## pgr_TSPeuclidean

• ``` pgr_TSPeuclidean ``` - Aproximation using metric algorithm.

Availability:

• Version 3.2.1

• Metric Algorithm from Boost library

• Simulated Annealing Algorithm no longer supported

• The Simulated Annealing Algorithm related parameters are ignored: max_processing_time, tries_per_temperature, max_changes_per_temperature, max_consecutive_non_changes, initial_temperature, final_temperature, cooling_factor, randomize

• Version 3.0.0

• Name change from pgr_eucledianTSP

• Version 2.3.0

• New Official function

### Description

#### Problem Definition

The travelling salesperson problem (TSP) asks the following question:

Given a list of cities and the distances between each pair of cities, which is the shortest possible route that visits each city exactly once and returns to the origin city?

#### General Characteristics

• This problem is an NP-hard optimization problem.

• Metric Algorithm is used

• Implementation generates solutions that are twice as long as the optimal tour in the worst case when:

• Graph is undirected

• Graph is fully connected

• Graph where traveling costs on edges obey the triangle inequality.

• On an undirected graph:

• The traveling costs are symmetric:

• Traveling costs from ``` u ``` to ``` v ``` are just as much as traveling from ``` v ``` to ``` u ```

#### Characteristics

• Duplicated identifiers with different coordinates are not allowed

• The coordinates are quite the same for the same identifier, for example

```1, 3.5, 1
1, 3.499999999999 0.9999999
```
• The coordinates are quite different for the same identifier, for example

```2 , 3.5, 1.0
2 , 3.6, 1.1
```
• Any duplicated identifier will be ignored. The coordinates that will be kept is arbitrarly.

### Signatures

Summary

```pgr_TSPeuclidean(Coordinates SQL, [start_id], [end_id])
RETURNS SETOF (seq, node, cost, agg_cost)
```
Example :

With default values

```SELECT * FROM pgr_TSPeuclidean(
\$\$
SELECT id, st_X(the_geom) AS x, st_Y(the_geom)AS y  FROM edge_table_vertices_pgr
\$\$);
seq  node       cost         agg_cost
-----+------+----------------+---------------
1     1               0              0
2     2               1              1
3     8   1.41421356237  2.41421356237
4     7               1  3.41421356237
5    14   1.58113883008  4.99535239246
6    15             1.5  6.49535239246
7    13             0.5  6.99535239246
8    17             1.5  8.49535239246
9    12   1.11803398875  9.61338638121
10     9               1  10.6133863812
11    16  0.583095189485  11.1964815707
12     6  0.583095189485  11.7795767602
13    11               1  12.7795767602
14    10               1  13.7795767602
15     5               1  14.7795767602
16     4    2.2360679775  17.0156447377
17     3               1  18.0156447377
18     1   1.41421356237     19.4298583
(18 rows)

```

### Parameters

Parameter

Type

Default

Description

Coordinates SQL

``` TEXT ```

An SQL query, described in the Coordinates SQL section

start_vid

``` BIGINT ```

``` 0 ```

The first visiting vertex

• When 0 any vertex can become the first visiting vertex.

end_vid

``` BIGINT ```

``` 0 ```

Last visiting vertex before returning to ``` start_vid ``` .

• When ``` 0 ``` any vertex can become the last visiting vertex before returning to ``` start_vid ``` .

• When ``` NOT 0 ``` and ``` start_vid = 0 ``` then it is the first and last vertex

### Inner query

#### Coordinates SQL

Coordinates SQL : an SQL query, which should return a set of rows with the following columns:

Column

Type

Description

id

``` ANY-INTEGER ```

Identifier of the starting vertex.

x

``` ANY-NUMERICAL ```

X value of the coordinate.

y

``` ANY-NUMERICAL ```

Y value of the coordinate.

### Result Columns

Returns SET OF ``` (seq, node, cost, agg_cost) ```

Column

Type

Description

seq

``` INTEGER ```

Row sequence.

node

``` BIGINT ```

Identifier of the node/coordinate/point.

cost

``` FLOAT ```

Cost to traverse from the current ``` node ``` to the next ``` node ``` in the path sequence.

• ``` 0 ``` for the last row in the tour sequence.

agg_cost

``` FLOAT ```

Aggregate cost from the ``` node ``` at ``` seq = 1 ``` to the current node.

• ``` 0 ``` for the first row in the tour sequence.

Example :

Test 29 cities of Western Sahara

This example shows how to make performance tests using University of Waterloo’s example data using the 29 cities of Western Sahara dataset

Creating a table for the data and storing the data

```CREATE TABLE wi29 (id BIGINT, x FLOAT, y FLOAT, geom geometry);
INSERT INTO wi29 (id, x, y) VALUES
(1,20833.3333,17100.0000),
(2,20900.0000,17066.6667),
(3,21300.0000,13016.6667),
(4,21600.0000,14150.0000),
(5,21600.0000,14966.6667),
(6,21600.0000,16500.0000),
(7,22183.3333,13133.3333),
(8,22583.3333,14300.0000),
(9,22683.3333,12716.6667),
(10,23616.6667,15866.6667),
(11,23700.0000,15933.3333),
(12,23883.3333,14533.3333),
(13,24166.6667,13250.0000),
(14,25149.1667,12365.8333),
(15,26133.3333,14500.0000),
(16,26150.0000,10550.0000),
(17,26283.3333,12766.6667),
(18,26433.3333,13433.3333),
(19,26550.0000,13850.0000),
(20,26733.3333,11683.3333),
(21,27026.1111,13051.9444),
(22,27096.1111,13415.8333),
(23,27153.6111,13203.3333),
(24,27166.6667,9833.3333),
(25,27233.3333,10450.0000),
(26,27233.3333,11783.3333),
(27,27266.6667,10383.3333),
(28,27433.3333,12400.0000),
(29,27462.5000,12992.2222);
```

Adding a geometry (for visual purposes)

```UPDATE wi29 SET geom = ST_makePoint(x,y);
```

Getting a total cost of the tour, compare the value with the length of an optimal tour is 27603, given on the dataset

```SELECT *
FROM pgr_TSPeuclidean(\$\$SELECT * FROM wi29\$\$)
WHERE seq = 30;
seq  node      cost         agg_cost
-----+------+---------------+---------------
30     1  2266.91173136  28777.4854127
(1 row)

```

Getting a geometry of the tour

```WITH
tsp_results AS (SELECT seq, geom FROM pgr_TSPeuclidean(\$\$SELECT * FROM wi29\$\$) JOIN wi29 ON (node = id))
SELECT ST_MakeLine(ARRAY(SELECT geom FROM tsp_results ORDER BY seq));
st_makeline
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Visualy, The first image is the optimal solution and the second image is the solution obtained with ``` pgr_TSPeuclidean ``` .