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.

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`

- first by
- 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 \(0\)

- If there is no path from \(v\) to \(u\):
- 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 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: \(O((E + V) * \log V)\)

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 \(\Delta y = y_1 - y_0\)

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 |