## 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.

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

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