Efficient Algorithms for Geometric Optimization PANKAJ K. AGARWAL Duke University AND MICHA SHARIR Tel Aviv University We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LP-type problems and their efficient solution. We then describe a wide range of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems. Categories and Subject Descriptors: A.1 [General]: Introductory and Survey; F.2.2 [Theory of Computation]: Analysis of Algorithms and Problems—Geometrical problems and computations; I.1.2 [Computing Methodologies]: Algorithms—Analysis of algorithms General Terms: Algorithms, Design Additional Key Words and Phrases: Clustering, collision detection, linear programming, matrix searching, parametric searching, proximity problems, prune- and-search, randomized algorithms 1. INTRODUCTION variables subject to a large number of inequality (and equality) constraints. Combinatorial optimization typically Many problems can be formulated as deals with problems of maximizing or combinatorial optimization problems, minimizing a function of one or more which has made this a very active area Both authors are supported by a grant from the U.S.-Israeli Binational Science Foundation. P. K. Agarwal has also been supported by National Science Foundation Grant CCR-93-01259, Army Research Office MURI grant DAAH04-96-1-0013, a Sloan fellowship, and an NYI award and matching funds from Xerox Corp. M. Sharir has also been supported by NSF Grants CCR-94-24398 and CCR-93-11127, a Max-Planck Research Award, and a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary version of this article appeared as: P. K. Agarwal and M. Sharir, Algorithmic techniques for geometric optimization, in Computer Science Today: Recent Trends and Developments, LNCS 1000, J. van Leeuwen, Ed., 1995, pp. 234–253. Authors’ addresses: P. K. Agarwal, Center for Geometric Computing, Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129; M. Sharir, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012. Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. © 1999 ACM 0360-0300/99/1200–0412 $5.00 ACM Computing Surveys, Vol. 30, No. 4, December 1998 Algorithms for Geometric Optimization • 413 of research during the past half century. attempts to replace parametric search- In many applications, the underlying ing by alternative techniques, including optimization problem involves a con- randomization,1 expander graphs,2 geo- stant number of variables and a large metric cuttings [Agarwal et al. 1993c; number of constraints that are induced Bro¨nnimann and Chazelle 1994], and by a given collection of geometric ob- matrix searching.3 We present these al- jects; we refer to such problems as geo- ternative techniques in Section 3. metric-optimization problems. In such Almost concurrently with the develop- cases one expects that faster and sim- ment of the parametric-searching tech- pler algorithms can be developed by ex- nique, Megiddo [1983b, 1984] devised ploiting the geometric nature of the another ingenious technique for solving problem. Much work has been done on linear programming and several related geometric-optimization problems during optimization problems. This technique, the last 20 years, and many new elegant now known as decimation or prune-and- and sophisticated techniques have been search, was later refined and extended developed and successfully applied to by Dyer [1984], Clarkson [1986], and them. The aim of this article is to sur- others. The technique can be viewed as vey the main techniques and applica- an optimized version of parametric tions of this kind. searching, in which certain special The first part of this survey describes properties of the problem allow one to several general techniques that have improve further the efficiency of the led to efficient algorithms for a variety algorithm. For example, this technique of geometric-optimization problems, the yields linear-time deterministic algo- most notable of which is linear pro- rithms for linear programming and for gramming. The second part lists many several related problems, including the geometric applications of these tech- smallest-enclosing-ball problem, when niques and discusses some of them in the dimension is fixed. (However, the more detail. dependence of the running time of these The first technique that we present is algorithms on the dimension is at best called parametric searching. Although exponential.) We illustrate the tech- restricted versions of parametric nique in Section 4 by applying it to searching existed earlier (see, e.g., Eis- linear programming. ner and Severance [1976]), the full-scale In the past decade, randomized algo- technique was presented by Megiddo rithms have been developed for a vari- [1979, 1983a] in the late 1970s and ety of problems in computational geom- early 1980s. The technique was origi- etry and in other fields; see, for nally motivated by so-called parametric- example, the books by Mulmuley [1994] optimization problems in combinatorial and by Motwani and Raghavan [1995]. optimization, and did not receive much Clarkson [1995] and Seidel [1991] give attention by the computational geome- randomized algorithms for linear pro- try community until the late 1980s. In gramming, whose expected time is lin- the last decade, though, it has become ear in any fixed dimension, which are one of the major techniques for solving much simpler than their earlier deter- geometric-optimization problems effi- ministic counterparts. The dependence ciently. We outline the technique in de- on the dimension of the running time of tail in Section 2, first exemplifying it on the slope-selection problem [Cole et al. 1 Please see Agarwal and Sharir [1996a], Chan 1989], and then presenting various ex- [1998], Clarkson and Shor [1989], and Matous˘ek tensions of the technique. [1991b]. Despite its power and versatility, 2 Please see Ajtai and Megiddo [1996], Katz [1995], and Katz and Sharir [1993, 1997]. parametric searching has certain draw- 3 Please see Frederickson [1991], Frederickson backs, which we discuss next. Conse- and Johnson [1982, 1983, 1984], and Glozman et quently, there have been several recent al. [1995]. ACM Computing Surveys, Vol. 30, No. 4, December 1998 414 • P. K. Agarwal and M. Sharir these algorithms is better (although contains a given planar point set), still exponential). Actually, Clarkson’s placement and intersection of polygons technique is rather general, and is also and polyhedra (e.g., finding the largest applicable to a variety of other geomet- similar copy of a convex polygon that ric-optimization problems. We describe fits inside a given polygonal environ- this technique in Section 5. ment), and query-type problems (e.g., Further significant progress in linear the ray-shooting problem, in which we programming was made in the begin- want to preprocess a given collection of ning of the 1990s, when new random- objects, so that the first object hit by a ized algorithms for linear programming query ray can then be determined effi- were obtained independently by Kalai ciently). [1992], and by Matous˘ek et al. [1996; Numerous nongeometric-optimization Sharir and Welzl 1992] (these two algo- problems have also benefited from the rithms are essentially dual versions of techniques presented here (see Agar- the same technique). The expected num- wala and Ferna´ndez-Baca [1996], Co- ber of arithmetic operations performed hen and Megiddo [1993], Frederickson by these algorithms is subexponential [1991], Gusfield et al. [1994], and Nor- in the input size, and is still linear in ton et al. [1992] for a sample of such any fixed dimension, so the operations applications), but we focus only on geo- constitute an important step toward the metric applications. still open goal of obtaining strongly Although the common theme of most polynomial algorithms for linear pro- of the applications reviewed here is that gramming. (Recall that the polynomial- they can be solved efficiently using time algorithms by Khachiyan [1980] parametric-searching, prune-and-search, and by Karmarkar [1984] are not LP-type, or related techniques, each of strongly polynomial, as the number of them requires a problem-specific, and arithmetic operations performed by often fairly sophisticated, approach. For these algorithms depends on the size of example, the heart of a typical applica- the coefficients of the input con- tion of parametric searching is the de- straints.) This new technique is pre- sign