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.

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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