[Paper Review] An Efficient Algorithm for Positioning Tiles in the Sloan Digital Sky Survey

Michael R. Blanton, R. Lupton+4 more|CERN Bulletin|May 30, 2001

Astronomy and Astrophysical ResearchPhysics and Astronomy387 citations

TL;DR

This paper presents an efficient algorithm for positioning tiles and allocating fibers in the Sloan Digital Sky Survey (SDSS), addressing fiber collision constraints and non-uniform target distributions. Using a polynomial-time assignment method and a heuristic for tile placement, the approach achieves >90% efficiency and >92% completeness for all targets, with >99% for non-colliding targets, enabling near-optimal spectroscopic coverage across 10,000 square degrees.

ABSTRACT

The Sloan Digital Sky Survey (SDSS) will observe around 10^6 spectra from targets distributed over an area of about 10,000 square degrees, using a multi-object fiber spectrograph which can simultaneously observe 640 objects in a circular field-of-view (referred to as a ``tile'') 1.49 degrees in radius. No two fibers can be placed closer than 55'' during the same observation; multiple targets closer than this distance are said to ``collide.'' We present here a method of allocating fibers to desired targets given a set of tile centers which includes the effects of collisions and which is nearly optimally efficient and uniform. Because of large-scale structure in the galaxy distribution (which form the bulk of the SDSS targets), a naive covering the sky with equally-spaced tiles does not yield uniform sampling. Thus, we present a heuristic for perturbing the centers of the tiles from the equally-spaced distribution which provides more uniform completeness. For the SDSS sample, we can attain a sampling rate greater than 92% for all targets, and greater than 99% for the set of targets which do not collide with each other, with an efficiency greater than 90% (defined as the fraction of available fibers assigned to targets).

Motivation & Objective

• To address the challenge of efficiently assigning 640 fibers per tile to astronomical targets while respecting a 55'' minimum separation (fiber collision) constraint.
• To minimize the number of tiles required to cover the full 10,000 square degree survey area, reducing survey cost and observing time.
• To achieve uniform sky coverage despite large-scale structure in galaxy clustering, which causes non-uniform target distribution.
• To develop a near-optimal, computationally feasible method for tile placement and fiber allocation that scales to ~2,000 tiles and ~1 million targets.
• To define and utilize 'decollied' targets to improve completeness and enable accurate simulation of survey results.

Proposed method

• Uses a polynomial-time algorithm to optimally assign targets to tiles given fixed tile centers, even with overlapping fields of view and fiber collision constraints.
• Applies a friends-of-friends algorithm to group targets within 55'' of each other, treating each group as a unit for collision resolution.
• Employs a heuristic method from Lupton et al. (1998) to perturb tile centers from a regular grid, improving uniformity of coverage in regions of non-uniform target density.
• Introduces the concept of 'decollied' targets—those assigned to a tile only if they are not in conflict with others—enabling higher overall completeness.
• Processes the sky in overlapping 'chunks' (rectangular regions in spherical coordinates), allowing local tiling with global consistency.
• Tracks tile assignments via tileId, collisionGroup, and a three-bit mask (ASSIGNED, DECOLLIDED, COVERED) for each target.

Experimental results

Research questions

• RQ1 How can fiber assignment to targets be optimized when tiles overlap and fiber collisions are enforced?
• RQ2 What tile placement strategy minimizes the number of tiles needed while ensuring uniform sky coverage despite galaxy clustering?
• RQ3 How can the efficiency and completeness of the spectroscopic survey be maximized under practical constraints of fiber size and target distribution?
• RQ4 To what extent can a heuristic method approximate the optimal solution for an NP-complete tile placement problem in a large-scale survey?
• RQ5 How can decollided targets be defined and used to improve survey completeness and simulation fidelity?

Key findings

• The algorithm achieves a fiber assignment efficiency of >90%, defined as the fraction of available fibers assigned to targets.
• The survey attains a target completeness of >92% for all targets and >99% for non-colliding targets, significantly exceeding the initial goal of 99% for the collision-free subset.
• The heuristic tile placement method reduces coverage non-uniformity caused by large-scale structure in galaxy distributions, improving overall survey uniformity.
• The method enables nearly optimal fiber assignment in polynomial time, even with complex target distributions and overlapping fields of view.
• The use of decollided targets allows for accurate simulation of survey results and simplifies data analysis.
• The approach is scalable and has been validated on both SDSS commissioning data and simulations, confirming robust performance across different sky regions.