TY - JOUR
AB - This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry. Among these applications we find problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points. All results are either combinatorial or computational in nature. They are based on the combinatorial analysis in two companion papers [PS] and [E2] and a divide-and-conquer algorithm for computing envelopes described in this paper.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas J
AU - Sharir, Micha
ID - 4081
IS - 1
JF - Discrete & Computational Geometry
TI - The upper envelope of piecewise linear functions: Algorithms and applications
VL - 4
ER -
TY - JOUR
AB - Sweeping a collection of figures in the Euclidean plane with a straight line is one of the novel algorithmic paradigms that have emerged in the field of computational geometry. In this paper we demonstrate the advantages of sweeping with a topological line that is not necessarily straight. We show how an arrangement of n lines in the plane can be swept over in O(n2) time and O(n) space by a such a line. In the process each element, i.e., vertex, edge, or region, is visited once in a consistent ordering. Our technique makes use of novel data structures which exhibit interesting amortized complexity behavior; the result is an algorithm that improves upon all its predecessors either in the space or the time bounds, as well as being eminently practical. Numerous applications of the technique to problems in computational geometry are given—many through the use of duality transforms. Examples include solving visibility problems, detecting degeneracies in configurations, computing the extremal shadows of convex polytopes, and others. Even though our basic technique solves a planar problem, its applications include several problems in higher dimensions.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas J
ID - 4082
IS - 1
JF - Journal of Computer and System Sciences
TI - Topologically sweeping an arrangement
VL - 38
ER -
TY - JOUR
AB - It is shown that, given a set S of n points in $R^3 $, one can always find three planes that form an eight-partition of S, that is, a partition where at most ${n / 8}$ points of S lie in each of the eight open regions. This theorem is used to define a data structure, called an octant tree, for representing any point set in $R^3 $. An octant tree for n points occupies $O(n)$ space and can be constructed in polynomial time. With this data structure and its refinements, efficient solutions to various range query problems in two and three dimensions can be obtained, including (1) half-space queries: find all points of S that lie to one side of any given plane; (2) polyhedron queries: find all points that lie inside (outside) any given polyhedron; and (3) circle queries in $R^2 $: for a planar set S, find all points that lie inside (outside) any given circle. The retrieval time for all these queries is $T(n) = O(n^\alpha + m)$, where $\alpha = 0.8988$ (or 0.8471 in case (3)), and m is the size of the output. This performance is the best currently known for linear-space data structures that can be deterministically constructed in polynomial time.
AU - Yao, F. Frances
AU - Dobkin, David P
AU - Herbert Edelsbrunner
AU - Paterson,Michael S
ID - 4083
IS - 2
JF - SIAM Journal on Computing
TI - Partitioning space for range queries
VL - 18
ER -
TY - JOUR
AB - A tour of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in . It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour.
In this paper, we present an algorithm that either reports that no necklace-tour exists or outputs a necklace-tour of a given set of n points in O(n2 log n) time. If a tour is given, then we can test in O(n2) time whether or not this tour is a necklace-tour. Both algorithms can be generalized to ƒ-factors of point sets in the plane. The complexity results rely on a combinatorial analysis of certain intersection graphs of disks defined for finite sets of points in the plane.
AU - Herbert Edelsbrunner
AU - Rote,Günter
AU - Welzl, Emo
ID - 4084
IS - 2
JF - Theoretical Computer Science
TI - Testing the necklace condition for shortest tours and optimal factors in the plane
VL - 66
ER -
TY - CONF
AB - Let C be a cell complex in d-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope in d + 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in_front/behind relation defined for the faces of C with respect to any fixed viewpoint x is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.
AU - Herbert Edelsbrunner
ID - 4085
TI - An acyclicity theorem for cell complexes in d dimension
ER -
TY - JOUR
AB - This note proves that the maximum number of faces (of any dimension) of the upper envelope of a set ofn possibly intersectingd-simplices ind+1 dimensions is (n d (n)). This is an extension of a result of Pach and Sharir [PS] who prove the same bound for the number ofd-dimensional faces of the upper envelope.
AU - Herbert Edelsbrunner
ID - 4086
IS - 4
JF - Discrete & Computational Geometry
TI - The upper envelope of piecewise linear functions: Tight bounds on the number of faces
VL - 4
ER -
TY - CONF
AB - This paper offers combinatorial results on extremum problems concerning the number of tetrahedra in a tetrahedrization of n points in general position in three dimensions, i.e. such that no four points are coplanar. It also presents an algorithm that in O(nlog n) time constructs a tetrahedrization of a set of n points consisting of at most 3n–11 tetrahedra.
AU - Herbert Edelsbrunner
AU - Preparata, Franco P
AU - West, Douglas B
ID - 4087
TI - Tetrahedrizing point sets in three dimensions
VL - 358
ER -
TY - JOUR
AB - Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n 2) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with (n) space1 and (n 3/2) preprocessing time, we can answer face queries in (n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path , form a data structure from which we can find the convex hull of any subpath of quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in (n 1/3)+O(K) time, given(n 4/3) space and (n 5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time (m 2/3 n 2/3+n), which is nearly optimal.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas
AU - Hershberger, John
AU - Seidel, Raimund
AU - Sharir, Micha
AU - Snoeyink, Jack
AU - Welzl, Emo
ID - 4088
IS - 1
JF - Discrete & Computational Geometry
TI - Implicitly representing arrangements of lines or segments
VL - 4
ER -
TY - JOUR
AB - Motivated by a number of motion-planning questions, we investigate in this paper some general topological and combinatorial properties of the boundary of the union ofn regions bounded by Jordan curves in the plane. We show that, under some fairly weak conditions, a simply connected surface can be constructed that exactly covers this union and whose boundary has combinatorial complexity that is nearly linear, even though the covered region can have quadratic complexity. In the case where our regions are delimited by Jordan acrs in the upper halfplane starting and ending on thex-axis such that any pair of arcs intersect in at most three points, we prove that the total number of subarcs that appear on the boundary of the union is only (n(n)), where(n) is the extremely slowly growing functional inverse of Ackermann's function.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas
AU - Hershberger, John
AU - Pach, János
AU - Pollack, Richard
AU - Seidel, Raimund
AU - Sharir, Micha
AU - Snoeyink, Jack
ID - 4089
IS - 1
JF - Discrete & Computational Geometry
TI - On arrangements of Jordan arcs with three intersections per pair
VL - 4
ER -
TY - CONF
AU - Chazelle, Bernard
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas J
AU - Sharir, Micha
ID - 4092
TI - A singly exponential stratification scheme for real semi-algebraic varieties and its applications
VL - 372
ER -