pgr_withPoints - Proposed

pgr_withPoints - Returns the shortest path in a graph with additional temporary vertices.

Warning

Proposed functions for next mayor release.

  • They are not officially in the current release.

  • They will likely officially be part of the next mayor release:

    • The functions make use of ANY-INTEGER and ANY-NUMERICAL

    • Name might not change. (But still can)

    • Signature might not change. (But still can)

    • Functionality might not change. (But still can)

    • pgTap tests have being done. But might need more.

    • Documentation might need refinement.

_images/boost-inside.jpeg

Boost Graph Inside

Availability

  • Version 3.2.0

    • New proposed function:

      • pgr_withPoints(Combinations)

  • Version 2.2.0

    • New proposed function

Description

Modify the graph to include points defined by points_sql. Using Dijkstra algorithm, find the shortest path(s)

The main characteristics are:

  • Process is done only on edges with positive costs.

  • Vertices of the graph are:

    • positive when it belongs to the edges_sql

    • negative when it belongs to the points_sql

  • Values are returned when there is a path.

    • When the starting vertex and ending vertex are the same, there is no path. - The agg_cost the non included values (v, v) is 0

    • When the starting vertex and ending vertex are the different and there is no path: - The agg_cost the non included values (u, v) is ∞

  • For optimization purposes, any duplicated value in the start_vids or end_vids are ignored.

  • The returned values are ordered: - start_vid ascending - end_vid ascending

  • Running time: \(O(|start\_vids|\times(V \log V + E))\)

Signatures

Summary

pgr_withPoints(Edges SQL, Points SQL, start vid, end vid, [options])
pgr_withPoints(Edges SQL, Points SQL, start vid, end vids, [options])
pgr_withPoints(Edges SQL, Points SQL, start vids, end vid, [options])
pgr_withPoints(Edges SQL, Points SQL, start vids, end vids, [options])
pgr_withPoints(Edges SQL, Points SQL, Combinations SQL, [options])
options: [directed, driving_side, details])
RETURNS SET OF (seq, path_seq, [start_pid], [end_pid], node, edge, cost, agg_cost)
OR EMTPY SET

One to One

pgr_withPoints(Edges SQL, Points SQL, start vid, end vid, [options])
options: [directed, driving_side, details])
RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost)
OR EMTPY SET
Example:

From point \(1\) to vertex \(10\) with details

SELECT * FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  -1, 10,
  details => true);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
   1 |        1 |   -1 |    1 |  0.6 |        0
   2 |        2 |    6 |    4 |  0.7 |      0.6
   3 |        3 |   -6 |    4 |  0.3 |      1.3
   4 |        4 |    7 |    8 |    1 |      1.6
   5 |        5 |   11 |    9 |    1 |      2.6
   6 |        6 |   16 |   16 |    1 |      3.6
   7 |        7 |   15 |    3 |    1 |      4.6
   8 |        8 |   10 |   -1 |    0 |      5.6
(8 rows)

One to Many

pgr_withPoints(Edges SQL, Points SQL, start vid, end vids, [options])
options: [directed, driving_side, details])
RETURNS SET OF (seq, path_seq, end_pid, node, edge, cost, agg_cost)
OR EMTPY SET
Example:

From point \(1\) to point \(3\) and vertex \(7\) on an undirected graph

SELECT * FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  -1, ARRAY[-3, 7],
  directed => false);
 seq | path_seq | end_pid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
   1 |        1 |      -3 |   -1 |    1 |  0.6 |        0
   2 |        2 |      -3 |    6 |    4 |    1 |      0.6
   3 |        3 |      -3 |    7 |   10 |    1 |      1.6
   4 |        4 |      -3 |    8 |   12 |  0.6 |      2.6
   5 |        5 |      -3 |   -3 |   -1 |    0 |      3.2
   6 |        1 |       7 |   -1 |    1 |  0.6 |        0
   7 |        2 |       7 |    6 |    4 |    1 |      0.6
   8 |        3 |       7 |    7 |   -1 |    0 |      1.6
(8 rows)

Many to One

pgr_withPoints(Edges SQL, Points SQL, start vids, end vid, [options])
options: [directed, driving_side, details])
RETURNS SET OF (seq, path_seq, start_pid, node, edge, cost, agg_cost)
OR EMTPY SET
Example:

From point \(1\) and vertex \(6\) to point \(3\)

SELECT * FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  ARRAY[-1, 6], -3);
 seq | path_seq | start_pid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
   1 |        1 |        -1 |   -1 |    1 |  0.6 |        0
   2 |        2 |        -1 |    6 |    4 |    1 |      0.6
   3 |        3 |        -1 |    7 |   10 |    1 |      1.6
   4 |        4 |        -1 |    8 |   12 |  0.6 |      2.6
   5 |        5 |        -1 |   -3 |   -1 |    0 |      3.2
   6 |        1 |         6 |    6 |    4 |    1 |        0
   7 |        2 |         6 |    7 |   10 |    1 |        1
   8 |        3 |         6 |    8 |   12 |  0.6 |        2
   9 |        4 |         6 |   -3 |   -1 |    0 |      2.6
(9 rows)

Many to Many

pgr_withPoints(Edges SQL, Points SQL, start vids, end vids, [options])
options: [directed, driving_side, details])
RETURNS SET OF (seq, path_seq, start_pid, end_pid, node, edge, cost, agg_cost)
OR EMTPY SET
Example:

From point \(1\) and vertex \(6\) to point \(3\) and vertex \(1\)

SELECT * FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  ARRAY[-1, 6], ARRAY[-3, 1]);
 seq | path_seq | start_pid | end_pid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
   1 |        1 |        -1 |      -3 |   -1 |    1 |  0.6 |        0
   2 |        2 |        -1 |      -3 |    6 |    4 |    1 |      0.6
   3 |        3 |        -1 |      -3 |    7 |   10 |    1 |      1.6
   4 |        4 |        -1 |      -3 |    8 |   12 |  0.6 |      2.6
   5 |        5 |        -1 |      -3 |   -3 |   -1 |    0 |      3.2
   6 |        1 |        -1 |       1 |   -1 |    1 |  0.6 |        0
   7 |        2 |        -1 |       1 |    6 |    4 |    1 |      0.6
   8 |        3 |        -1 |       1 |    7 |    7 |    1 |      1.6
   9 |        4 |        -1 |       1 |    3 |    6 |    1 |      2.6
  10 |        5 |        -1 |       1 |    1 |   -1 |    0 |      3.6
  11 |        1 |         6 |      -3 |    6 |    4 |    1 |        0
  12 |        2 |         6 |      -3 |    7 |   10 |    1 |        1
  13 |        3 |         6 |      -3 |    8 |   12 |  0.6 |        2
  14 |        4 |         6 |      -3 |   -3 |   -1 |    0 |      2.6
  15 |        1 |         6 |       1 |    6 |    4 |    1 |        0
  16 |        2 |         6 |       1 |    7 |    7 |    1 |        1
  17 |        3 |         6 |       1 |    3 |    6 |    1 |        2
  18 |        4 |         6 |       1 |    1 |   -1 |    0 |        3
(18 rows)

Combinations

pgr_withPoints(Edges SQL, Points SQL, Combinations SQL, [options])
options: [directed, driving_side, details])
RETURNS SET OF (seq, path_seq, start_pid, end_pid, node, edge, cost, agg_cost)
OR EMTPY SET
Example:

Two combinations

From point \(1\) to vertex \(10\), and from vertex \(6\) to point \(3\) with right side driving.

SELECT * FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  'SELECT * FROM (VALUES (-1, 10), (6, -3)) AS combinations(source, target)',
  driving_side => 'r', details => true);
 seq | path_seq | start_pid | end_pid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
   1 |        1 |        -1 |      10 |   -1 |    1 |  0.4 |        0
   2 |        2 |        -1 |      10 |    5 |    1 |    1 |      0.4
   3 |        3 |        -1 |      10 |    6 |    4 |  0.7 |      1.4
   4 |        4 |        -1 |      10 |   -6 |    4 |  0.3 |      2.1
   5 |        5 |        -1 |      10 |    7 |    8 |    1 |      2.4
   6 |        6 |        -1 |      10 |   11 |    9 |    1 |      3.4
   7 |        7 |        -1 |      10 |   16 |   16 |    1 |      4.4
   8 |        8 |        -1 |      10 |   15 |    3 |    1 |      5.4
   9 |        9 |        -1 |      10 |   10 |   -1 |    0 |      6.4
  10 |        1 |         6 |      -3 |    6 |    4 |  0.7 |        0
  11 |        2 |         6 |      -3 |   -6 |    4 |  0.3 |      0.7
  12 |        3 |         6 |      -3 |    7 |   10 |    1 |        1
  13 |        4 |         6 |      -3 |    8 |   12 |  0.6 |        2
  14 |        5 |         6 |      -3 |   -3 |   -1 |    0 |      2.6
(14 rows)

Parameters

Column

Type

Description

Edges SQL

TEXT

Edges SQL as described below

Points SQL

TEXT

Points SQL as described below

Combinations SQL

TEXT

Combinations SQL as described below

start vid

BIGINT

Identifier of the starting vertex of the path. Negative value is for point’s identifier.

start vids

ARRAY[BIGINT]

Array of identifiers of starting vertices. Negative values are for point’s identifiers.

end vid

BIGINT

Identifier of the ending vertex of the path. Negative value is for point’s identifier.

end vids

ARRAY[BIGINT]

Array of identifiers of ending vertices. Negative values are for point’s identifiers.

Optional parameters

Column

Type

Default

Description

directed

BOOLEAN

true

  • When true the graph is considered Directed

  • When false the graph is considered as Undirected.

With points optional parameters

Parameter

Type

Default

Description

driving_side

CHAR

b

Value in [r, l, b] indicating if the driving side is:

  • r for right driving side.

  • l for left driving side.

  • b for both.

details

BOOLEAN

false

  • When true the results will include the points that are in the path.

  • When false the results will not include the points that are in the path.

Inner Queries

Edges SQL

Column

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)

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

Points SQL

Parameter

Type

Default

Description

pid

ANY-INTEGER

value

Identifier of the point.

  • Use with positive value, as internally will be converted to negative value

  • If column is present, it can not be NULL.

  • If column is not present, a sequential negative value will be given automatically.

edge_id

ANY-INTEGER

Identifier of the “closest” edge to the point.

fraction

ANY-NUMERICAL

Value in <0,1> that indicates the relative postition from the first end point of the edge.

side

CHAR

b

Value in [b, r, l, NULL] indicating if the point is:

  • In the right r,

  • In the left l,

  • In both sides b, NULL

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

Result Columns

Returns set of (seq, path_seq [, start_pid] [, end_pid], node, edge, cost, agg_cost)

Column

Type

Description

seq

INTEGER

Sequential value starting from 1.

path_seq

INTEGER

Relative position in the path.

  • 1 For the first row of the path.

start_pid

BIGINT

Identifier of a starting vertex/point of the path.

  • When positive is the identifier of the starting vertex.

  • When negative is the identifier of the starting point.

  • Returned on Many to One and Many to Many

end_pid

BIGINT

Identifier of an ending vertex/point of the path.

  • When positive is the identifier of the ending vertex.

  • When negative is the identifier of the ending point.

  • Returned on One to Many and Many to Many

node

BIGINT

Identifier of the node in the path from start_pid to end_pid.

  • When positive is the identifier of the a vertex.

  • When negative is the identifier of the a point.

edge

BIGINT

Identifier of the edge used to go from node to the next node in the path sequence.

  • -1 for the last row of the path.

cost

FLOAT

Cost to traverse from node using edge to the next node in the path sequence.

  • 0 For the first row of the path.

agg_cost

FLOAT

Aggregate cost from start_vid to node.

  • 0 For the first row of the path.

Additional Examples

Use pgr_findCloseEdges in the Points SQL.

Find the routes from vertex \(1\) to the two closest locations on the graph of point (2.9, 1.8).

SELECT * FROM pgr_withPoints(
  $e$ SELECT * FROM edges $e$,
  $p$ SELECT edge_id, round(fraction::numeric, 2) AS fraction, side
      FROM pgr_findCloseEdges(
        $$SELECT id, geom FROM edges$$,
        (SELECT ST_POINT(2.9, 1.8)),
        0.5, cap => 2)
  $p$,
  1, ARRAY[-1, -2]);
 seq | path_seq | end_pid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
   1 |        1 |      -2 |    1 |    6 |    1 |        0
   2 |        2 |      -2 |    3 |    7 |    1 |        1
   3 |        3 |      -2 |    7 |    8 |  0.9 |        2
   4 |        4 |      -2 |   -2 |   -1 |    0 |      2.9
   5 |        1 |      -1 |    1 |    6 |    1 |        0
   6 |        2 |      -1 |    3 |    7 |    1 |        1
   7 |        3 |      -1 |    7 |    8 |    1 |        2
   8 |        4 |      -1 |   11 |    9 |    1 |        3
   9 |        5 |      -1 |   16 |   16 |    1 |        4
  10 |        6 |      -1 |   15 |    3 |    1 |        5
  11 |        7 |      -1 |   10 |    5 |  0.8 |        6
  12 |        8 |      -1 |   -1 |   -1 |    0 |      6.8
(12 rows)

  • Point \(-1\) corresponds to the closest edge from point (2.9,1.8).

  • Point \(-2\) corresponds to the next close edge from point (2.9,1.8).

Usage variations

All the examples are about traveling from point \(1\) and vertex \(5\) to points \(\{2, 3, 6\}\) and vertices \(\{10, 11\}\)

SELECT *
FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  ARRAY[5, -1], ARRAY[-2, -3, -6, 10, 11],
  driving_side => 'r', details => true);
 seq | path_seq | start_pid | end_pid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
   1 |        1 |        -1 |      -6 |   -1 |    1 |  0.4 |        0
   2 |        2 |        -1 |      -6 |    5 |    1 |    1 |      0.4
   3 |        3 |        -1 |      -6 |    6 |    4 |  0.7 |      1.4
   4 |        4 |        -1 |      -6 |   -6 |   -1 |    0 |      2.1
   5 |        1 |        -1 |      -3 |   -1 |    1 |  0.4 |        0
   6 |        2 |        -1 |      -3 |    5 |    1 |    1 |      0.4
   7 |        3 |        -1 |      -3 |    6 |    4 |  0.7 |      1.4
   8 |        4 |        -1 |      -3 |   -6 |    4 |  0.3 |      2.1
   9 |        5 |        -1 |      -3 |    7 |   10 |    1 |      2.4
  10 |        6 |        -1 |      -3 |    8 |   12 |  0.6 |      3.4
  11 |        7 |        -1 |      -3 |   -3 |   -1 |    0 |        4
  12 |        1 |        -1 |      -2 |   -1 |    1 |  0.4 |        0
  13 |        2 |        -1 |      -2 |    5 |    1 |    1 |      0.4
  14 |        3 |        -1 |      -2 |    6 |    4 |  0.7 |      1.4
  15 |        4 |        -1 |      -2 |   -6 |    4 |  0.3 |      2.1
  16 |        5 |        -1 |      -2 |    7 |    8 |    1 |      2.4
  17 |        6 |        -1 |      -2 |   11 |    9 |    1 |      3.4
  18 |        7 |        -1 |      -2 |   16 |   15 |  0.4 |      4.4
  19 |        8 |        -1 |      -2 |   -2 |   -1 |    0 |      4.8
  20 |        1 |        -1 |      10 |   -1 |    1 |  0.4 |        0
  21 |        2 |        -1 |      10 |    5 |    1 |    1 |      0.4
  22 |        3 |        -1 |      10 |    6 |    4 |  0.7 |      1.4
  23 |        4 |        -1 |      10 |   -6 |    4 |  0.3 |      2.1
  24 |        5 |        -1 |      10 |    7 |    8 |    1 |      2.4
  25 |        6 |        -1 |      10 |   11 |    9 |    1 |      3.4
  26 |        7 |        -1 |      10 |   16 |   16 |    1 |      4.4
  27 |        8 |        -1 |      10 |   15 |    3 |    1 |      5.4
  28 |        9 |        -1 |      10 |   10 |   -1 |    0 |      6.4
  29 |        1 |        -1 |      11 |   -1 |    1 |  0.4 |        0
  30 |        2 |        -1 |      11 |    5 |    1 |    1 |      0.4
  31 |        3 |        -1 |      11 |    6 |    4 |  0.7 |      1.4
  32 |        4 |        -1 |      11 |   -6 |    4 |  0.3 |      2.1
  33 |        5 |        -1 |      11 |    7 |    8 |    1 |      2.4
  34 |        6 |        -1 |      11 |   11 |   -1 |    0 |      3.4
  35 |        1 |         5 |      -6 |    5 |    1 |    1 |        0
  36 |        2 |         5 |      -6 |    6 |    4 |  0.7 |        1
  37 |        3 |         5 |      -6 |   -6 |   -1 |    0 |      1.7
  38 |        1 |         5 |      -3 |    5 |    1 |    1 |        0
  39 |        2 |         5 |      -3 |    6 |    4 |  0.7 |        1
  40 |        3 |         5 |      -3 |   -6 |    4 |  0.3 |      1.7
  41 |        4 |         5 |      -3 |    7 |   10 |    1 |        2
  42 |        5 |         5 |      -3 |    8 |   12 |  0.6 |        3
  43 |        6 |         5 |      -3 |   -3 |   -1 |    0 |      3.6
  44 |        1 |         5 |      -2 |    5 |    1 |    1 |        0
  45 |        2 |         5 |      -2 |    6 |    4 |  0.7 |        1
  46 |        3 |         5 |      -2 |   -6 |    4 |  0.3 |      1.7
  47 |        4 |         5 |      -2 |    7 |    8 |    1 |        2
  48 |        5 |         5 |      -2 |   11 |    9 |    1 |        3
  49 |        6 |         5 |      -2 |   16 |   15 |  0.4 |        4
  50 |        7 |         5 |      -2 |   -2 |   -1 |    0 |      4.4
  51 |        1 |         5 |      10 |    5 |    1 |    1 |        0
  52 |        2 |         5 |      10 |    6 |    4 |  0.7 |        1
  53 |        3 |         5 |      10 |   -6 |    4 |  0.3 |      1.7
  54 |        4 |         5 |      10 |    7 |    8 |    1 |        2
  55 |        5 |         5 |      10 |   11 |    9 |    1 |        3
  56 |        6 |         5 |      10 |   16 |   16 |    1 |        4
  57 |        7 |         5 |      10 |   15 |    3 |    1 |        5
  58 |        8 |         5 |      10 |   10 |   -1 |    0 |        6
  59 |        1 |         5 |      11 |    5 |    1 |    1 |        0
  60 |        2 |         5 |      11 |    6 |    4 |  0.7 |        1
  61 |        3 |         5 |      11 |   -6 |    4 |  0.3 |      1.7
  62 |        4 |         5 |      11 |    7 |    8 |    1 |        2
  63 |        5 |         5 |      11 |   11 |   -1 |    0 |        3
(63 rows)

Passes in front or visits with right side driving.

For point \(6\) and vertex \(11\).

SELECT (start_pid || ' -> ' || end_pid ||' at ' || path_seq || 'th step')::TEXT AS path_at,
  CASE WHEN edge = -1 THEN ' visits'
      ELSE ' passes in front of'
  END as status,
  CASE WHEN node < 0 THEN 'Point'
      ELSE 'Vertex'
  END as is_a,
  abs(node) as id
FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  ARRAY[5, -1], ARRAY[-2, -3, -6, 10, 11],
  driving_side => 'r', details => true)
WHERE node IN (-6, 11);
       path_at        |       status        |  is_a  | id
----------------------+---------------------+--------+----
 -1 -> -6 at 4th step |  visits             | Point  |  6
 -1 -> -3 at 4th step |  passes in front of | Point  |  6
 -1 -> -2 at 4th step |  passes in front of | Point  |  6
 -1 -> -2 at 6th step |  passes in front of | Vertex | 11
 -1 -> 10 at 4th step |  passes in front of | Point  |  6
 -1 -> 10 at 6th step |  passes in front of | Vertex | 11
 -1 -> 11 at 4th step |  passes in front of | Point  |  6
 -1 -> 11 at 6th step |  visits             | Vertex | 11
 5 -> -6 at 3th step  |  visits             | Point  |  6
 5 -> -3 at 3th step  |  passes in front of | Point  |  6
 5 -> -2 at 3th step  |  passes in front of | Point  |  6
 5 -> -2 at 5th step  |  passes in front of | Vertex | 11
 5 -> 10 at 3th step  |  passes in front of | Point  |  6
 5 -> 10 at 5th step  |  passes in front of | Vertex | 11
 5 -> 11 at 3th step  |  passes in front of | Point  |  6
 5 -> 11 at 5th step  |  visits             | Vertex | 11
(16 rows)

Passes in front or visits with left side driving.

For point \(6\) and vertex \(11\).

SELECT (start_pid || ' => ' || end_pid ||' at ' || path_seq || 'th step')::TEXT AS path_at,
  CASE WHEN edge = -1 THEN ' visits'
      ELSE ' passes in front of'
  END as status,
  CASE WHEN node < 0 THEN 'Point'
        ELSE 'Vertex'
    END as is_a,
    abs(node) as id
FROM pgr_withPoints(
  'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
  'SELECT pid, edge_id, fraction, side from pointsOfInterest',
  ARRAY[5, -1], ARRAY[-2, -3, -6, 10, 11],
  driving_side => 'l', details => true)
WHERE node IN (-6, 11);
       path_at        |       status        |  is_a  | id
----------------------+---------------------+--------+----
 -1 => -6 at 3th step |  visits             | Point  |  6
 -1 => -3 at 3th step |  passes in front of | Point  |  6
 -1 => -2 at 3th step |  passes in front of | Point  |  6
 -1 => -2 at 5th step |  passes in front of | Vertex | 11
 -1 => 10 at 3th step |  passes in front of | Point  |  6
 -1 => 10 at 5th step |  passes in front of | Vertex | 11
 -1 => 11 at 3th step |  passes in front of | Point  |  6
 -1 => 11 at 5th step |  visits             | Vertex | 11
 5 => -6 at 4th step  |  visits             | Point  |  6
 5 => -3 at 4th step  |  passes in front of | Point  |  6
 5 => -2 at 4th step  |  passes in front of | Point  |  6
 5 => -2 at 6th step  |  passes in front of | Vertex | 11
 5 => 10 at 4th step  |  passes in front of | Point  |  6
 5 => 10 at 6th step  |  passes in front of | Vertex | 11
 5 => 11 at 4th step  |  passes in front of | Point  |  6
 5 => 11 at 6th step  |  visits             | Vertex | 11
(16 rows)

See Also

Indices and tables