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35
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 416 (115 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 75 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
MinimumLink Paths Among Obstacles in the Plane
 ALGORITHMICA
, 1992
"... Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that computes the link distance (and a correspon ..."
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Cited by 53 (6 self)
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Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimumlink path) between two points in time O(E#(n) log² n) (and space O(E)), where n is the total number of edges of the obstacles, E is the size of the visibility graph, and #(n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted at s) of minimumlink paths from s to all obstacle vertices. This leads to a method of solving the query version of our problem (for query points t).
The Union Of Convex Polyhedra In Three Dimensions
, 1997
"... . We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with## k 3 + kn#(k)) union complexity. We also describe a ..."
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Cited by 38 (24 self)
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. We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with## k 3 + kn#(k)) union complexity. We also describe a rather simple randomized incremental algorithm for computing the boundary of the union in O(k 3 + kn log k log n) expected time. Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539793250755 1. Combinatorial bounds. Let P = {P 1 , . . . , P k } be a family of k convex polyhedra in 3space, let n i be the number of faces of P i , and let n = # k i=1 n i . Put U = # P. By the combinatorial complexity of a polyhedral set we mean the total number of its vertices, edges, and faces. Our main result is the followin...
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
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Cited by 33 (6 self)
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We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...
Efficiently approximating polygonal paths in three and higher dimensions
 Algorithmica
, 1998
"... Abstract. We present efficient algorithms for solving polygonalpath approximation problems in three and higher dimensions. Given an nvertex polygonal curve P in R d, d ≥ 3, we approximate P by another polygonal curve P ′ of m ≤ n vertices in R d such that the vertex sequence of P ′ is an ordered s ..."
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Cited by 23 (5 self)
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Abstract. We present efficient algorithms for solving polygonalpath approximation problems in three and higher dimensions. Given an nvertex polygonal curve P in R d, d ≥ 3, we approximate P by another polygonal curve P ′ of m ≤ n vertices in R d such that the vertex sequence of P ′ is an ordered subsequence of the vertices of P. The goal is either to minimize the size m of P ′ for a given error tolerance ε (called the min # problem), or to minimize the deviation error ε between P and P ′ for a given size m of P ′ (called the minε problem). Our techniques enable us to develop efficient nearquadratictime algorithms in three dimensions and subcubictime algorithms in four dimensions for solving the min # and minε problems. We discuss extensions of our solutions to ddimensional space, where d> 4, and for the L1 and L∞ metrics. Key Words. Curve approximation, Parametric searching. 1. Introduction. In
On the Union of Fat Wedges and Separating a Collection of Segments by a Line
"... We call a line ` a separator for a set S of objects in the plane if ` avoids all the objects and partitions S into two nonempty subsets, one consisting of objects lying above ` and the other of objects lying below `. In this paper we present an O(n log n) time algorithm for finding a separator li ..."
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Cited by 23 (10 self)
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We call a line ` a separator for a set S of objects in the plane if ` avoids all the objects and partitions S into two nonempty subsets, one consisting of objects lying above ` and the other of objects lying below `. In this paper we present an O(n log n) time algorithm for finding a separator line for a set of n segments, provided the ratio between the diameter of the set of segments and the length of the smallest segment is bounded. No subquadratic algorithms are known for the general case. Our algorithm is based on the recent results of [13], concerning the union of `fat ' triangles, but we also include an analysis which improves the bounds obtained in [13].
The Common Exterior of Convex Polygons in the Plane
 Comput. Geom. Theory Appl
, 1997
"... We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: 1. The maximum complexity of the entire common exterior is \Theta(nff( ..."
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Cited by 19 (8 self)
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We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: 1. The maximum complexity of the entire common exterior is \Theta(nff(k) + k 2 ). 1 2. The maximum complexity of a single cell of the common exterior is \Theta(nff(k)). 3. The complexity of m distinct cells in the common exterior is O(m 2=3 k 2=3 log 1=3 ( k 2 m )+ n log k) and can be \Omega\Gamma m 2=3 k 2=3 + nff(k)) in the worst case. 1 Introduction In this paper we establish several combinatorial bounds on the complexity of the common exterior (namely, the complement of the union) of a collection of k convex polygons in the plane, with a total of n edges. The arrangement of such a collection of polygons can be viewed as a special case of an arrangement of n segments, but we prefer to regard it as a generalization of an arrangement of k...
On the Boundary Complexity of the Union of Fat Triangles
, 2000
"... A triangle is said to be fat if its smallest angle is at least > 0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of n fat triangles in the plane determines at most O n log 2 holes. This improves on some earlier bo ..."
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Cited by 18 (4 self)
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A triangle is said to be fat if its smallest angle is at least > 0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of n fat triangles in the plane determines at most O n log 2 holes. This improves on some earlier bounds of Efrat, Rote, Sharir, Matousek et al. Solving a problem of Agarwal and Bern, we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc.
Geometric Lower Bounds for Parametric Matroid Optimization
, 1998
"... We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: ksets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in suc ..."
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Cited by 16 (2 self)
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We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: ksets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: #(nr 1/3 ) for a general nelement matroid with rank r , and #(m#(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was #(n log r) for uniform matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 / log # n) for uniform matroids were also known. 1 Introduction In this paper we study connections between combinatorial geometry and matroid optimization theory, as represented by the following problem. Parametric matroid optimization. Given a matroid for which the elements have weights that vary as a linear function of a parameter t , what is the sequence of minimum weight bases over the range of values o...