# All Pairs - Family of Functions¶

The following functions work on all vertices pair combinations

## Introduction¶

The main characteristics are:

• It does not return a path.

• Returns the sum of the costs of the shortest path for each pair of nodes in the graph.

• Process is done only on edges with positive costs.

• Boost returns a $$V \times V$$ matrix, where the infinity values. Represent the distance between vertices for which there is no path.

• We return only the non infinity values in form of a set of (start_vid, end_vid, agg_cost).

• Let be the case the values returned are stored in a table, so the unique index would be the pair: (start_vid, end_vid).

• For the undirected graph, the results are symmetric.

• The agg_cost of (u, v) is the same as for (v, u).

• When start_vid = end_vid, the agg_cost = 0.

• Recommended, use a bounding box of no more than 3500 edges.

## Parameters¶

Parameter

Type

Default

Description

Edges SQL

TEXT

Edges SQL as described below.

### Optional parameters¶

Column

Type

Default

Description

directed

BOOLEAN

true

• When true the graph is considered Directed

• When false the graph is considered as Undirected.

## Inner Queries¶

### Edges SQL¶

Column

Type

Default

Description

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)

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.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

## Result Columns¶

Set of (start_vid, end_vid, agg_cost)

Column

Type

Description

start_vid

BIGINT

Identifier of the starting vertex.

end_vid

BIGINT

Identifier of the ending vertex.

agg_cost

FLOAT

Aggregate cost from start_vid to end_vid.

## Performance¶

The following tests:
• non server computer

• with AMD 64 CPU

• 4G memory

• trusty

• posgreSQL version 9.3

### Data¶

The following data was used

BBOX="-122.8,45.4,-122.5,45.6"
wget --progress=dot:mega -O "sampledata.osm"      "https://www.overpass-api.de/api/xapi?*[bbox=][@meta]"

Data processing was done with osm2pgrouting-alpha

createdb portland
psql -c "create extension postgis" portland
psql -c "create extension pgrouting" portland
osm2pgrouting -f sampledata.osm -d portland -s 0


### Results¶

Test:

One

This test is not with a bounding box The density of the passed graph is extremely low. For each <SIZE> 30 tests were executed to get the average The tested query is:

SELECT count(*) FROM pgr_floydWarshall(
'SELECT gid as id, source, target, cost, reverse_cost
FROM ways where id <=  <SIZE>');

SELECT count(*) FROM pgr_johnson(
'SELECT gid as id, source, target, cost, reverse_cost
FROM ways where id <=  <SIZE>');


The results of this tests are presented as:

SIZE:

is the number of edges given as input.

EDGES:

is the total number of records in the query.

DENSITY:

is the density of the data $$\dfrac{E}{V \times (V-1)}$$.

OUT ROWS:

is the number of records returned by the queries.

Floyd-Warshall:

is the average execution time in seconds of pgr_floydWarshall.

Johnson:

is the average execution time in seconds of pgr_johnson.

SIZE

EDGES

DENSITY

OUT ROWS

Floyd-Warshall

Johnson

500

500

0.18E-7

1346

0.14

0.13

1000

1000

0.36E-7

2655

0.23

0.18

1500

1500

0.55E-7

4110

0.37

0.34

2000

2000

0.73E-7

5676

0.56

0.37

2500

2500

0.89E-7

7177

0.84

0.51

3000

3000

1.07E-7

8778

1.28

0.68

3500

3500

1.24E-7

10526

2.08

0.95

4000

4000

1.41E-7

12484

3.16

1.24

4500

4500

1.58E-7

14354

4.49

1.47

5000

5000

1.76E-7

16503

6.05

1.78

5500

5500

1.93E-7

18623

7.53

2.03

6000

6000

2.11E-7

20710

8.47

2.37

6500

6500

2.28E-7

22752

9.99

2.68

7000

7000

2.46E-7

24687

11.82

3.12

7500

7500

2.64E-7

26861

13.94

3.60

8000

8000

2.83E-7

29050

15.61

4.09

8500

8500

3.01E-7

31693

17.43

4.63

9000

9000

3.17E-7

33879

19.19

5.34

9500

9500

3.35E-7

36287

20.77

6.24

10000

10000

3.52E-7

38491

23.26

6.51

Test:

Two

This test is with a bounding box The density of the passed graph higher than of the Test One. For each <SIZE> 30 tests were executed to get the average The tested edge query is:

WITH
buffer AS (
SELECT ST_Buffer(ST_Centroid(ST_Extent(the_geom)), SIZE) AS geom
FROM ways),
bbox AS (
SELECT ST_Envelope(ST_Extent(geom)) as box FROM buffer)
SELECT gid as id, source, target, cost, reverse_cost
FROM ways where the_geom && (SELECT box from bbox);


The tested queries

SELECT count(*) FROM pgr_floydWarshall(<edge query>)
SELECT count(*) FROM pgr_johnson(<edge query>)


The results of this tests are presented as:

SIZE:

is the size of the bounding box.

EDGES:

is the total number of records in the query.

DENSITY:

is the density of the data $$\dfrac{E}{V \times (V-1)}$$.

OUT ROWS:

is the number of records returned by the queries.

Floyd-Warshall:

is the average execution time in seconds of pgr_floydWarshall.

Johnson:

is the average execution time in seconds of pgr_johnson.

SIZE

EDGES

DENSITY

OUT ROWS

Floyd-Warshall

Johnson

0.001

44

0.0608

1197

0.10

0.10

0.002

99

0.0251

4330

0.10

0.10

0.003

223

0.0122

18849

0.12

0.12

0.004

358

0.0085

71834

0.16

0.16

0.005

470

0.0070

116290

0.22

0.19

0.006

639

0.0055

207030

0.37

0.27

0.007

843

0.0043

346930

0.64

0.38

0.008

996

0.0037

469936

0.90

0.49

0.009

1146

0.0032

613135

1.26

0.62

0.010

1360

0.0027

849304

1.87

0.82

0.011

1573

0.0024

1147101

2.65

1.04

0.012

1789

0.0021

1483629

3.72

1.35

0.013

1975

0.0019

1846897

4.86

1.68

0.014

2281

0.0017

2438298

7.08

2.28

0.015

2588

0.0015

3156007

10.28

2.80

0.016

2958

0.0013

4090618

14.67

3.76

0.017

3247

0.0012

4868919

18.12

4.48