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

## Parameters¶

Column

Type

Description

Edges SQL

TEXT

Edges SQL as described below

Combinations SQL

TEXT

Combinations SQL as described below

start vid

BIGINT

Identifier of the starting vertex of the path.

start vids

ARRAY[BIGINT]

Array of identifiers of starting vertices.

end vid

BIGINT

Identifier of the ending vertex of the path.

end vids

ARRAY[BIGINT]

Array of identifiers of ending vertices.

### Optional parameters¶

Column

Type

default

Description

directed

BOOLEAN

true

• When true the graph is considered Directed

• When false the graph is considered as Undirected.

### 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(dx, dy))

• 2: h(v) abs(min(dx, dy))

• 3: h(v) = dx * dx + dy * dy

• 4: h(v) = sqrt(dx * dx + dy * dy)

• 5: h(v) = abs(dx) + abs(dy)

factor

FLOAT

1

For units manipulation. $$factor > 0$$. See Factor

epsilon

FLOAT

1

For less restricted results. $$epsilon >= 1$$.

## Inner queries¶

### Edges SQL¶

Parameter

Type

Default

Description

id

ANY-INTEGER

Identifier of the edge.

source

ANY-INTEGER

Identifier of the first end point vertex of the edge.

target

ANY-INTEGER

Identifier of the second end point vertex of the edge.

cost

ANY-NUMERICAL

Weight of the edge (source, target)

• When negative: edge (source, target) does not exist, therefore it’s not part of the graph.

reverse_cost

ANY-NUMERICAL

-1

Weight of the edge (target, source),

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

x1

ANY-NUMERICAL

X coordinate of source vertex.

y1

ANY-NUMERICAL

Y coordinate of source vertex.

x2

ANY-NUMERICAL

X coordinate of target vertex.

y2

ANY-NUMERICAL

Y coordinate of target vertex.

Where:

ANY-INTEGER

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

### Combinations SQL¶

Parameter

Type

Description

source

ANY-INTEGER

Identifier of the departure vertex.

target

ANY-INTEGER

Identifier of the arrival vertex.

Where:

ANY-INTEGER

SMALLINT, INTEGER, BIGINT

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