pgr_findCloseEdges
¶
pgr_findCloseEdges
- Finds the close edges to a point geometry.
Availability
Version 3.4.0
New proposed signatures:
pgr_findCloseEdges
(One point)pgr_findCloseEdges
(Many points)
Description¶
pgr_findCloseEdges
- An utility function that finds the closest edge to a
point geometry.
The geometries must be in the same coordinate system (have the same SRID).
The code to do the calculations can be obtained for further specific adjustments needed by the application.
EMTPY SET
is returned on dryrun executions
Signatures¶
Summary
[cap, partial, dryrun]
(edge_id, fraction, side, distance, geom, edge)
One point¶
[cap, partial, dryrun]
(edge_id, fraction, side, distance, geom, edge)
- Example:
With default values
Default:
cap => 1
Maximum one row answer.
Default:
partial => true
With less calculations as possible.
Default:
dryrun => false
Process query
Returns
values on
edge_id
,fraction
,side
columns.NULL
ondistance
,geom
,edge
columns.
SELECT *
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT geom FROM pointsOfInterest WHERE pid = 5),
0.5);
edge_id | fraction | side | distance | geom | edge
---------+----------+------+----------+------+------
5 | 0.8 | l | | |
(1 row)
Many points¶
[cap, partial, dryrun]
(edge_id, fraction, side, distance, geom, edge)
- Example:
Find at most \(2\) edges close to all vertices on the points of interest table.
One answer per point, as small as possible.
SELECT edge_id, round(fraction::numeric, 2) AS fraction, side, ST_AsText(geom) AS original_point
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT array_agg(geom) FROM pointsOfInterest),
0.5);
edge_id | fraction | side | original_point
---------+----------+------+----------------
1 | 0.40 | l | POINT(1.8 0.4)
6 | 0.30 | r | POINT(0.3 1.8)
12 | 0.60 | l | POINT(2.6 3.2)
15 | 0.40 | r | POINT(4.2 2.4)
5 | 0.80 | l | POINT(2.9 1.8)
4 | 0.70 | r | POINT(2.2 1.7)
(6 rows)
Columns edge_id
, fraction
, side
and geom
are returned with
values.
geom
contains the original point geometry to assist on deterpartialing to which
point geometry the row belongs to.
Parameters¶
Parameter |
Type |
Description |
---|---|---|
|
Edges SQL as described below. |
|
point |
|
The point geometry |
points |
|
An array of point geometries |
tolerance |
|
Max distance between geometries |
Optional parameters¶
Parameter |
Type |
Default |
Description |
---|---|---|---|
|
|
\(1\) |
Limit output rows |
|
|
|
|
|
|
|
|
Inner Queries¶
Edges SQL¶
Column |
Type |
Description |
---|---|---|
|
ANY-INTEGER |
Identifier of the edge. |
|
|
The |
Result Columns¶
Returns set of (edge_id, fraction, side, distance, geom, edge)
Column |
Type |
Description |
---|---|---|
|
|
Identifier of the edge.
|
|
|
Value in <0,1> that indicates the relative postition from the first end-point of the edge. |
|
|
Value in
|
|
|
Distance from point to edge.
|
|
|
|
|
|
|
One point results
The green nodes is the original point
The geometry
geom
is a point on the \(sp \rightarrow ep\) edge.The geometry
edge
is a line that connects the original point withgeom
Many point results
The green nodes are the original points
The geometry
geom
, marked as g1 and g2 are the original pointsThe geometry
edge
, marked as edge1 and edge2 is a line that connects the original point with the closest point on the \(sp \rightarrow ep\) edge.
Additional Examples¶
One point examples¶
At most two answers¶
cap => 2
Maximum two row answer.
Default:
partial => true
With less calculations as possible.
Default:
dryrun => false
Process query
SELECT *
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT geom FROM pointsOfInterest WHERE pid = 5),
0.5, cap => 2);
edge_id | fraction | side | distance | geom | edge
---------+--------------------+------+---------------------+------+------
5 | 0.8 | l | 0.10000000000000009 | |
8 | 0.8999999999999999 | r | 0.19999999999999996 | |
(2 rows)
Understanding the result
NULL
ongeom
,edge
edge_id
identifier of the edge close to the original pointTwo edges are withing \(0.5\) distance units from the original point: \({5, 8}\)
For edge \(5\):
fraction
: The closest point from the original point is at the \(0.8\) fraction of the edge \(5\).side
: The original point is located to the left side of edge \(5\).distance
: The original point is located \(0.1\) length units from edge \(5\).
For edge \(8\):
fraction
: The closest point from the original point is at the \(0.89..\) fraction of the edge \(8\).side
: The original point is located to the right side of edge \(8\).distance
: The original point is located \(0.19..\) length units from edge \(8\).
One answer, all columns¶
Default:
cap => 1
Maximum one row answer.
partial => false
Calculate all columns
Default:
dryrun => false
Process query
SELECT edge_id, round(fraction::numeric, 2) AS fraction, side, round(distance::numeric, 3) AS distance,
ST_AsText(geom) AS new_point,
ST_AsText(edge) AS original_to_new_point
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT geom FROM pointsOfInterest WHERE pid = 5),
0.5, partial => false);
edge_id | fraction | side | distance | new_point | original_to_new_point
---------+----------+------+----------+--------------+---------------------------
5 | 0.80 | l | 0.100 | POINT(3 1.8) | LINESTRING(2.9 1.8,3 1.8)
(1 row)
Understanding the result
edge_id
identifier of the edge closest to the original pointFrom all edges within \(0.5\) distance units from the original point: \({5}\) is the closest one.
For edge \(5\):
fraction
: The closest point from the original point is at the \(0.8\) fraction of the edge \(5\).side
: The original point is located to the left side of edge \(5\).distance
: The original point is located \(0.1\) length units from edge \(5\).geom
: Contains the geometry of the closest point on edge \(5\) from the original point.edge
: Contains theLINESTRING
geometry of the original point to the closest point on on edge \(5\)geom
At most two answers with all columns¶
cap => 2
Maximum two row answer.
partial => false
Calculate all columns
Default:
dryrun => false
Process query
SELECT edge_id, round(fraction::numeric, 2) AS fraction, side, round(distance::numeric, 3) AS distance,
ST_AsText(geom) AS new_point,
ST_AsText(edge) AS original_to_new_point
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT geom FROM pointsOfInterest WHERE pid = 5),
0.5, cap => 2, partial => false);
edge_id | fraction | side | distance | new_point | original_to_new_point
---------+----------+------+----------+--------------+---------------------------
5 | 0.80 | l | 0.100 | POINT(3 1.8) | LINESTRING(2.9 1.8,3 1.8)
8 | 0.90 | r | 0.200 | POINT(2.9 2) | LINESTRING(2.9 1.8,2.9 2)
(2 rows)
Understanding the result:
edge_id
identifier of the edge close to the original pointTwo edges are withing \(0.5\) distance units from the original point: \({5, 8}\)
For edge \(5\):
fraction
: The closest point from the original point is at the \(0.8\) fraction of the edge \(5\).side
: The original point is located to the left side of edge \(5\).distance
: The original point is located \(0.1\) length units from edge \(5\).geom
: Contains the geometry of the closest point on edge \(5\) from the original point.edge
: Contains theLINESTRING
geometry of the original point to the closest point on on edge \(5\)geom
For edge \(8\):
fraction
: The closest point from the original point is at the \(0.89..\) fraction of the edge \(8\).side
: The original point is located to the right side of edge \(8\).distance
: The original point is located \(0.19..\) length units from edge \(8\).geom
: Contains the geometry of the closest point on edge \(8\) from the original point.edge
: Contains theLINESTRING
geometry of the original point to the closest point on on edge \(8\)geom
One point dry run execution¶
Returns
EMPTY SET
.partial => true
Is ignored
Because it is a dry run excecution, the code for all calculations are shown on the PostgreSQL
NOTICE
.
dryrun => true
Do not process query
Generate a PostgreSQL
NOTICE
with the code used to calculate all columnscap
and original point are used in the code
SELECT *
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT geom FROM pointsOfInterest WHERE pid = 5),
0.5,
dryrun => true);
NOTICE:
WITH
edges_sql AS (SELECT id, geom FROM edges),
point_sql AS (SELECT '01010000003333333333330740CDCCCCCCCCCCFC3F'::geometry AS point)
SELECT
id::BIGINT AS edge_id,
ST_LineLocatePoint(geom, point) AS fraction,
CASE WHEN ST_Intersects(ST_Buffer(geom, 0.5, 'side=right endcap=flat'), point)
THEN 'r'
ELSE 'l' END::CHAR AS side,
geom <-> point AS distance,
ST_ClosestPoint(geom, point) AS new_point,
ST_MakeLine(point, ST_ClosestPoint(geom, point)) AS new_line
FROM edges_sql, point_sql
WHERE ST_DWithin(geom, point, 0.5)
ORDER BY geom <-> point LIMIT 1
edge_id | fraction | side | distance | geom | edge
---------+----------+------+----------+------+------
(0 rows)
Many points examples¶
At most two answers per point¶
cap => 2
Maximum two row answer.
Default:
partial => true
With less calculations as possible.
Default:
dryrun => false
Process query
SELECT edge_id, round(fraction::numeric, 2) AS fraction, side, round(distance::numeric, 3) AS distance,
ST_AsText(geom) AS geom_is_original, edge
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT array_agg(geom) FROM pointsOfInterest),
0.5, cap => 2);
edge_id | fraction | side | distance | geom_is_original | edge
---------+----------+------+----------+------------------+------
1 | 0.40 | l | 0.200 | POINT(1.8 0.4) |
6 | 0.30 | r | 0.200 | POINT(0.3 1.8) |
12 | 0.60 | l | 0.200 | POINT(2.6 3.2) |
11 | 1.00 | l | 0.447 | POINT(2.6 3.2) |
15 | 0.40 | r | 0.200 | POINT(4.2 2.4) |
9 | 1.00 | l | 0.447 | POINT(4.2 2.4) |
5 | 0.80 | l | 0.100 | POINT(2.9 1.8) |
8 | 0.90 | r | 0.200 | POINT(2.9 1.8) |
4 | 0.70 | r | 0.200 | POINT(2.2 1.7) |
8 | 0.20 | r | 0.300 | POINT(2.2 1.7) |
(10 rows)
Understanding the result
NULL
onedge
edge_id
identifier of the edge close to a original point (geom
)Two edges at most withing \(0.5\) distance units from each of the original points:
For
POINT(1.8 0.4)
andPOINT(0.3 1.8)
only one edge was found.For the rest of the points two edges were found.
For point
POINT(2.9 1.8)
Edge \(5\) is before \(8\) therefore edge \(5\) has the shortest distance to
POINT(2.9 1.8)
.For edge \(5\):
fraction
: The closest point from the original point is at the \(0.8\) fraction of the edge \(5\).side
: The original point is located to the left side of edge \(5\).distance
: The original point is located \(0.1\) length units from edge \(5\).
For edge \(8\):
fraction
: The closest point from the original point is at the \(0.89..\) fraction of the edge \(8\).side
: The original point is located to the right side of edge \(8\).distance
: The original point is located \(0.19..\) length units from edge \(8\).
One answer per point, all columns¶
Default:
cap => 1
Maximum one row answer.
partial => false
Calculate all columns
Default:
dryrun => false
Process query
SELECT edge_id, round(fraction::numeric, 2) AS fraction, side, round(distance::numeric, 3) AS distance,
ST_AsText(geom) AS geom_is_original,
ST_AsText(edge) AS original_to_new_point
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT array_agg(geom) FROM pointsOfInterest),
0.5, partial => false);
edge_id | fraction | side | distance | geom_is_original | original_to_new_point
---------+----------+------+----------+------------------+---------------------------
1 | 0.40 | l | 0.200 | POINT(1.8 0.4) | LINESTRING(1.8 0.4,2 0.4)
6 | 0.30 | r | 0.200 | POINT(0.3 1.8) | LINESTRING(0.3 1.8,0.3 2)
12 | 0.60 | l | 0.200 | POINT(2.6 3.2) | LINESTRING(2.6 3.2,2.6 3)
15 | 0.40 | r | 0.200 | POINT(4.2 2.4) | LINESTRING(4.2 2.4,4 2.4)
5 | 0.80 | l | 0.100 | POINT(2.9 1.8) | LINESTRING(2.9 1.8,3 1.8)
4 | 0.70 | r | 0.200 | POINT(2.2 1.7) | LINESTRING(2.2 1.7,2 1.7)
(6 rows)
Understanding the result
edge_id
identifier of the edge closest to the original pointFrom all edges within \(0.5\) distance units from the original point: \({5}\) is the closest one.
For the original point
POINT(2.9 1.8)
Edge \(5\) is the closest edge to the original point
fraction
: The closest point from the original point is at the \(0.8\) fraction of the edge \(5\).side
: The original point is located to the left side of edge \(5\).distance
: The original point is located \(0.1\) length units from edge \(5\).geom
: Contains the geometry of the original pointPOINT(2.9 1.8)
edge
: Contains theLINESTRING
geometry of the original point (geom
) to the closest point on on edge.
Many points dry run execution¶
Returns
EMPTY SET
.partial => true
Is ignored
Because it is a dry run excecution, the code for all calculations are shown on the PostgreSQL
NOTICE
.
dryrun => true
Do not process query
Generate a PostgreSQL
NOTICE
with the code used to calculate all columnscap
and original point are used in the code
SELECT *
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT array_agg(geom) FROM pointsOfInterest),
0.5,
dryrun => true);
NOTICE:
WITH
edges_sql AS (SELECT id, geom FROM edges),
point_sql AS (SELECT unnest('{0101000000CDCCCCCCCCCCFC3F9A9999999999D93F:0101000000333333333333D33FCDCCCCCCCCCCFC3F:0101000000CDCCCCCCCCCC04409A99999999990940:0101000000CDCCCCCCCCCC10403333333333330340:01010000003333333333330740CDCCCCCCCCCCFC3F:01010000009A99999999990140333333333333FB3F}'::geometry[]) AS point),
results AS (
SELECT
id::BIGINT AS edge_id,
ST_LineLocatePoint(geom, point) AS fraction,
CASE WHEN ST_Intersects(ST_Buffer(geom, 0.5, 'side=right endcap=flat'), point)
THEN 'r'
ELSE 'l' END::CHAR AS side,
geom <-> point AS distance,
point,
ST_MakeLine(point, ST_ClosestPoint(geom, point)) AS new_line
FROM edges_sql, point_sql
WHERE ST_DWithin(geom, point, 0.5)
ORDER BY geom <-> point),
prepare_cap AS (
SELECT row_number() OVER (PARTITION BY point ORDER BY point, distance) AS rn, *
FROM results)
SELECT edge_id, fraction, side, distance, point, new_line
FROM prepare_cap
WHERE rn <= 1
edge_id | fraction | side | distance | geom | edge
---------+----------+------+----------+------+------
(0 rows)
Find at most two routes to a given point¶
Using pgr_withPoints - Proposed
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 geom FROM pointsOfInterest WHERE pid = 5),
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)
A point of interest table¶
Handling points outside the graph.
Points of interest¶
Some times the applications work “on the fly” starting from a location that is not a vertex in the graph. Those locations, in pgRrouting are called points of interest.
The information needed in the points of interest is pid
, edge_id
,
side
, fraction
.
On this documentation there will be some 6 fixed points of interest and they will be stored on a table.
Column |
Description |
---|---|
|
A unique identifier. |
|
Identifier of the edge nearest edge that allows an arrival to the point. |
|
Is it on the left, right or both sides of the segment |
|
Where in the segment is the point located. |
|
The geometry of the points. |
|
The geometry of the points moved on top of the segment. |
CREATE TABLE pointsOfInterest(
pid BIGSERIAL PRIMARY KEY,
edge_id BIGINT,
side CHAR,
fraction FLOAT,
geom geometry,
newPoint geometry);
CREATE TABLE
Points of interest geometry¶
Inserting the data of the points of interest:
INSERT INTO pointsOfInterest (geom) VALUES
(ST_POINT(1.8, 0.4)),
(ST_POINT(4.2, 2.4)),
(ST_POINT(2.6, 3.2)),
(ST_POINT(0.3, 1.8)),
(ST_POINT(2.9, 1.8)),
(ST_POINT(2.2, 1.7));
INSERT 0 6
Points of interest fillup¶
Using pgr_findCloseEdges Calculating for visual purposes the points over the graph.
UPDATE pointsOfInterest AS p SET
edge_id = q.edge_id,
side = q.side,
fraction = q.fraction,
newPoint = ST_endPoint(q.edge)
FROM (SELECT * FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges$$,
(SELECT array_agg(geom) FROM pointsOfInterest),
0.5, partial => false)) AS q
WHERE p.geom = q.geom;
UPDATE 6
A special case to arrive from both sides of the edge.¶
UPDATE pointsOfInterest SET side = 'b'
WHERE pid = 6;
UPDATE 1
Points of interest data¶
SELECT pid, edge_id, side, fraction,
ST_AsText(geom), ST_AsText(newPoint)
FROM pointsOfInterest
ORDER BY pid;
pid | edge_id | side | fraction | st_astext | st_astext
-----+---------+------+--------------------+----------------+--------------
1 | 1 | l | 0.4 | POINT(1.8 0.4) | POINT(2 0.4)
2 | 15 | r | 0.3999999999999999 | POINT(4.2 2.4) | POINT(4 2.4)
3 | 12 | l | 0.6000000000000001 | POINT(2.6 3.2) | POINT(2.6 3)
4 | 6 | r | 0.3 | POINT(0.3 1.8) | POINT(0.3 2)
5 | 5 | l | 0.8 | POINT(2.9 1.8) | POINT(3 1.8)
6 | 4 | b | 0.7 | POINT(2.2 1.7) | POINT(2 1.7)
(6 rows)
Connecting disconnected components¶
To get the graph connectivity:
SELECT * FROM pgr_connectedComponents(
'SELECT id, source, target, cost, reverse_cost FROM edges'
);
seq | component | node
-----+-----------+------
1 | 1 | 1
2 | 1 | 3
3 | 1 | 5
4 | 1 | 6
5 | 1 | 7
6 | 1 | 8
7 | 1 | 9
8 | 1 | 10
9 | 1 | 11
10 | 1 | 12
11 | 1 | 13
12 | 1 | 14
13 | 1 | 15
14 | 1 | 16
15 | 1 | 17
16 | 1 | 18
17 | 2 | 2
18 | 2 | 4
(18 rows)
In this example, the component \(2\) consists of vertices \(\{2, 4\}\) and both vertices are also part of the dead end result set.
This graph needs to be connected.
Note
With the original graph of this documentation, there would be 3 components as the crossing edge in this graph is a different component.
Prepare storage for connection information¶
ALTER TABLE vertices ADD COLUMN component BIGINT;
ALTER TABLE
ALTER TABLE edges ADD COLUMN component BIGINT;
ALTER TABLE
Save the vertices connection information¶
UPDATE vertices SET component = c.component
FROM (SELECT * FROM pgr_connectedComponents(
'SELECT id, source, target, cost, reverse_cost FROM edges'
)) AS c
WHERE id = node;
UPDATE 18
Save the edges connection information¶
UPDATE edges SET component = v.component
FROM (SELECT id, component FROM vertices) AS v
WHERE source = v.id;
UPDATE 20
Get the closest vertex¶
Using pgr_findCloseEdges the closest vertex to component \(1\) is vertex \(4\). And the closest edge to vertex \(4\) is edge \(14\).
SELECT edge_id, fraction, ST_AsText(edge) AS edge, id AS closest_vertex
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges WHERE component = 1$$,
(SELECT array_agg(geom) FROM vertices WHERE component = 2),
2, partial => false) JOIN vertices USING (geom) ORDER BY distance LIMIT 1;
edge_id | fraction | edge | closest_vertex
---------+----------+--------------------------------------+----------------
14 | 0.5 | LINESTRING(1.999999999999 3.5,2 3.5) | 4
(1 row)
The edge
can be used to connect the components, using the fraction
information about the edge \(14\) to split the connecting edge.
Connecting components¶
There are three basic ways to connect the components
From the vertex to the starting point of the edge
From the vertex to the ending point of the edge
From the vertex to the closest vertex on the edge
This solution requires the edge to be split.
The following query shows the three ways to connect the components:
WITH
info AS (
SELECT
edge_id, fraction, side, distance, ce.geom, edge, v.id AS closest,
source, target, capacity, reverse_capacity, e.geom AS e_geom
FROM pgr_findCloseEdges(
$$SELECT id, geom FROM edges WHERE component = 1$$,
(SELECT array_agg(geom) FROM vertices WHERE component = 2),
2, partial => false) AS ce
JOIN vertices AS v USING (geom)
JOIN edges AS e ON (edge_id = e.id)
ORDER BY distance LIMIT 1),
three_options AS (
SELECT
closest AS source, target, 0 AS cost, 0 AS reverse_cost,
capacity, reverse_capacity,
ST_X(geom) AS x1, ST_Y(geom) AS y1,
ST_X(ST_EndPoint(e_geom)) AS x2, ST_Y(ST_EndPoint(e_geom)) AS y2,
ST_MakeLine(geom, ST_EndPoint(e_geom)) AS geom
FROM info
UNION
SELECT closest, source, 0, 0, capacity, reverse_capacity,
ST_X(geom) AS x1, ST_Y(geom) AS y1,
ST_X(ST_StartPoint(e_geom)) AS x2, ST_Y(ST_StartPoint(e_geom)) AS y2,
ST_MakeLine(info.geom, ST_StartPoint(e_geom))
FROM info
/*
UNION
-- This option requires splitting the edge
SELECT closest, NULL, 0, 0, capacity, reverse_capacity,
ST_X(geom) AS x1, ST_Y(geom) AS y1,
ST_X(ST_EndPoint(edge)) AS x2, ST_Y(ST_EndPoint(edge)) AS y2,
edge
FROM info */
)
INSERT INTO edges
(source, target,
cost, reverse_cost,
capacity, reverse_capacity,
x1, y1, x2, y2,
geom)
(SELECT
source, target, cost, reverse_cost, capacity, reverse_capacity,
x1, y1, x2, y2, geom
FROM three_options);
INSERT 0 2
Checking components¶
Ignoring the edge that requires further work. The graph is now fully connected as there is only one component.
SELECT * FROM pgr_connectedComponents(
'SELECT id, source, target, cost, reverse_cost FROM edges'
);
seq | component | node
-----+-----------+------
1 | 1 | 1
2 | 1 | 2
3 | 1 | 3
4 | 1 | 4
5 | 1 | 5
6 | 1 | 6
7 | 1 | 7
8 | 1 | 8
9 | 1 | 9
10 | 1 | 10
11 | 1 | 11
12 | 1 | 12
13 | 1 | 13
14 | 1 | 14
15 | 1 | 15
16 | 1 | 16
17 | 1 | 17
18 | 1 | 18
(18 rows)
See Also¶
Indices and tables