Data Collection for the Sloan Digital Sky Survey - A Network-Flow Heuristic

This note describes a combinatorial optimization problem arising in the Sloan Digital Sky Survey and an effective heuristic for the problem that has been implemented and will be used in the Survey. The heuristic is based on network flow theory. 1 The Sloan Digital Sky Survey The Sloan Digital Sky Survey [is] a joint project of the Astrophysical Research Consortium... The go+ of the project, which is scheduled to begin m 1997 and take five years, is to make a much better map of the universe than is currently available. The volume of the universe to be surveyed will be 100 times larger than the volume of previous surveys. The number of galaxies with known distances is ex ected to increase by a factor of 100 to 1 OOf 000 galaxies and the number of quasars t; i&ease to 100,000. The Sloan Foundation... has contributed $8 million to the $18 million capital costs of the project... In order to do the survey, ARC is designing and building a special purpose 2.5 meter (looinch) telescope at its Apache Point Observatory... [The Sky Survey will proceed in two phases. In the first phase, a two-dimensional ma of the sky will be made. For the secon % phase, the] million brightest stars and the one hundred thousand brightest quasars will be selected for spectroscopic analysis from the two-dimensional map.. . [29] To gather the spectroscopic data in the second phase, the telescope will be pointed repeatedly at ‘Astrophysics Department, Princeton University, Princeton, NJ 08540. E-mail : rhlQastro .princeton. edu. tMathematics Department, Princeton University, Princeton, NJ 08540. Email : fmaley@haverf ord. edu. t Computer Science Department, Dartmouth College, Hanover, NH 03755. Parts of this research were done while at: AT&T Bell Laboratories, Murray Hill, NJ 07974; the School of ORIE, Cornell University, Ithaca NY 14853 (on &a Tardos’ NSF PYI grant DDM-9157199); and the Dept’s of Astrophysics and Computer Science, Princeton University, Princeton, NJ 08540. E-mail : leal.YoungQdartmouth.edu. the sky to take a series of “snapshots”. Each snapshot will capture data for up to 660 galaxies (and quasars) in the circular portion of the sky visible through the telescope. For each captured galaxy, light from that galaxy will enter the telescope and travel through an optical fiber to a spectral analyzer. The optical fibers (one for each galaxy) will be held in place by a “plug plate” drilled to hold the up to 660 fibers, each aligned to accept the light of its respective galaxy [7]. 1.1 A Capacitated Covering Problem. A main factor determining the cost of the survey will be the number of snapshots to be taken. This paper concerns the following problem: given the desired galaxies, determine a minimum-size set of snapshots that captures them. Formally: Given a collection of points on the unit sphere, a radius T, and a capacity c, find a small set of discs of radius T (located on the sphere) such that each given point can be assigned to a disc containing it, with no disc being assigned more than c points. The sphere corresponds to the view-sphere centered at the telescope. The “galaxies” are the images of the actual galaxies projected on the view-sphere. Each disc represents one snapshot to be taken through the telescope; the galaxies assigned to that disc correspond to those for which data will be collected in that snapshot. The capacity c is the maximum number of galaxies for which spectral data can be gathered in a single snapshot (due to limitations in packing the optical fibers). The problem is NP-hard [23]. However, the instances we need to solve exhibit some structure. In this paper we describe an effective heuristic algorithm for the problem that will be used in the Survey and we report some computational results.