We present the "Near Optimal IsoSurface Extraction" (NOISE) algorithm for rapidly extracting isosurfaces from structured and unstructured grids. Using the span space, a new representation of the underlying domain, we develop an isosurface extraction algorithm with a worst case complexity of O( p n + k) for the search phase, where n is the size of the data set and k is the number of cells intersected by the isosurface. The memory requirement is kept at O(n) while the preprocessing step is O(n log n). We utilize the span space representation as a tool for comparing isosurface extraction methods on structured and unstructured grids. We also present a fast triangulation scheme for generating and displaying unstructured tetrahedral grids. Keywords---Isosurface extraction, unstructured grids, span space, kd-trees I. Introduction Isosurface extraction is a powerful tool for investigating scalar fields within volumetric data sets. The position of an isosurface, as well as its relation to ...

Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random k-d trees, quadtrees and union-end trees under various models of randomization. For example, for the random binary search tree constructed from a random permutation of 1,..., n, it is shown that H„/(c log (n)) tends to 1 in probability and in the mean as n- oo, where H „ is the height of the tree, and c =4.31107... is a solution of the equation c log (2e / = 1. In addition, we ~c ~ show that H „-clog (n) = O (/log (n) loglog (n)) in probability.

String matching over a long text can be significantly speeded up with an index structure formed by preprocessing the text. For very long texts, the size of such an index can be a problem. This paper presents the first sublinear-size index structure. The new structure is based on Lempel-Ziv parsing of the text and has size linear in N, the size of the Lempel-Ziv parse. For a text of length n, N = O(n = log n) and can be still smaller if the text is compressible. With the new index structure, all occurrences of a pattern string of length m can be found in time O(m 2

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We consider the problem of dynamic range searching in tree structures that do not replicate data. We propose a new dynamic structure, called the O-tree, that achieves a query time complexity of O(n (d\Gamma1)=d ) on n d-dimensional points and an amortized insertion/deletion time complexity of O(log n). We show that this structure is optimal when data is not replicated. In addition to optimal query and insertion/deletion times, the O-tree also supports exact match queries in worst-case logarithmic time. 1 Introduction Given a set S of d-dimensional points, a range query q is specified by d 1-dimensional intervals [q s i ; q e i ], one for each dimension i, and retrieves all points p = (p 1 ; p 2 ; : : : p d ) in S such that h8i 2 f1; : : : ; dg : q s i p i q e i i. This type of searching in multidimensional space has important applications in geographic information systems, image databases, and computer graphics. Several structures such as the range trees [3], P-range trees [29...

In this paper we investigate automated methods for externalizing internal memory data structures. We consider a class of balanced trees that we call weight-balanced partitioning trees (or wp-trees) for indexing a set of points in R d . Well-known examples of wp-trees include kd- trees, BBD-trees, pseudo-quad-trees, and BAR-trees. Given an efficient external wp-tree construction algorithm, we present a general framework for automatically obtaining a dynamic external data structure. Using this framework together with a new general construction (bulk loading) technique of independent interest, we obtain data structures with guaranteed good update performance in terms of I/O transfers. Our approach gives considerably improved construction and update I/O bounds for e.g. external kd-trees and BBD-trees.

We describe a new approach for cluster-based drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) time, where n is the number of points. We show that the BAR tree can be used to recursively divide a graph embedded in the plane into subgraphs of roughly equal size, such that the drawing of each subgraph has a balanced aspect ratio. As a result, we obtain a representation of a graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. The overall running time of the algorithm is O(n log n+m+D0(G)), where n and m are the number of vertices and edges of the graph G, andD0(G) is the time it takes to obtain an initial embedding of G in the plane. In particular, if the graph is planar each layer is a graph drawn with straight lines and without crossings on the n×n grid and the running time reduces to O(n log n).

Dynamic Queries is a querying technique for doing range search on multi-key data sets. It is a direct manipulation mechanism where the query is formulated using graphical widgets and the results are displayed graphically in real time. This paper evaluates four data structures, the multilist, the grid file, k-d tree and the quad tree used to organize data in high speed storage for dynamic queries. The effect of factors like size, distribution and dimensionality of data on the storage overhead and the speed of search is explored. A way of estimating the storage and the search overheads using analytical models is presented. These models are verified to be correct by empirical data. Results indicate that multilists are suitable for small (few thousand points) data sets irrespective of the data distribution. For large data sets the grid files are excellent for uniformly distributed data, and trees are good for skewed data distributions. There was no significant difference in performance bet...

Abstract. The research goal of highly versatile document analysis systems, capable of performing useful functions on the great majority of document images, seems to be receding, even in the face of decades of research. One family of nearly universally applicable capabilities includes document image content extraction tools able to locate regions containing handwriting, machine-print text, graphics, line-art, logos, photographs, noise, etc. To solve this problem in its full generality requires coping with a vast diversity of document and image types. The severity of the methodological problems is suggested by the lack of agreement within the R&D community on even what is meant by a representative set of samples in this context. Even when this is agreed, it is often not clear how sufficiently large sets for training and testing can be collected and ground truthed. Perhaps this can be alleviated by discovering a principled way to amplify sample sets using synthetic variations. We will then need classification methodologies capable of learning automatically from these huge sample sets in spite of their poorly parameterized—or unparameterizable—distributions. Perhaps fast expected-time approximate k-nearest neighbors classifiers are a good solution, even if they tend to require enormous data structures: hashed k-d trees seem promising. We discuss these issues and report recent progress towards their resolution.

. We analyze the expected time complexity of range searching with k-d trees in all dimensions when the data points are uniformly distributed in the unit hypercube. The partial match results of Flajolet and Puech are reproved using elementary probabilistic methods. In addition, we give asymptotic expected time analysis for orthogonal and convex range search, as well as nearest neighbor search. We disprove a conjecture by Bentley that nearest neighbor search for a given random point in the k-d tree can be done in O(1) expected time. Keywords and phrases. k-d trees, partial match query, range search, expected time, probabilistic analysis of algorithms, data structures, nearest neoghbor search. Research of the authors was sponsored by NSERC grant A3456. The third author received a DGAPA-UNAM Scholarship. x1. Introduction The k-d tree, or k-dimensional binary search tree, was proposed by Bentley in 1975. It is a binary tree in which each record contains k keys, right and left pointers to ...