Further Results on Arithmetic Filters for Geometric Predicates

An efficient technique to solve precision problems consists in using exact computations. For geometric predicates, using systematically expensive exact computations can be avoided by the use of filters. The predicate is first evaluated using rounding computations, and an error estimation gives a certificate of the validity of the result. In this note, we studies the statistical efficiency of filters for cosphericity predicate with an assumption of regular distribution of the points. We prove that the expected value of the polynomial corresponding to the in sphere test is greater than epsilon with probability O(epsilon log 1/epsilon) improving the results of a previous paper by the same authors.Comment: 7 pages 2 figures presented at the 15th European Workshop Comput. Geom., 113--116, 1999 improve previous results (in other paper

A Probabilistic Analysis of the Power of Arithmetic Filters

The assumption of real-number arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In most cases, rounded computations yield a reliable result, but sometimes rounded arithmetic introduces errors which may invalidate the algorithms. The rounded arithmetic may produce an incorrect result only if the exact absolute value of the algebraic expression is smaller than some (small) varepsilon, which represents the largest error that may arise in the evaluation of the expression. The threshold varepsilon depends on the structure of the expression and on the adopted computer arithmetic, assuming that the input operands are error-free. A pair (arithmetic engine,threshold) is an "arithmetic filter". In this paper we develop a general technique for assessing the efficacy of an arithmetic filter. The analysis consists of evaluating both the threshold and the probability of failure of the filter. To exemplify the approach, under the assumption that the input points be chosen randomly in a unit ball or unit cube with uniform density, we analyze the two important predicates "which-side" and "insphere". We show that the probability that the absolute values of the corresponding determinants be no larger than some positive value V, with emphasis on small V, is Theta(V) for the which-side predicate, while for the insphere predicate it is Theta(V^(2/3)) in dimension 1, O(sqrt(V)) in dimension 2, and O(sqrt(V) ln(1/V)) in higher dimensions. Constants are small, and are given in the paper.Comment: 22 pages 7 figures Results for in sphere test inproved in cs.CG/990702

Accurate and precise aggregation counting

AbstractAggregation counting is any procedure designed to solve the following problem: a number n of agents produces a fixed length binary message, and a central station produces an estimate of n from the bit-by-bit OR of the messages, which is therefore duplicate-insensitive. Such procedures are applicable to a situation where each of n independent sensors broadcasts the message to be used to estimate the count. A mathematically brilliant solution to this problem, due to Flajolet and Martin (1985) [1], is unfortunately affected by substantial bias and error. In this note we outline an alternative approach, which uses the Flajolet–Martin technique as a preparatory step and substantially reduces both error and bias. Specifically, the standard deviation of the count estimate drops from ∼110% to ∼20% of the estimated value

Routing through a rectangle

In this paper we present an O(Nlog N) algorithm for routing through a rectangle. Consider an n by m rectangular grid and a set of N two-terminal nets. A net is a pair of points on the boundary of the rectangle. A layout is a set of edge-disjoint paths, one for each net. Our algorithm constructs a layout, if there is one, in time O(NlogN) which contrasts favorably to the area of the layout which might be as large as N^{2}. The layout constructed can be wired using 4 layers of interconnect with only O(N) contact-cuts. A partial extension to multi-terminal nets is also discussed

An NC1 parallel 3D convex hull algorithm

In this paper we present an O(log n) time paridlel algorithm for computing the convex hull of n points in!)?3. This algorithm uses O (nl+a) processors on a CREW PRAM, for any constant O < cr <1. So far, all adequately documented parallel algorithms proposed for this problem use time at least 0(log2! n). In addition, the algorithm presented here is the first parallel algorithm for the three-dimensional convex hull problem that is not based on the serial divide-and-conquer algorithm of Preparat a and Hong, whose crucial operation is the merging of the convex hulls of two linearly separated point sets. The contributions of this paper are therefore (i) an O (log n) time parallel algorithm for the threedimensional convex hull problem, and (ii) a parallel algorithm for this problem that does not follow the traditional divide-and-conquer paradigm.

A New Approach to Planar Point Location

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Group / DAAB-07-72-C-0259National Science Foundation / MSC76-1732

On Multitransmission Networks

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB 07-67-C-019

A Study of Nordstrom-Robinson Optimum Code

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB-07-67-C-0199NSF / GK-233