Applications of random sampling in computational geometry,II

We use random sampling for several new geometric algorithms. The algorithms are Las Vegas, and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requiresO(A+n logn) expected time, whereA is the number of intersecting pairs reported. The algorithm requiresO(n) space in the worst case. Another algorithm computes the convex hull ofn points inE ^d inO(n logn) expected time ford=3, andO(n ^[d/2]) expected time ford>3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set ofn points inE ³ inO(n logn) expected time, and on the way computes the intersection ofn unit balls inE ³. We show thatO(n logA) expected time suffices to compute the convex hull ofn points inE ³, whereA is the number of input points on the surface of the hull. Algorithms for halfspace range reporting are also given. In addition, we give asymptotically tight bounds for (k)-sets, which are certain halfspace partitions of point sets, and give a simple proof of Lee's bounds for high-order Voronoi diagrams.     

Efficient partition trees

We prove a theorem on partitioning point sets inE ^d (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space,O(n logn) deterministic preprocessing time, andO(n ^1?1/d (logn) ^O(1)) query time. WithO(nlogn) preprocessing time, where δ is an arbitrary positive constant, a more complicated data structure yields query timeO(n ^1?1/d (log logn) ^O(1)). This attains the lower bounds due to Chazelle [C1] up to polylogarithmic factors, improving and simplifying previous results of Chazelleet al. [CSW]. The partition result implies that, forr ^d≤n ^1?δ, a (1/r)-approximation of sizeO(r ^d) with respect to simplices for ann-point set inE ^d can be computed inO(n logr) deterministic time. A (1/r)-cutting of sizeO(r ^d) for a collection ofn hyperplanes inE ^d can be computed inO(n logr) deterministic time, provided thatr≤n ^1/(2d?1).     

Some simple in-place merging algorithms

The theoretical presentation and analysis is given for two families of simple in-place merging algorithms and their limiting cases. The first family merges stably inO(k·n) time andO(n ^1/k ) additional space with a limiting case running inO(n logn) time and constant space. The second family merges unstably inO (k ·n) time andO(log ^k n) space with a limiting case running inO(nG(n)) time and constant space. HereG(n) is the leastk such thatF(k) n whereF(0)=1 andF(i)=2 ^F(i–1) fori1. Each algorithm gives rise to a corresponding merge sort.     

Halfspace range search: An algorithmic application ofk-sets

Given a fixed setS ofn points inE ³ and a query plane, the halfspace range search problem asks for the retrieval of all points ofS on a chosen side of. We prove that withO(n(logn)⁸ (loglogn)⁴) storage it is possible to solve this problem inO(k+logn) time, wherek is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the total number ofj-sets (j=1, ...,k) realized by a set ofn points inE ³ isO(nk ⁵); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.Supported in part by NSF grants MCS 83-03925 and the Office of Naval Research and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146 and ARPA Order No. 4786.Supported in part by Joint Services Electronics Program under Contract N00014-79-C-0424.     

Euclidean minimum spanning trees and bichromatic closest pairs

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inE ^d in timeO(F _d (N,N) log ^d N), whereF _d (n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inE ^d . IfF _d (N,N)=Ω(N ^1+ε), for some fixed ɛ>0, then the running time improves toO(F _d (N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)^2/3+m log² n+n log² m) inE ₃, which yields anO(N ^4/3 log^4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE ³. Ind≥4 dimensions we obtain expected timeO((nm)^{1−1/([d/2]+1)+ε}+m logn+n logm) for the bichromatic closest pair problem andO(N ^{2−2/([d/2]+1)ε}) for the Euclidean minimum spanning tree problem, for any positive ɛ. The first, second, and fourth authors acknowledge support from the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center under NSF Grant STC 88-09648. The second author's work was supported by the National Science Foundation under Grant CCR-8714565. The third author's work was supported by the Deutsche Forschungsgemeinschaft under Grant A1 253/1-3, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”. The last two authors' work was also partially supported by the ESPRIT II Basic Research Action of the EC under Contract No. 3075 (project ALCOM).     

Augmenting Edge-Connectivity over the Entire Range inÕ(nm) Time

For a given undirected graphG = (V, E, c_G) with edges weighted by nonnegative realsc_G:E → R ⁺ , let Λ_G(k) stand for the minimum amount of weights which needs to be added to makeG k-edge-connected, and letG*(k) be the resulting graph obtained fromG. This paper first shows that function Λ_Gover the entire rangek [0, +∞] can be computed inO(nm + n² log n) time, and then shows that allG*(k) in the entire range can be obtained fromO(n log n) weighted cycles, and such cycles can be computed inO(nm + n² log n) time, wherenandmare the numbers of vertices and edges, respectively.     

Quasi-optimal range searching in spaces of finite VC-dimension

The range-searching problems that allow efficient partition trees are characterized as those defined by range spaces of finite Vapnik-Chervonenkis dimension. More generally, these problems are shown to be the only ones that admit linear-size solutions with sublinear query time in the arithmetic model. The proof rests on a characterization of spanning trees with a low stabbing number. We use probabilistic arguments to treat the general case, but we are able to use geometric techniques to handle the most common range-searching problems, such as simplex and spherical range search. We prove that any set ofn points inE ^d admits a spanning tree which cannot be cut by any hyperplane (or hypersphere) through more than roughlyn ^1–1/d edges. This result yields quasi-optimal solutions to simplex range searching in the arithmetic model of computation. We also look at polygon, disk, and tetrahedron range searching on a random access machine. Givenn points inE ², we derive a data structure of sizeO(n logn) for counting how many points fall inside a query convexk-gon (for arbitrary values ofk). The query time isO(kn logn). Ifk is fixed once and for all (as in triangular range searching), then the storage requirement drops toO(n). We also describe anO(n logn)-size data structure for counting how many points fall inside a query circle inO(n log² n) query time. Finally, we present anO(n logn)-size data structure for counting how many points fall inside a query tetrahedron in 3-space inO(n ^2/3 log² n) query time. All the algorithms are optimal within polylogarithmic factors. In all cases, the preprocessing can be done in polynomial time. Furthermore, the algorithms can also handle reporting within the same complexity (adding the size of the output as a linear term to the query time).Portions of this work have appeared in preliminary form in Partition trees for triangle counting and other range searching problems (E. Welzl),Proc. 4th Ann. ACM Symp. Comput. Geom. (1988), 23–33, and Tight Bounds on the Stabbing Number of Spanning Trees in Euclidean Space (B. Chazelle), Comput. Sci. Techn. Rep. No. CS-TR-155-88, Princeton University, 1988. Bernard Chazelle acknowledges the National Science Foundation for supporting this research in part under Grant CCR-8700917. Emo Welzl acknowledges the Deutsche Forschungsgemeinschaft for supporting this research in part under Grant We 1265/1-1.     

Triangulating point sets in space

A setP ofn points inR ^d is called simplicial if it has dimensiond and contains exactlyd + 1 extreme points. We show that whenP containsn interior points, there is always one point, called a splitter, that partitionsP intod + 1 simplices, none of which contain more thandn/(d + 1) points. A splitter can be found inO(d ⁴ +nd ²) time. Using this result, we give anO(nd ⁴ log_1+1/d n) algorithm for triangulating simplicial point sets that are in general position. InR ³ we give anO(n logn +k) algorithm for triangulating arbitrary point sets, wherek is the number of simplices produced. We exhibit sets of 2n + 1 points inR ³ for which the number of simplices produced may vary between (n – 1)² + 1 and 2n – 2. We also exhibit point sets for which every triangulation contains a quadratic number of simplices.Research supported by the Natural Science and Engineering Research Council grant A3013 and the F.C.A.R. grant EQ1678.     

Constructing arrangements optimally in parallel

We give two optimal parallel algorithms for constructing the arrangement ofn lines in the plane. The first nethod is quite simple and runs inO(log² n) time usingO(n ²) work, and the second method, which is more sophisticated, runs inO(logn) time usingO(n ²) work. This second result solves a well-known open problem in parallel computational geometry, and involves the use of a new algorithmic technique, the construction of an -pseudocutting. Our results immediately imply that the arrangement ofn hyperplanes in ^d inO(logn) time usingO(n ^d) work, for fixedd, can be optimally constructed. Our algorithms are for the CREW PRAM.This research was supported by the National Science Foundation under Grants CCR-8810568 and CCR-9003299, and by the NSF and DARPA under Grant CCR-8908092.     

Approximation algorithms for projective clustering

We consider the following two instances of the projective clustering problem: Given a set S of n points in and an integer k>0, cover S by k slabs (respectively d-cylinders) so that the maximum width of a slab (respectively the maximum diameter of a d-cylinder) is minimized. Let w^* be the smallest value so that S can be covered by k slabs (respectively d-cylinders), each of width (respectively diameter) at most w^*. This paper contains three main results: (i) For d=2, we present a randomized algorithm that computes O(klogk) strips of width at most w^* that cover S. Its expected running time is O(nk²log⁴n) if k²logkn; for larger values of k, the expected running time is O(n^2/3k^8/3log^14/3n). (ii) For d=3, a cover of S by O(klogk) slabs of width at most w^* can be computed in expected time O(n^3/2k^9/4polylog(n)). (iii) We compute a cover of by O(dklogk) d-cylinders of diameter at most 8w^* in expected time O(dnk³log⁴n). We also present a few extensions of this result.     

Cutting hyperplanes for divide-and-conquer

Givenn hyperplanes inE ^d, a (1/r)-cutting is a collection of simplices with disjoint interiors, which together coverE ^d and such that the interior of each simplex intersects at mostn/r hyperplanes. We present a deterministic algorithm for computing a (1/r)-cutting ofO(r ^d) size inO(nr ^d–1) time. If we require the incidences between the hyperplanes and the simplices of the cutting to be provided, then the algorithm is optimal. Our method is based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes. We mention several other applications of our result, e.g., counting segment intersections, Hopcroft's line/point incidence problem, linear programming in fixed dimension.This research was supported in part by the National Science Foundation under Grant CCR-9002352.     

On the expected number of k-sets

Given a setS ofn points inR ^d , a subsetX of sized is called ak-simplex if the hyperplane aff(X) has exactlyk points on one side. We studyE _d (k,n), the expected number of k-simplices whenS is a random sample ofn points from a probability distributionP onR ^d . WhenP is spherically symmetric we prove thatE _d (k, n)≤cn ^d−1 WhenP is uniform on a convex bodyK⊂R ² we prove thatE ₂ (k, n) is asymptotically linear in the rangecn≤k≤n/2 and whenk is constant it is asymptotically the expected number of vertices on the convex hull ofS. Finally, we construct a distributionP onR ² for whichE ₂((n−2)/2,n) iscn logn. The authors express gratitude to the NSF DIMACS Center at Rutgers and Princeton. The research of I. Bárány was supported in part by Hungarian National Science Foundation Grants 1907 and 1909, and W. Steiger's research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

Efficient Algorithms for Finding a Core of a Tree with a Specified Length

Acoreof a graphGis a pathPinGthat is central with respect to the property of minimizingd(P)=∑_v∈V(G)d(v, P), whered(v, P) is the distance from vertexvto pathP. This paper presents efficient algorithms for finding a core of a tree with a specified length. The sequential algorithm runs inO(n log n) time, wherenis the size of the tree. The parallel algorithm runs inO(log²n) time usingO(n) processors on an EREW PRAM model.     

On neighbouring matrices with quadratic elementary divisors

Summary Algorithms are presented which compute theQR factorization of an order-n Toeplitz matrix inO(n ²) operations. The first algorithm computes onlyR explicitly, and the second computes bothQ andR. The algorithms are derived from a well-known procedure for performing the rank-1 update ofQR factors, using the shift-invariance property of the Toeplitz matrix. The algorithms can be used to solve the Toeplitz least-squares problem, and can be modified to solve Toeplitz systems inO(n) space.     

Range Searching and Point Location among Fat Objects

We present a data structure that can store a set of disjoint fat objects ind-space such that point location and bounded-size range searching with arbitrarily shaped ranges can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or nonconvex (fat) polytopes. The multipurpose data structure supports point location and range searching queries in timeO(log^d−1 n) and requiresO(n log^d−1 n) storage, afterO(n log^d−1 n log log n) preprocessing. The data structure and query algorithm are rather simple.     

An optimal convex hull algorithm in any fixed dimension

We present a deterministic algorithm for computing the convex hull ofn points inE ^d in optimalO(n logn+n ^⌞d/2⌟ ) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n ^⌜d/2⌝ ) time. This research was supported in part by the National Science Foundation under Grant CCR-9002352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. A preliminary version of this paper has appeared in “An optimal convex hull algorithm and new results on cuttings”,Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science, October 1991, pp. 29–38. The convex hull algorithm given in the present paper, although similar in spirit, is considerably simpler than the one given in the proceedings.     

A fast and simple randomized parallel algorithm for the maximal independent set problem

A simple parallel randomized algorithm to find a maximal independent set in a graph G = (V, E) on n vertices is presented. Its expected running time on a concurrent-read concurrent-write PRAM with O(|E|d_max) processors is O(log n), where d_max denotes the maximum degree. On an exclusive-read exclusive-write PRAM with O(|E|) processors the algorithm runs in O(log²n). Previously, an O(log⁴n) deterministic algorithm was given by Karp and Wigderson for the EREW-PRAM model. This was recently (independently of our work) improved to O(log²n) by M. Luby. In both cases randomized algorithms depending on pairwise independent choices were turned into deterministic algorithms. We comment on how randomized combinatorial algorithms whose analysis only depends on d-wise rather than fully independent random choices (for some constant d) can be converted into deterministic algorithms. We apply a technique due to A. Joffe (1974) and obtain deterministic construction in fast parallel time of various combinatorial objects whose existence follows from probabilistic arguments.     

Cutting hyperplane arrangements

We consider a collectionH ofn hyperplanes in E ^d (where the dimensiond is fixed). An ε-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all E ^d and such that the interior of any simplex is intersected by at mostεn hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(r ^d ) simplices (which is asymptotically optimal). Forr≤n ^1−σ, where δ>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn) ^O(1) r ^d−1). In the plane we achieve a time boundO(nr) forr≤n ^1−δ, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For ann point setX⊆E ^d and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ϒ R; R is a simplex}), In timeO(n(logn) ^O(1) r ^d−1 +r ^O(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ε-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly. A preliminary version of this paper appeared inProceedings of the Sixth ACM Symposium on Computational Geometry, Berkeley, 1990, pp. 1–9. Work on this paper was supported by DIMACS Center.     

A sparse graph almost as good as the complete graph on points inK dimensions

A setV ofn points ink-dimensional space induces a complete weighted undirected graph as follows. The points are the vertices of this graph and the weight of an edge between any two points is the distance between the points under someL _p metric. Let ε≤1 be an error parameter and letk be fixed. We show how to extract inO(n logn+ε ^−k log(1/ε)n) time a sparse subgraphG=(V, E) of the complete graph onV such that: (a) for any two pointsx, y inV, the length of the shortest path inG betweenx andy is at most (1+∈) times the distance betweenx andy, and (b)|E|=O(ε^−k n).     

Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T) and a set A ? XA\in\mathcal{X} of positive measure, the set of integers n such that A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tⁿx)f(T²ⁿx)? f(T^knx)  dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)} , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d^*(E)>0 and for all {n ? \mathbbZ\colon d^*(E?(E+n)?(E+2n)?(E+3n)) > d^*(E)⁴-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}