m |
111319 |

Analysis 3

Working with cost/reverse_cost as time in seconds, x/y in lat/lon: Factor: would depend on the location of the points and on the average speed say 25m/s is the speed.

Latitude |
Conversion |
Factor |
---|---|---|

45 |
1 longitude degree is (78846.81m)/(25m/s) |
3153 s |

0 |
1 longitude degree is (111319.46 m)/(25m/s) |
4452 s |