We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...

We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

We introduce the concept of a sensitive E-approx-imation, and use it to derive a more efficient algorithm for computing &-nets. We define and investigate prod-uct range spaces, for which we establish sampling the-orems analogous to the standard finite VC-dimensional case. This generalizes and simplifies results from previ-ous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(n log3 n) deterministic algorithm for com-puting the diameter of an n-point set in 3-dimensional space.

We consider the problem of computing the lower envelope (the minimum) of n constant degree algebraic functions of one variable. The lower envelope has size O(nfi(n)) where fi(n) is a nearly constant function, and it can easily be computed in time O(nfi(n) log n) by a simple deterministic divide-and-conquer algorithm [45]. We give an alternative simple (module a derandomization black box) approach using divide-and-conquer based on cuttings that results in a deterministic sequential algorithm that runs in the same time bound. This algorithm uses derandomization tools by now standard. This approach however allows us to obtain the following results: ffl A deterministic sequential algorithm that is output sensitive and runs in time O(n log f) if f n ffl , or O(nfi(f) log f) = O(nfi(n) log n) otherwise, where f is the size of the output; ffl a randomized parallel EREW algorithm that runs in time O(log n) and uses nearly optimal work O(nfi 2 (n) log n) with n-polynomial probability...

We describe an algorithm for computing the intersection of n balls of equal radius in R³ which runs in time O(n lg² n). The algorithm can be parallelized so that the comparisons that involve the radius of the balls are performed in O(lg³ n) batches. Using parametric search, these algorithms are used to obtain an algorithm for computing the diameter of a set of n points in R³ (the maximum distance between any pair) which runs in time O(n lg^5 n). The algorithms are deterministic and elementary; this is in contrast with the running time O(n log n) in both cases that can be achieved using randomization [3], and the running times O(n lg n) and O(n lg³ n) using deterministic geometric sampling [2, 1].

This chapter reviews randomization algorithms developed in the last few years to solve a wide range of geometric optimization problems. We rst review a number of general techniques, including randomized binary search, randomized linear-programming algorithms, and random sampling. Next, we describe several applications of these and other techniques, including facility location, proximity problems, statistical estimators, nearest neighbor searching, and Euclidean TSP.

The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.

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We provide a lower bound construction showing that the union of unit balls in R 3 has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir. 1 Introduction The union of a set of n balls in R 3 has quadratic complexity \Theta(n 2 ), even if they all have the same radius. All the already known constructions have balls scattered around, however, and Sharir posed the problem whether a quadratic complexity could be achieved if all the balls (of same radius) contained the origin. In this note, we show a construction of n unit balls, all containing the origin, whose union has complexity \Theta(n 2 ). As a trivial observation, we observe that the centers are arbitrarily close to the origin in our construction. In fact, if the centers are forced to be at least pairwise " apart, for some constant " ? 0, then no more than O( 1 " 3 ) can meet in a single point, and hence the union has complexity at most O( 1 " 3 n) = O " (n). It is an interes...

gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions. A. NUMBER THEORY 1. It is known [35, Vol. 1, Section 581 that n(2x) < 257(x) for sufficiently large x. Is it true that (1) n(x + y) < B(X) + n(y)? Ungrir has verified the inequality for y 5 41. Hardy and Littlewood [29, p. 691 have proved that (2) P(X + Y)- n(x) < cy/log y. In the same paper, they discuss many interesting conjectures. They put h-0 sup [R(x+ Y) x-00- n(x)1 = p(y), and they conjecture that p(y)> y/log y, and that perhaps n(y)- p(y)- oo as y--) 00. Hardy and Littlewood deduce (2) by Bruds method. A very difficult conjecture, weaker than (1) but much stronger than (2), is that corresponding to each s> 0 there exists a yE such that, for y> yE, dx + Y)- dy) < (1 + dy/log y-It has not yet been disproved that p(y) = 1 for all y. If p(y)> 1 for some y, then lim inf (h+l- pn) < 00. 2. About seventy years ago, Piltz [38] conjectured that, for each c> 0, pn+l- pn = O(n&). Cram & conjectured [7, p. 241 that pn+l- pn = O((log n)?. If lim sup (p,+l- p,)/(log n) ” = 1, then, for each E> 0, infinitely many of the intervals [n, n + (1- s)(log n)“] contain no primes, but for n> ns, there is a prime between n and n + (1 + E)(log n)“. The Riemann hypothesis implies that prl+1- pn < n&+1/2 [35, Vol. 1, p. 3381. Thus the old conjecture that there is always a prime between two consecutive squares already goes beyond the Riemann hypothesis. 292 PAUL ERDOS The first big achievement in this direction is due to Hoheisel [32], who showed that pn+i- Pn < n lv6. Ingham [34, p. 2561 proved that 1- 6 can be taken to be 5/8. In the opposite direction, I have proved [9, p. 1241 that, for a certain c> 0 and for infinitely many n, Pnfl