Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linear-space data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a re-examination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a re-examination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are non-uniform spatially or temporally (in which case one desires data structures that adapt to s...

For a set S of n line segments in the plane, we give the first work-optimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log 2 n) time using O(n log n + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairly simple divide-and-conquer alternative to the optimal sequential “plane-sweep ” algorithm of Chazelle and Edelsbrunner. Moreover, our method can be used to output all k intersecting pairs while using only O(n) working space, which solves an open problem posed by Chazelle and Edelsbrunner. We also describe a sequential algorithm for computing a single face in an arrangement of n line segments that runs in O(n 2 (n) log n) time, which improves on a previous O(n log 2 n) time algorithm. For collections of simplices in IR d, we give methods for constructing a set of m = O(n d,1 log c n+k) cells of constant descriptive complexity that covers their arrangement, where c> 1 is a constant and k is the number of faces in the arrangement. The construction is performed sequentially in O(m) time, or in O(log n) time using O(m) work in the EREW PRAM model. The covering can be augmented to answer point location queries in O(log n) time. In addition to supplying the first parallel methods for these problems, we improve on the previous best sequential methods by reducing the query times (from O(log 2 n) in IR 3 and O(log 3 n) in IR d, d>3), and also the size and construction cost of the covering (from O(n d,1+ + k)). 1

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We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of non-optimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottom-up preprocessing phase previous to the top-down construction phase. 1 Introduction Polygon triangulation is a classic problem in comp...

. In this article we introduce (combinatorial) multi--color discrepancy and generalize some classical results from 2--color discrepancy theory to c colors. We give a recursive method that constructs c--colorings from approximations to the 2--color discrepancy. This method works for a large class of theorems like the six--standard--deviation theorem of Spencer, the Beck--Fiala theorem and the results of Matousek, Welzl and Wernisch for bounded VC--dimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of two--color discrepancy even if c is a power of 2. For the linear discrepancy version of the Beck--Fiala theorem the recursive approach also fails. Here we extend the method of floating colors to multi--colorings and prove multi--color versions of the the Beck--Fiala theorem and the Barany--Grunberg theorem. 1 Introduction Combinatorial discrepancy theory deals with the problem of partitioning the vertices of a hypergraph (set--...

We study the complexity of the motion planning problem for a bounded-reach robot in the situation where the n obstacles in its workspace satisfy two of the realistic models proposed in the literature, namely unclutteredness and small simple-cover complexity. We show that the maximum complexity of the free space of a robot with f degrees of freedom in the plane is #(n f/2 + n) for uncluttered environments as well as environments with small simple-cover complexity. The maximum complexity of the free space of a robot moving in a three-dimensional uncluttered environment is #(n 2f/3 +n). All these bounds fit nicely between the #(n) bound for the maximum free-space complexity for low-density environments and the #(n f ) bound for unrestricted environments. Surprisingly---because contrary to the situation in the plane---the maximum free-space complexity is #(n f ) for a three-dimensional environment with small simple-cover complexity. 1 Introduction It is well known that ...

A set of points G is a -guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This definition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: a d-dimensional scene satisfying the new model's requirements is known to have a linear-size binary space partition. We propose several algorithms for computing -guarding sets, and evaluate them experimentally. One of them appears to be quite practical. 1. Introduction Recently de Berg et al. [4] brought together several of the realistic input models that have been proposed in the literature, namely fatness, low density, unclutteredness, and small simple-cover 1 Supported by the ESPRIT IV LTR Project No. 21957 (CGAL). 2 Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. 3 Supported by the Netherlands' Organization for Scientific Research...

We give fast and efficient methods for constructing... time using linear work on an EREW PRAM.

We show that several discrepancy-like problems can be solved in NC 2 nearly achieving the corresponding sequential bounds. For example, given a set system (X; S), where X is a ground set and S ` 2 X , a set R ` X can be computed in NC 2 so that, for each S 2 S, the discrepancy jjR " Sj \Gamma jR " Sjj is O( p jSj log jSj). Previous NC algorithms could only achieve O( p jSj 1+ffl log jSj), while ours matches the probabilistic bound achieved sequentially by the method of conditional probabilities within a multiplicative factor 1 + o(1). Other problems whose NC solution we improve are lattice approximation, ffl-approximations of range spaces of bounded VC-exponent, sampling in geometric configuration spaces, and approximation of integer linear programs. 1 Introduction Problem and previous work. Discrepancy is an important concept in combinatorics, see e.g. [1, 5], and theoretical computer science, see e.g. [27, 23, 9]. It attempts to capture the idea of a good sample from ...

We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle's algorithm and of simple non-optimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991). As in Chazelle's algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.