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 -