In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, as a worst-case quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach, we consider the important classical problem of proximity queries in 2 and 3 dimensions, and develop a new technique for the efficient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique gives both low degree and fast query time, and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations and arithmetic degree.

This paper presents a prediction-and-verification segmentation scheme using attention images from multiple fixations. The scheme has two major components: (a) a hierarchical quasi-Voronoi diagram which organizes training attention images for prediction of the segmentation masks; (b) a learning-based function approximation scheme to verify the segmentation result. A major advantage of this scheme is that it can handle a large number of different deformable objects presented in complex backgrounds. The scheme is also relatively efficient since the segmentation is guided by the past knowledge through a prediction-and-verification scheme. The system was tested to segment hands in sequences of intensity images, where each sequence represents a hand sign in American Sign Language. The experimental result showed a 95% correct segmentation rate with a 3% false rejection rate. Categories: 2D segmentation, hand sign recognition, visual learning, nearest neighbor, and feature derivatio...

We address a longstanding open problem of [10, 9], and present a general transformation that transforms any pointer based data structure to be confluently persistent. Such transformations for fully persistent data structures are given in [10], greatly improving the performance compared to the naive scheme of simply copying the inputs. Unlike fully persistent data structures, where both the naive scheme and the fully persistent scheme of [10] are feasible, we show that the naive scheme for confluently persistent data structures is itself infeasible (requires exponential space and time). Thus, prior to this paper there was no feasible method for implementing confluently persistent data structures at all. Our methods give an exponential reduction in space and time compared to the naive method, placing confluently persistent data structures in the realm of possibility.

We present a data structure that can store a set of disjoint fat objects in d-space such that point location and bounded-size range searching with arbitrarily-shaped ranges can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or non-convex polytopes. The multi-purpose data structure supports point location and range searching queries in time O(log d\Gamma1 n) and requires O(n log d\Gamma1 n) storage, after O(n log d\Gamma1 n log log n) preprocessing. The data structure and query algorithm are rather simple. 1 Introduction Fatness turns out to be an interesting phenomenon in computational geometry. Several papers present surprising combinatorial complexity reductions [3, 15, 22, 26, 32] and efficiency gains for algorithms [1, 4, 19, 28, 33] if the objects under consideration have a certain fatness. Fat objects are compact to some extent, rather than long and thin. Fatness is a realistic assumption, since in many practical instances of ...

Fractional cascading is a technique designed to allow efficient sequential search in a graph with catalogs of total size n. The search consists of locating a key in the catalogs along a path. In this paper we show how to preprocess a variety of fractional cascaded data structures whose underlying graph is a tree so that searching can be done efficiently in parallel. The preprocessing takes O(log n) time with n/log n processors on an EREW PRAM. For a balanced binary tree cooperative search along root-to-leaf paths can be done in O((logn)/logp) time using p processors on a CREW PRAM.

The Nearest Neighbor Search problem is defined as follows: given a set P of n points, preprocess the points so as to efficiently answer queries that require finding the closest point in P to a query point q. If we are willing to settle for a point that is almost as close as the nearest neighbor, then we can relax the problem to the approximate Nearest Neighbor Search. Nearest Neighbor Search (exact or approximate) is an integral component in a wide range of applications that include multimedia databases, computational biology, data mining, and information retrieval. The common thread in all these applications is similarity search: given a database of objects, we want to return the object in the database that is most similar to a query object. The objects are mapped onto points in a high dimensional metric space , and similarity search reduces to a nearest neighbor search. The dimension of the underlying space may be in the order of a few hundreds, or thousands; therefore, we r...

Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems that arise in various disciplines such as pattern recognition, computer graphics, computer vision, robotics, VLSI layout, operations research, statistics, etc. In contrast with the classical approach to proving mathematical theorems about geometry-related problems, this discipline emphasizes the computational aspect of these problems and attempts to exploit the underlying geometric properties possible, e.g., the metric space, to derive efficient algorithmic solutions. The classical theorem, for instance, that a set S is convex if and only if for any 0 ff 1 the convex combination ffp + (1 \Gamma<F

With the transformation of the Internet to a commercial infrastructure, the ability to provide differentiated services to users with widely varying requirements is rapidly becoming as important as meeting the massive increases in bandwidth demand. Hence, while deploying routers, switches, and transmission systems of ever increasing capacity, Internet Service Providers would also like to provide customer specific differentiated services using the same shared network infrastructure. In this paper, we describe router architectures that can support the two trends of rising bandwidth demand and rising demand for differentiated services. We focus on router mechanisms that can support differentiated services at a level not being contemplated in proposals currently under consideration due to concern regarding their implementability at high-speeds. We consider the types of differentiated services that service providers may want to offer and then discuss the mechanisms needed in route...

We consider dynamic data structures in which updates rebuild a static solution. Space bounds for persistent versions of these structures can often be reduced by using an offline persistent data structure in place of the static solution. We apply this technique to decomposable search problems, to dynamic linear programming, and to maintaining the minimum spanning tree in a dynamic graph. Our algorithms admit trade-offs of update time vs. query time, and of time vs. space.

Abstract. In this paper a new approach is proposed to point-location in a three-dimensional cellcomplex 7, which may be viewed as a nontrivial generalization of a corresponding two-dimensional technique due to Sarnak and Tarjan. Specifically, in a space-sweep of 7), the intersections of the sweep-plane with P occurring in a given slab, i.e., between two consecutive vertices, are topologically conformal planar subdivisions. If the sweep direction is viewed as time, the descriptions of the various slabs are distinct "versions " of a two-dimensional point-location data structure, dynamically updated each time a vertex is swept. Combining the persistence-addition technique of Driscoll, Sarnak, Sleator, and Tarjan [J. Comput. System. Sci., 38 (1989), pp. 86-124] with the recently discovered dynamic structure for planar point-location in monotone subdivisions, a method with query time O(log N) and space O(N log N) for point-location in a convex cell-complex with N facets is obtained. Key words, point location, convex cell complex, computational geometry, analysis of algorithms AMS(MOS) subject classifications. 68U05, 68Q25, 68P05, 68P10