... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.

We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t; report all K points of S that lie inside R at time t: We first present an indexing structure that, for any given constant e> 0; uses OðN=BÞ disk blocks and answers a query in OððN=BÞ 1=2þe þ K=BÞ I/Os, where B is the block size. It can also report all the points of S that lie inside R during a given time interval. A point can be inserted or deleted, or the trajectory of a point can be changed, in Oðlog 2 B NÞ I/Os. Next, we present a general approach that improves the query time if the queries arrive in chronological order, by allowing the index to evolve over time. We obtain a tradeoff between the query time and the number of times the index needs to be updated as the points move. We also describe an indexing scheme in which the number of I/Os required to answer a query depends monotonically on the difference between the query time stamp t and the current time. Finally, we develop an efficient indexing scheme to answer approximate

Abstract. Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following F-range searching problem: Given P, build a data structure for efficient answering of queries of the form, &quot;Given a 7 ~ F, count (or report) the points of P lying in 7.&quot; Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1- x/b+~) query time, where d < b < 2d- 3 and ~> 0 is an arbitrarily small constant. The acutal value of b is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of F-range searching problem, including improved ray shooting among triangles in ~3 1.

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: Let V ` f0; 1g n have Vapnik-Chervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn=k) log(n=k)) d . This new result has applications in the theory of empirical processes. 1 The author gratefully acknowledges the support of the Mathematical Sciences Research Institute at UC Berkeley and ONR grant N00014-91-J-1162. 1 1 Statement of Results Let n be natural number greater than zero. Let V ` f0; 1g n . For a sequence of indices I = (i 1 ; . . . ; i k ), with 1 i j n, let V j I denote the projection of V onto I, i.e. V j I = f(v i 1 ; . . . ; v i k ) : (v 1 ; . . . ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The Vapnik-Chervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] (t...

Let P be a set of n points in R d (where d is a small fixed positive integer), and let \Gamma be a collection of subsets of R d , each of which is defined by a constant number of bounded degree polynomials. We consider the following \Gamma-range searching problem: Given P , build a data structure for efficient answering of queries of the form `Given a fl 2 \Gamma, count (or report) the points of P lying in fl'. Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1\Gamma1=b+ffi ) query time, where d b 2d \Gamma 3 and ffi ? 0 is an arbitrarily small constant. The actual value of b is related to the problem of partitioning arrangements of algebraic surfaces into constant-complexity cells. We present some of the applications of \Gamma-range searching problem, including improved ray shooting among triangles in R³.

We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in fixed dimension into a lineartime deterministic one. The constant of proportionality is d O(d) , which is better than for previously known such algorithms. We show that the algorithm works in a fairly general abstract setting, which allows us to solve various other problems (such as finding the maximum volume ellipsoid inscribed into the intersection of n halfspaces) in linear time.

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 DAAH04-96-1-0013, 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 CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

We present a technique for converting RNC algorithms into NC algorithms. Our approach is based on a parallel implementation of the method of conditional probabilities. This method was used to convert probabilistic proofs of existence of combinatorial structures into polynomial time deterministic algorithms. It has the apparent drawback of being extremely sequential in nature. We show certain general conditions under which it is possible to use this technique for devising deterministic parallel algorithms. We use our technique to devise an NC algorithm for the set balancing problem. This problem turns out to be a useful tool for parallel algorithms. Using our de-randomization method and the set balancing algorithm, we provide an NC algorithm for the lattice approximation problem. We also use the lattice approximation problem to bootstrap the set balancing algorithm, and the result is a more processor efficient algorithm. The set balancing algorithm also yields an NC algorithm for near-optimal edge coloring of simple graphs. Our methods also extend to the parallelization of various algorithms in computational geometry that rely upon the random sampling technique of Clarkson. Finally, our methods apply to constructing certain combinatorial structures, e.g. ...

In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.