An (n; s) Davenport-Schinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between Davenport-Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

Wireless ad-hoc sensor networks have recently emerged as a premier research topic. They have great longterm economic potential, ability to transform our lives, and pose many new system-building challenges. Sensor networks also pose a number of new conceptual and optimization problems. Some, such as location, deployment, and tracking, are fundamental issues, in that many applications rely on them for needed information. In this paper, we address one of the fundamental problems, namely coverage. Coverage in general, answers the questions about quality of service (surveillance) that can be provided by a particular sensor network. We first define the coverage problem from several points of view including deterministic, statistical, worst and best case, and present examples in each domain. By combining computational geometry and graph theoretic techniques, specifically the Voronoi diagram and graph search algorithms, we establish the main highlight of the paper - optimal polynomial time worst and average case algorithm for coverage calculation. We also present comprehensive experimental results and discuss future research directions related to coverage in sensor networks. I.

We address the problem of designing data structures that allow efficient search for approximate nearest neighbors. More specifically, given a database consisting of a set of vectors in some high dimensional Euclidean space, we want to construct a space-efficient data structure that would allow us to search, given a query vector, for the closest or nearly closest vector in the database. We also address this problem when distances are measured by the L 1 norm, and in the Hamming cube. Significantly improving and extending recent results of Kleinberg, we construct data structures whose size is polynomial in the size of the database, and search algorithms that run in time nearly linear or nearly quadratic in the dimension (depending on the case; the extra factors are polylogarithmic in the size of the database). Computer Science Department, Technion --- IIT, Haifa 32000, Israel. Email: eyalk@cs.technion.ac.il y Bell Communications Research, MCC-1C365B, 445 South Street, Morristown, NJ ...

We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two- and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.

We present a randomized strategy for maintaining balance in dynamically changing search trees that has optimal expected behavior. In particular, in the expected case a search or an update takes logarithmic time, with the update requiring fewer than two rotations. Moreover, the update time remains logarithmic, even if the cost of a rotation is taken to be proportional to the size of the rotated subtree. Finger searches and splits and joins can be performed in optimal expected time also. We show that these results continue to hold even if very little true randomness is available, i.e. if only a logarithmic number of truely random bits are available. Our approach generalizes naturally to weighted trees, where the expected time bounds for accesses and updates again match the worst case time bounds of the best deterministic methods. We also discuss ways of implementing our randomized strategy so that no explicit balance information is maintained. Our balancing strategy and our alg...

We present an isocontouringalgorithm which is near-optimal for real-time interaction and modification of isovalues in large datasets. A preprocessing step selects a subset S of the cells which are considered as seed cells. Given a particular isovalue, all cells in S which intersect the given isocontour are extracted using a high-performance range search. Each connected component is swept out using a fast isocontour propagation algorithm. The computational complexity for the repeated action of seed point selection and isocontour propagation is O(logn 0 + k), where n 0 is the size of S and k is the size of the output. In the worst case, n 0 = O(n), where n is the number of cells, while in practical cases, n 0 is smaller than n by one to two orders of magnitude. The general case of seed set construction for a convex complex of cells is described, in addition to a specialized algorithm suitable for meshes of regular topology, including rectilinear and curvilinear meshes. Keyword...

Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives data structures for this problem when the sites and queries are in a metric space. One data structure, D(S), uses a divide-and-conquer recursion. The other data structure, M(S, Q), is somewhat like a skiplist. Both are simple and implementable. The data structures are analyzed when the metric space obeys a certain sphere-packing bound, and when the sites and query points are random and have distributions with an exchangeability property. This property implies, for example, that query point q is a random element of S ∪ {q}. Under these conditions, the preprocessing and space bounds for the algorithms are close to linear in n. They depend also on the sphere-packing bound, and on the logarithm of the distance ratio Υ(S) of S, the ratio of the distance between the farthest pair of points in S to the distance between the closest pair. The data structure M(S, Q) requires as input data an additional set Q, taken to be representative of the query points. The resource bounds of M(S, Q) have a dependence on the distance ratio of S ∪ Q. While M(S, Q) can return wrong answers, its failure probability can be bounded, and is decreasing in a parameter K. Here K ≤ |Q|/n is chosen when building M(S, Q). The expected query time for M(S, Q) is O(K log n) log Υ(S ∪ Q), and the resource bounds increase linearly in K. The data structure D(S) has expected O(log n) O(1) query time, for fixed distance ratio. The preprocessing algorithm for M(S, Q) can be used to solve the all-nearest-neighbor problem for S in O(n(log n) 2 (log Υ(S)) 2) expected time. 1

Abstract. Massive data sets with terabytes of data are becoming commonplace. There is an increasing demand for algorithms and data structures that provide fast response times to queries on such data sets. In this paper, we describe a context for algorithmic work relevant to massive data sets and a framework for evaluating such work. We consider the use of "synopsis" data structures, which use very little space and provide fast (typically approximated) answers to queries. The design and analysis of effective synopsis data structures o er many algorithmic challenges. We discuss a number of concrete examples of synopsis data structures, and describe fast algorithms for keeping them up-to-date in the presence of online updates to the data sets.

In interactive data mining it is advantageous to have condensed representations of data that can be used to efficiently answer different queries. In this paper we show how frequent sets can be used as a condensed representation for answering various types of queries. Given a table r with 0/1 values and a threshold oe, a frequent set of r is a set X of columns of r such that at least a fraction oe of the rows of r have a 1 in all the columns of X. Finding frequent sets is a first step in finding association rules, and there exists several efficient algorithms for finding the frequent sets. We show that frequent sets have wider applications than just finding association rules. We show that using the inclusion-exclusion principle one can obtain approximate confidences of arbitrary boolean rules. We derive bounds for the errors in the confidences, and show that information collected during the computation of frequent sets can also be used to provide individual error bounds for each clause...