2003 Apr 8 1

Indexing the Sky

Clive Page

2003 Apr 8 2

2003 Apr 8 3

Formats of Raw Data

• Radio:

– Complex visibility for each polarisation at set of points sampling the (u,v) plane.

• Infra-red, Optical, Ultra-violet:

– Images from 1k×1k to 18k×20k, collected every few seconds or few minutes.

• X-ray, Gamma-ray:

– Lists of detected photons (x, y, time, energy) typically accumulated for several hours.

2003 Apr 8 4

Formats of Reduced Data

• Images

• Time-series

• Spectra

• Source Catalogues:

– Vital to cross-identify sources from different wavebands, basis for many subsequent data mining investigations.

– Problem: can be large, examples:

USNO-B 1,045,913,669 rows 30 columns

1st XMM-Newton catalogue 56,711 rows 379 columns

2003 Apr 8 5

Required Functionality

• SELECT sources in given small patch of sky (circle, rectangle, or polygon)

• JOIN two tables e.g. from different wavebands to find corresponding sources

– Principal matching criterion is positional match - typically overlap of error-circles.

2003 Apr 8 6

Problems handling source catalogues• Positions use spherical-polar coordinates (RA, Dec)

– Right Ascension corresponds to geographic longitude

– Declination corresponds to geographic latitude

• There are singularities at the poles and distortions in the scales everywhere except at the equator.

• RA wraps from 24 hours (360 degrees) to zero.

• Two-dimensional indexing is really needed.

• All source positions are imprecise points have an error radius.

• Distances between points must use a great-circle distance function not cartesian distance.

2003 Apr 8 7

Indexing Possibilities

1. Use simple B-tree on one spatial axis only

2. Use 1-d to 2-d mapping function then B-tree

3. Use spatial index such as R-tree

2003 Apr 8 8

(1) Index one spatial axis only

• For example consider USNO-B: a table of a billion rows.

• Typical search/join uses a radius of say 3 arc-seconds.

• Probability of finding a source in a circle of radius 3 arc-seconds in a random position is around 17%, so most searches find 0 or 1 rows.

• An index on just one coordinate (say Dec) will effectively search a strip 360° wide by 6 arc-seconds high, and will find some 10,000 rows matching. These have to be scanned sequentially to find at most one matching row.

• Conclusion: a true 2-d index can gain five orders of magnitude in efficiency.

2003 Apr 8 9

(2) 2-d to 1-d mapping

• Cover the space with cells (pixels) and number them.

• Create conventional B-tree on resulting set of integers.

• Each point maps to an integer.

• Areas map to a list of integers:

– Ideally a small spatial area maps to a small range of integers so one can do a range search using the B-tree.

– Various space-filling curves such as the Z-ordering index and Peano Curve have been used in the hope that this works…

2003 Apr 8 10

Z-order mapping function

2003 Apr 8 11

Space-filling Curves

• All have same failing:

– At some places in the grid a high-order bit flips and the range of integers becomes huge.

– Tests confirm this defect: the worst-case performance is rather poor.

• Simple cartesian grids also unsuited to spherical-polar coordinates as there are too many tiny pixels near the poles.

2003 Apr 8 12

Covering the sky evenly with pixels

• Hierarchical Equal Area iso-Latitude Pixelisation (HEALPix) – invented at European Southern Observatory.

• Hierarchical Triangular Mesh (HTM) – invented at Johns Hopkins University

• Can use either algorithm – call it pixel-code or PCODE for short

– Do not try to conduct spatial range search using range of PCODE values.

2003 Apr 8 13

Hierarchical Equal Area iso-Latitude Pixelisation (HEALPix)

2003 Apr 8 14

Hierarchical Triangular Mesh (HTM)

2003 Apr 8 15

Spatial Join using PCODE

Table CAT1 has columns

• ID1

• RA

• DEC

• POSERR

• MAGNITUDE

• etc

Table CAT2 has columns

• ID2

• RA

• DEC

• POSERR

• FLUX

• etc

2003 Apr 8 16

Create additional tables with PCODE values

Table CAT1 has columns

• ID1 – primary key

• RA

• DEC

• POSERR

• MAGNITUDE

• Etc

Table P1 has columns

• ID1

• PCODE1 – primary key

• Table CAT2 has columns

• ID2 – primary key

• RA

• DEC

• POSERR

• FLUX

• Etc

Table P2 has columns

• ID2

• PCODE2 – primary key

2003 Apr 8 17

JOIN the two PCODE tables

Note: tables P1, P2 have extra rows when error-circles overlap more than one pixel.

• Join P1 and P2 on PCODE1=PCODE2 making a table PJOIN with just two columns: ID1 and ID2.

• Use SELECT DISTINCT to remove any duplicates

• Table PJOIN identifies pixels which may contain sources with overlapping error circles (or they may just be near but not overlapping)

• Create B-tree index on PJOIN(ID1)

2003 Apr 8 18

Use PJOIN table to match catalogue rows

• Three-way join then produces required results, e.g.

SELECT cols FROM CAT1, PJOIN, CAT2

WHERE CAT1.ID1=PJOIN.ID1

AND PJOIN.ID2=CAT2.ID2

AND (2 * asin(sqrt(pow(sin((cat1.dec-cat2.dec)/2),2) + cos(cat1.dec) * cos(cat2.dec) * pow(sin((cat1.ra-cat2.ra)/2),2))) <= cat1.poserr+cat2.poserr) ;

2003 Apr 8 19

(3) True Multi-dimensional Indexing

• Hot topic of research in computer science departments for more than 20 years

• Very many algorithms have been proposed:– BANG file, BV-tree, Buddy tree, Cell tree, G-tree, GBD-tree, Gridfile, hB-

tree, kd-tree, LSD-tree, P-tree, PK-tree, PLOP hashing, Pyramid tree, Q0-tree, Quadtree, R-tree, SKD-tree, SR-tree, SS-tree, TV-tree, UB-tree, Z-order index.

– So many alternatives, but none of them provides a good general solution, like the B-tree in 1-D indexing.

• R-tree indexing is built into several modern DBMS.

2003 Apr 8 20

Spatial Options in current DBMS

Commercial:

DB2 Spatial Extender – multi-level grid file

Ingres None

Oracle Spatial Option – R-tree (?)

SQL Server None

Sybase Spatial Option (Boeing SQS) – R-tree

Open Source:

MySQL R-tree in V4.1 (beta, documentation lacking)

Interbase None

PostgreSQL R-tree

2003 Apr 8 21

Using R-trees

Used R-trees in Postgres – does what it says on the box.Problems/limitations include:• Object indexed by R-tree is a rectangular box, so must

draw a box outside each error circle• Boxes get rather extended (along RA axis) near poles• Need a subsequent filter to remove spurious matches where

rectangles overlap but circles do not.• R-tree indices are large, creation is slow (2 hours for table

of 3.5 million rows using Postgres). – Kalpakis et al. used Informix to load part of USNO-A2

and found data load and R-tree creation would have taken 39 days for the entire 500M row table.

2003 Apr 8 22

Comparison of PCODE and R-tree

• Advantages– PCODE join seems to be faster (but not yet

benchmarked with identical systems).– Takes up less disc space in total.– Can use any DBMS, not just those with an R-tree or

other spatial data option.• Disadvantages

– Additional tables and indices have to be created– More complex set of joins. – Needs external code as neither HTM or HEALPix can

be expressed as an SQL-callable function (they return a variable-length array of integers).

2003 Apr 8 23

Conclusions

• Indexing on just one spatial axis is simply too inefficient for large tables.

• R-trees are powerful and easy to use, but index creation times are a serious cause for concern.

• 2d1d mapping functions such as HTM or HEALPix are more complicated to use, but may be worthwhile for JOINs if they turn out to be faster.