Arrangement Ofn Research Articles

On the bichromatic k -set problem

We study a bichromatic version of the well-known k-set problem : given two sets R and B of points of total size n and an integer k , how many subsets of the form (R ∩ h ) ∪ ( B ∖ h ) can have size exactly k over all halfspaces h ? In the dual, the problem is asymptotically equivalent to determining the worst-case combinatorial complexity of the k-level in an arrangement of n halfspaces . Disproving an earlier conjecture by Linhart [1993], we present the first nontrivial upper bound for all k ≪ n in two dimensions: O ( nk 1/3 + n 5/6−ϵ k 2/3+2 ϵ + k 2 ) for any fixed ϵ<0. In three dimensions, we obtain the bound O ( nk 3/2 + n 0.5034 k 2.4932 + k 3 ). Incidentally, this also implies a new upper bound for the original k -set problem in four dimensions: O ( n 2 k 3/2 + n 1.5034 k 2.4932 + n k 3 ), which improves the best previous result for all k ≪ n 0.923 . Extensions to other cases, such as arrangements of disks, are also discussed.

Asymptotic statistics of the n-sided planar Poisson–Voronoi cell: II. Heuristics

We develop a set of heuristic arguments to explain several results on planarPoisson–Voronoi tessellations that were derived earlier at the cost of considerablemathematical effort. The results concern Voronoi cells having a large numbern of sides. The arguments start from an entropy balance applied to the arrangement ofn neighborsaround a central cell. This is followed by a simplified evaluation of the phase space integral for the probabilitypn that anarbitrary cell be n-sided. The limitations of the arguments are indicated. As a new applicationwe calculate the expected number of Gabriel (or full) neighbors of ann-sided cellin the large-n limit.

Parallel algorithms for arrangements

We give the first efficient parallel algorithms for solving the arrangement problem. We give a deterministic algorithm for the CREW PRAM which runs in nearly optimal bounds ofO (logn log* n) time andn 2/logn processors. We generalize this to obtain anO (logn log* n)-time algorithm usingn d/logn processors for solving the problem ind dimensions. We also give a randomized algorithm for the EREW PRAM that constructs an arrangement ofn lines on-line, in which each insertion is done in optimalO (logn) time usingn/logn processors. Our algorithms develop new parallel data structures and new methods for traversing an arrangement.

Parallel Algorithms for Arrangements

We give the first efficient parallel algorithms for solving the arrangement problem. We give a deterministic algorithm for the CREW PRAM which runs in nearly optimal bounds ofO (logn log* n) time andn 2/logn processors. We generalize this to obtain anO (logn log* n)-time algorithm usingn d /logn processors for solving the problem ind dimensions. We also give a randomized algorithm for the EREW PRAM that constructs an arrangement ofn lines on-line, in which each insertion is done in optimalO (logn) time usingn/logn processors. Our algorithms develop new parallel data structures and new methods for traversing an arrangement.

Almost tight upper bounds for the single cell and zone problems in three dimensions

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement ofn low-degree algebraic surface patches in 3-space. We show that this complexity isO(n 2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries. This extends several previous results, almost settles a 9-year-old open problem, and has applications to motion planning of general robot systems with three degrees of freedom. As a corollary of the above result, we show that the overall complexity of all the three-dimensional cells of an arrangement ofn low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-calledzone of σ in the arrangement) isO(n 2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries.

Arrangements of segments that share endpoints: Single face results

We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement ofn line segments determined byh endpoints isO(h logh). While the previous upper bound,O(n?(n)), is tight for segments with distinct endpoints, it is far from being optimal whenn=Ω(h2). Our results show that, in a sense, the fundamental combinatorial complexity of a face arises not as a result of the number ofsegments, but rather as a result of the number ofendpoints.

Castles in the air revisited

We show that the total number of faces bounding any one cell in an arrangement ofn (d−1)-simplices in ℝd isO(n d−1 logn), thus almost settling a conjecture of Pach and Sharir. We present several applications of this result, mainly to translational motion planning in polyhedral environments. We than extend our analysis to derive other results on complexity in arrangements of simplices. For example, we show that in such an arrangement the total number of vertices incident to the same cell on more than one “side” isO(n d−1 logn). We, also show that the number of repetitions of a “k-flap,” formed by intersectingd−k given simplices, along the boundary of the same cell, summed over all cells and allk-flaps, isO(n d−1 log2 n). We use this quantity, which we call theexcess of the arrangement, to derive bounds on the complexity ofm distinct cells of such an arrangement.

Algorithms for ham-sandwich cuts

Given disjoint setsP 1,P 2, ...,P d inR d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP i . We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets inR d−1 . With the current estimates, we get complexity close toO(n 3/2 ) ford=3, roughlyO(n 8/3 ) ford=4, andO(n d−1−a(d) ) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR 3 when the three sets are suitably separated.

Arrangements of oriented hyperplanes

An arrangement ofn oriented hyperplanes or half-spaces dividesEd into a certain number of convex cells. We study the numberck of cells which are covered by exactlyk half-spaces and derive an upper bound onck for givenn andd.

Oriented matroids with few mutations

This paper defines a connected sum operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement ofn projective planes in ?Pd-1 contains at leastn simplicial regions and every plane is adjacent to at leastd simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4n pseudoplanes in ?P3 with only 3n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region. Finally we disprove the strong-map conjecture of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.

On the total weight of arrangements of halfplanes

An arrangement ofn halfplanes (or oriented lines) divides the plane into a certain number of convex cells. The weight of a cell is the number of halfplanes containing it, Presumably the sum of the weights of all cells of an arrangement ofn halfplanes attains its maximum if the halfplanes define an (almost) regularn-gon. We show that this is true at least for oddn.

Ray shooting on triangles in 3-space

We present a uniform approach to problems involving lines in 3-space. This approach is based on mapping lines inR3 into points and hyperplanes in five-dimensional projective space (Plucker space). We obtain new results on the following problems: 1. Preprocessn triangles so as to answer efficiently the query: “Given a ray, which is the first triangle hit?” (Ray- shooting problem). We discuss the ray-shooting problem for both disjoint and nondisjoint triangles. 2. Construct the intersection of two nonconvex polyhedra in an output sensitive way with asubquadratic overhead term. 3. Construct the arrangement ofn intersecting triangles in 3-space in an output-sensitive way, with asubquadratic overhead term. 4. Efficiently detect the first face hit by any ray in a set of axis-oriented polyhedra. 5. Preprocessn lines (segments) so as to answer efficiently the query “Given two lines, is it possible to move one into the other without crossing any of the initial lines (segments)?” (Isotopy problem). If the movement is possible produce an explicit representation of it.

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 ɛ-pseudocutting. Our results immediately imply that the arrangement ofn hyperplanes in ℝd inO(logn) time usingO(n d) work, for fixedd, can be optimally constructed. Our algorithms are for the CREW PRAM.

The number of edges of many faces in a line segment arrangement

We show that the maximum number of edges boundingm faces in an arrangement ofn line segments in the plane isO(m2/3n2/3+nα(n)+nlogm). This improves a previous upper bound of Edelsbrunner et al. [5] and almost matches the best known lower bound which is Ω(m2/3n2/3+nα(n)). In addition, we show that the number of edges bounding anym faces in an arrangement ofn line segments with a total oft intersecting pairs isO(m2/3t1/3+nα(t/n)+nmin{logm,logt/n}), almost matching the lower bound of Ω(m2/3t1/3+nα(t/n)) demonstrated in this paper.

A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra

We present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties: For example, the algorithm finds thev vertices of a polyhedron inR d defined by a nondegenerate system ofn inequalities (or, dually, thev facets of the convex hull ofn points inR d, where each facet contains exactlyd given points) in timeO(ndv) andO(nd) space. Thev vertices in a simple arrangement ofn hyperplanes inR d can be found inO(n 2 dv) time andO(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.

Counting facets and incidences

We show thatm distinct cells in an arrangement ofn planes in ℝ3 are bounded byO(m 2/3 n+n 2) faces, which in turn yields a tight bound on the maximum number of facets boundingm cells in an arrangement ofn hyperplanes in ℝd, for everyd≥3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in ℝ3. We also present a simpler proof of theO(m 2/3 n d/3+n d−1) bound on the number of incidences betweenn hyperplanes in ℝd andm vertices of their arrangement.

Lower bounds on the length of monotone paths in arrangements

We show that the maximal number of turns of anx-monotone path in an arrangement ofn lines is Ω(n 5/3) and the maximal number of turns of anx-monotone path in arrangement ofn pseudolines is Ω(n 2/logn).

Partitioning arrangements of lines II: Applications

In this paper we present efficient deterministic algorithms for various problems involving lines or segments in the plane, using the partitioning algorithm described in a companion paper [A3]. These applications include: (i) anO(m2/3n2/3 · log2/3n · log?/3 (m/?n)+(m+n) logn) algorithm to compute all incidences betweenm points andn lines, where ? is a constant <3.33; (ii) anO(m2/3n2/3 · log5/3n · log?/3 (m/?n)+(m+n) logn) algorithm to computem faces in an arrangement ofn lines; (iii) anO(n4/3 log(?+2)/3n) algorithm to count the number of intersections in a set ofn segments; (iv) anO(n4/3 log(? + 2)/3n) algorithm to count red-blue intersections between two sets of segments, and (v) anO(n3/2 log?/3n) algorithm to compute spanning trees with low stabbing number for a set ofn points. We also present an algorithm that, given set ofn points in the plane, preprocesses it, in timeO(n?m log?+1/2n), into a data structure of sizeO(m) forn logn≤m≤n2, so that the number of points ofS lying inside a query triangle can be computed inO((n/?m) log3/2n) time.

Triangles in space or building (and analyzing) castles in the air

We show that the total combinatorial complexity of all non-convex cells in an arrangement ofn (possibly intersecting) triangles in 3-space isO(n 7/3 logn) and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement and an alternative less efficient, but still subcubic algorithm for calculating all non-convex cells, analyze some special cases of the problem where improved bounds (and faster algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.

The complexity of many cells in arrangements of planes and related problems

We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m2/3n logn +n2); we can calculatem such cells specified by a point in each, in worst-case timeO(m2/3n log3n+n2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m2/3n logn+n2), but this number is onlyO(m3/5??n4/5+2?+m+n logm), for any?>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m3/4??n3/4+3?+m) log2n+n logn logm) for any?>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m3/4??n3/4+3?+m] log2n+n logn logm) for any?>0. (v) The maximum number of facets (i.e., (d?1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m2/3nd/3 logn+nd?1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.