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Showing papers in "Discrete and Computational Geometry in 1991"


Journal ArticleDOI
TL;DR: In this paper, a deterministic algorithm for triangulating a simple polygon in linear time is presented. But the main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals.
Abstract: We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

592 citations


Journal ArticleDOI
TL;DR: Two randomized algorithms solve linear programs involvingm constraints ind variables in expected timeO(m) and constructs convex hulls ofn points in ℝd,d>3, in expectedTimeO(n[d/2]).
Abstract: We present two randomized algorithms. One solves linear programs involvingm constraints ind variables in expected timeO(m). The other constructs convex hulls ofn points in ?d,d>3, in expected timeO(n[d/2]). In both boundsd is considered to be a constant. In the linear programming algorithm the dependence of the time bound ond is of the formd!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.

265 citations


Journal ArticleDOI
TL;DR: A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram ofn independent and identically distributed points and it is shown that in this case, the complexity of the diagram is ∵(n) for fixedd.
Abstract: A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram ofn independent and identically distributed points. The method is applied to derive exact asymptotic bounds on the expected number of vertices of the Voronoi diagram of points chosen from the uniform distribution on the interior of ad-dimensional ball; it is shown that in this case, the complexity of the diagram is ?(n) for fixedd. An algorithm for constructing the Voronoid diagram is presented and analyzed. The algorithm is shown to require only ?(n) time on average for random points from ad-ball assuming a real-RAM model of computation with a constant-time floor function. This algorithm is asymptotically faster than any previously known and optimal in the average-case sense.

205 citations


Journal ArticleDOI
TL;DR: A randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)2/3+m log2n+n log2m) inE3, which yields anO(N4/3 log 4/3N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points in E3.
Abstract: We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inEd in timeO(Fd(N,N) logdN), whereFd(n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inEd. IfFd(N,N)=Ω(N1+?), for some fixed ?>0, then the running time improves toO(Fd(N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)2/3+m log2n+n log2m) inE3, which yields anO(N4/3 log4/3N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE3. Ind?4 dimensions we obtain expected timeO((nm)1?1/([d/2]+1)+?+m logn+n logm) for the bichromatic closest pair problem andO(N2?2/([d/2]+1)?) for the Euclidean minimum spanning tree problem, for any positive ?.

153 citations


Journal ArticleDOI
TL;DR: A new index for convex polytopes is introduced, a vector whose length is the dimension of the linear span of the flag vectors of poly topes, equivalent to the generalized Dehn-Sommerville equations.
Abstract: A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized Dehn-Sommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual.

142 citations


Journal ArticleDOI
TL;DR: It is proved that it can be decided whether translated copies of the polyomino can tile the plane, and every such tiling of the plane by translated Copy of the Polyomino is half-periodic.
Abstract: Given a polyomino, we prove that we can decide whether translated copies of the polyomino can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider. If such a tiling exists the polyomino is called anexact polyomino. Further, every such tiling of the plane by translated copies of the polyomino is half-periodic. Moreover, all the possible surroundings of an exact polyomino are described in a simple way.

119 citations


Journal ArticleDOI
Pravin M. Vaidya1
TL;DR: It is shown 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 for any two pointsx, y inV, the length of the shortest path inG betweenx andy is at most (1+∈) times the distance between x andy, and E|=O(ε−kn).
Abstract: 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 someLp metric. Let ?≤1 be an error parameter and letk be fixed. We show how to extract inO(n logn+??k log(1/?)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(??kn).

118 citations


Journal ArticleDOI
TL;DR: A deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices with asymptotically optimal running time is given, which has numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.
Abstract: We consider a collectionH ofn hyperplanes in Ed (where the dimensiond is fixed). An ?-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is intersected by at most?n hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices (which is asymptotically optimal). Forr≤n1??, where ?>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn)O(1)rd?1). In the plane we achieve a time boundO(nr) forr≤n1??, 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⊆Ed and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ? R; R is a simplex}), In timeO(n(logn)O(1)rd?1+rO(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ?-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.

113 citations


Journal ArticleDOI
TL;DR: A digital, three-dimensional, analogue of the Jordan curve theorem is proved, which is a digital topological formulation of theJordan-Brouwer theorem about surfaces that separate three- dimensional space into two connected components.
Abstract: Many applications of digital image processing now deal with three-dimensional images (the third dimension can be time or a spatial dimension). In this paper we develop a topological model for digital three space which can be useful in this context. In particular, we prove a digital, three-dimensional, analogue of the Jordan curve theorem. (The Jordan curve theorem states that a simple closed curve separates the real plane into two connected components.) Our theorem here is a digital topological formulation of the Jordan-Brouwer theorem about surfaces that separate three-dimensional space into two connected components.

107 citations


Journal ArticleDOI
TL;DR: It is shown that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph, and that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, which achieves the minimum number of crossings from among all such drawings.
Abstract: This paper presents a connection between the problem of drawing a graph with the minimum number of edge crossings, and the theory of arrangements of pseudolines, a topic well-studied by combinatorialists. In particular, we show that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph. Using some recent results of Goodman, Pollack, and Sturmfels, this yields that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, which achieves the minimum number of crossings from among all such drawings. While this result has no bearing on the P versus NP question, it is fairly negative with regard to applications. We also study the problem of drawing a graph with polygonal edges, to achieve the (unrestricted) minimum number of crossings. Here we obtain a tight bound on the smallest number of breakpoints which are required in the polygonal lines.

96 citations


Journal ArticleDOI
TL;DR: The formal power series ∑k≥0 dimℝ Ckr(Δ)λk is considered and it is shown that this always has the form P(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Γ).
Abstract: For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk We consider the formal power series ?k?0 dim? Ckr(Δ)?k and show, under mild conditions on Δ, that this always has the formP(?)/(1??)d+1, whereP(?) is a polynomial in ? with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP?(1)=(r+1)fd?10(Δ) We discuss how the polynomialP(?) and bases for the spacesCkr(Δ) can be effectively calculated by use of Grobner basis techniques of computational commutative algebra A further application is given to the theory of hyperplane arrangements

Journal ArticleDOI
TL;DR: This paper gives efficient, randomized algorithms for the following problems: construction of levels of order 1 tok in an arrangement of hyperplanes in any dimension and construction of higher-order Voronoi diagrams of order1 tokIn any dimension.
Abstract: This paper gives efficient, randomized algorithms for the following problems: (1) construction of levels of order 1 tok in an arrangement of hyperplanes in any dimension and (2) construction of higher-order Voronoi diagrams of order 1 tok in any dimension. A new combinatorial tool in the form of a mathematical series, called a ? series, is associated with an arrangement of hyperplanes inRd. It is used to study the combinatorial as well as algorithmic complexity of the geometric problems under consideration.

Journal ArticleDOI
TL;DR: It is shown that the abstract Voronoi diagram of n sites in the plane can be constructed in timeO(n logn) by a randomized algorithm based on Clarkson and Shor's randomized incremental construction technique.
Abstract: We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7].

Journal ArticleDOI
TL;DR: A tracking lemma and a loop-elimination theorem are proved, both of which are applicable to the case of arbitrary norms, and a path which intersects itself can be replaced by one which does not do so and which takes time less than or equal to that taken by the original path.
Abstract: Planning time-optimal motions has been a major focus of research in robotics. In this paper we consider the following problem: given an object in two-dimensional physical space, an initial point, and a final point, plan a time-optimal obstacle-avoiding motion for this object subject to bounds on the velocity and acceleration of the object. We give the first algorithm which solves the problem exactly in the case where the velocity and acceleration bounds are given in theL? norm. We further prove the following important results: a tracking lemma and a loop-elimination theorem, both of which are applicable to the case of arbitrary norms. The latter result implies that, with or without obstacles, a path which intersects itself can be replaced by one which does not do so and which takes time less than or equal to that taken by the original path.

Journal ArticleDOI
TL;DR: It is proved that for any setS ofn points in the plane and n3−α triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn3−3α/(c log5n) of the triangles.
Abstract: We prove that for any setS ofn points in the plane andn3?? triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn3?3?/(c log5n) of the triangles. This implies that any set ofn points in three-dimensional space defines at most $$\sqrt[3]{{(c/2)}}n^{8/3} \log ^{5/3} n$$ halving planes.

Journal ArticleDOI
TL;DR: A very simple method is pointed out of achievingO(ρnn!) simplices, where ρ<1 is a constant, which is the only previously published triangulation of then-cube usingo(n!) simplice.
Abstract: The only previously published triangulation of then-cube usingo(n!) simplices, due to Sallee, usesO(n?2n!) simplices. We point out a very simple method of achievingO(?nn!) simplices, where ?<1 is a constant.

Journal ArticleDOI
TL;DR: The sharp constants of the asymptotic expansion ofEn(f), En(v), andEn(V), asn tends to infinity are determined.
Abstract: Denote the expected number of facets and vertices and the expected volume of the convex hullPn ofn random points, selected independently and uniformly from the interior of a simpled-polytope byEn(f), En(v), andEn(V), respectively. In this note we determine the sharp constants of the asymptotic expansion ofEn(f), En(v), andEn(V), asn tends to infinity. Further, some results concerning the expected shape ofPn are given.

Journal Article
TL;DR: This survey paper describes the known bounds on the size of such a partitioning and presents some of the algorithms for computing a geometric partitioning.
Abstract: \indent In this survey paper we review some results related to {\em geometric partitioning}, i.e. given a set of objects in $ {\bf R}^{d}$, partition the space into few regions so that each region intersects a small number of objects. We first describe the known bounds on the size of such a partitioning and present some of the algorithms for computing a geometric partitioning. We then discuss several applications of geometric partitioning.

Journal Article
TL;DR: This paper presents a meta-analyses of the chiral stationary phase of the response of the immune system to various types of infectious disease.
Abstract: Note: Professor Pach's number: [092] Reference DCG-CHAPTER-2008-008 Record created on 2008-11-18, modified on 2017-05-12

Journal Article
TL;DR: This paper presents a meta-analyses of the response of the immune system to the presence of certain types of deposits, including those of EMTs.
Abstract: Note: Professor Pach's number: [084]; Also in: Proc. 3rd Canadian Conference on Computational Geometry, 1991, pp. 54-57 Reference DCG-CHAPTER-2008-009 Record created on 2008-11-18, modified on 2017-05-12

Journal ArticleDOI
TL;DR: A bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μd(n) for everyd is obtained, and it is asymptotically equal to 1/2(k+1)nk−1dn, ifk is fixed andn tends to infinity.
Abstract: For every polynomial mapf=(f1,?,fk): ?n??k, we consider the number of connected components of its zero set,B(Zf) and two natural "measures of the complexity off," that is the triple(n, k, d), d being equal to max(degree offi), and thek-tuple (Δ1,...,Δ4), Δk being the Newton polyhedron offi respectively. Our aim is to boundB(Zf) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom's bound μd(n)=d(2d?1)n?1. Considered as a polynomial ind, μd(n) has leading coefficient equal to 2n?1. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μd(n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)nk?1dn, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.

Journal ArticleDOI
TL;DR: Kertész solved the case that the regular triangular lattice gives the optimal packing of points in an infinite strip of width and proved, surprisingly, it is not difficult to prove.
Abstract: What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Toth a number of years ago. The answer is trivial for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai% abgsMiJoaakaaabaGaaG4maaWcbeaakiaac+cacaaIYaaaaa!3AF6! $$w \leqslant \sqrt 3 /2$$ and, surprisingly, it is not difficult to prove [M2] for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai% aa-1dacaWGUbWaaOaaaeaacaaIZaaaleqaaOGaai4laiaaikdaaaa!3AF2! $$w = n\sqrt 3 /2$$ , wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertesz [K] solved the case % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83DaG% qaaiaa+bcacaGF8aWaaOaaaeaacaaIYaaaleqaaaaa!392C! $$w< \sqrt 2 $$ . Here we fill the first gap, i.e., the maximal density is determined for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaie% aacaWFZaaaleqaaOGaai4laiaaikdacqGH8aapcaWG3bGaeyizIm6a% aOaaaeaacaaIZaaaleqaaaaa!3CCB! $$\sqrt 3 /2< w \leqslant \sqrt 3 $$ .

Journal ArticleDOI
TL;DR: In this paper, a polynomial-time algorithm for the problem of finding simple paths in a planar graph that is homotopic to a set of vertices in the space is presented.
Abstract: In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ?2, a subset {I1, ?,Ip} of the faces ofG, and pathsC1, ?,Ck inG, with endpoints on the boundary ofI1 ? ? ?Ip; find: pairwise disjoint simple pathsP1, ?,Pk inG so that, for eachi=1, ?,k, Pi is homotopic toCi in the space ?2\(I1 ? ? ?Ip) Moreover, we prove a theorem characterizing the existence of a solution to this problem Finally, we extend the algorithm to disjoint homotopic trees As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ?2 and pairwise disjoint setsW1, ?,Wk of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT1, ?,Tk ofG whereTi(i=1, ?, k)


Journal ArticleDOI
TL;DR: The minimum dimension needed to representK(m, n) as a “unit neighborhood graph” in Euclidean space is considered and some upper and lower bounds on this dimension are given.
Abstract: The minimum dimension needed to representK(m, n) as a "unit neighborhood graph" in Euclidean space is considered. Some upper and lower bounds on this dimension are given, and the exact values of the dimension are calculated form≤3,n≤10.

Journal ArticleDOI
TL;DR: It is shown that 2n−1 ≤σ(n)≤σor( n)≦4n−4 forn≥5 and that σor(n), the minimum number of ideal hyperbolic tetrahedra,≤2n for alln.
Abstract: Let ?(n) be the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volumen-cusped hyperbolic 3-manifold, orientable or not Let ?or(n) be the corresponding number when we restrict ourselves to orientable manifolds The correct values of ?(n) and ?or(n) and the corresponding manifolds are given forn=1,2,3,4 and 5 We then show that 2n?1≤?(n)≤?or(n)≤4n?4 forn?5 and that ?or(n)?2n for alln

Journal ArticleDOI
TL;DR: It is shown that the maximal number of turns of anx-monotone path in an arrangement ofn lines is Ω(n5/3) and the maximum number ofturns in arrangement of n pseudolines is Γ(n2/logn).
Abstract: We show that the maximal number of turns of anx-monotone path in an arrangement ofn lines is Ω(n5/3) and the maximal number of turns of anx-monotone path in arrangement ofn pseudolines is Ω(n2/logn).

Journal ArticleDOI
TL;DR: On-line packing methods which work under the restriction that during the packing process the authors are given each succeeding “potato” only when the preceding one has been packed, and every sequence of convex bodies of diameters at most 1 whose total volume does not exceed 1.
Abstract: We discuss packings of sequences of convex bodies of Euclideann-spaceEn in a box and particularly in a cube. Following an Auerbach-Banach-Mazur-Ulam problem from the well-knownScottish Book, results of this kind are called potato-sack theorems. We consider on-line packing methods which work under the restriction that during the packing process we are given each succeeding "potato" only when the preceding one has been packed. One of our on-line methods enables us to pack into the cube of sided>1 inEn every sequence of convex bodies of diameters at most 1 whose total volume does not exceed ( $$(d - 1)(\sqrt d - 1)^{2(n - 1)} /n!.$$ ). Asymptotically, asd??, this volume is as good as that given by the non-on-line methods previously known.

Journal ArticleDOI
Lee R. Nackman1, Vijay Srinivasan1
TL;DR: This work shows that bisectors of weakly linearly separable sets inEd have many properties of interest, and gives necessary and sufficient conditions for the existence of a particular continuous map from (a portion of) any linear separator to the bisector.
Abstract: A bisector of two sets is the set of points equidistant form them. Bisectors arise naturally in several areas of computational geometry. We show that bisectors of weakly linearly separable sets inEd have many properties of interest. Among these, the bisector of a restricted class of linearly separated sets is a homeomorphic image of the linear separator. We also give necessary and sufficient conditions for the existence of a particular continuous map from (a portion of) any linear separator to the bisector.

Journal ArticleDOI
TL;DR: The number of plane segments in a crystalline minimal surface S is bound in terms of its Euler characteristic, the number of line segments in its boundary, and a factor determined by the Wulff shapeW of its surface energy function.
Abstract: We bound the number of plane segments in a crystalline minimal surface S in terms of its Euler characteristic, the number of line segments in its boundary, and a factor determined by the Wulff shapeW of its surface energy function. A major technique in the proofs is to quantize the Gauss map ofS based on the Gauss map ofW. One thereby bounds the number of positive-curvature corners and the interior complexity ofS.