A* - 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.

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

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. \(epsilon >= 1\).

Parameter

Type

Default

Description

heuristic

INTEGER

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.

Advanced documentation¶

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