Showing papers by "Richard Cole published in 2005"

Dynamic LCA Queries on Trees

Abstract: We show how to maintain a data structure on trees which allows for the following operations, all in worst-case constant time: insertion of leaves and internal nodes, deletion of leaves, deletion of internal nodes with only one child, determining the least common ancestor of any two nodes. We also generalize the Dietz--Sleator "cup-filling" scheduling methodology, which may be of independent interest.

Fast window correlations over uncooperative time series

Abstract: Data arriving in time order (a data stream) arises in fields including physics, finance, medicine, and music, to name a few. Often the data comes from sensors (in physics and medicine for example) whose data rates continue to improve dramatically as sensor technology improves. Further, the number of sensors is increasing, so correlating data between sensors becomes ever more critical in order to distill knowlege from the data. In many applications such as finance, recent correlations are of far more interest than long-term correlation, so correlation over sliding windows (windowed correlation) is the desired operation. Fast response is desirable in many applications (e.g., to aim a telescope at an activity of interest or to perform a stock trade). These three factors -- data size, windowed correlation, and fast response -- motivate this work.Previous work [10, 14] showed how to compute Pearson correlation using Fast Fourier Transforms and Wavelet transforms, but such techniques don't work for time series in which the energy is spread over many frequency components, thus resembling white noise. For such "uncooperative" time series, this paper shows how to combine several simple techniques -- sketches (random projections), convolution, structured random vectors, grid structures, and combinatorial design -- to achieve high performance windowed Pearson correlation over a variety of data sets.

The complexity of the minimum k-Cover Problem

Abstract: The k-coversproblem (kCP asks us to compute a minimum cardinality set of strings given length k>1 that covers a given string. It was shown in a recent paper, by reduction to 3 -SAT, that the k-covers problem is NP-complete. In this paper we introduce a new problem, that we call the Relaxed Vertex Cover Problem (RVCP), which we show is a special case of Set Cover (SCP). We show further the kCP is equivalent to RVCP restricted to certain classes GXk of graphs that represent all strings x. We discuss approximate solutions of kCP and we state a number of conjectures and open problems related to kCP and GXk.

Finding tree structures by grouping symmetries

Abstract: The representation of objects in images as tree structures is of great interest to vision, as they can represent articulated objects such as people as well as other structured objects like arteries in human bodies, roads, circuit board patterns, etc. Tree structures are often related to the symmetry axis representation of shapes, which captures their local symmetries. Algorithms have been introduced to detect (i) open contours in images in quadratic time (ii) closed contours in images in cubic time, and (iii) tree structures from contours in quadratic time. The algorithms are based on dynamic programming and single source shortest path algorithms. However, in this paper, we show that the problem of finding tree structures in images in a principled manner is a much harder problem. We argue that the optimization problem of finding tree structures in images is essentially equivalent to a variant of the Steiner tree problem, which is NP-hard. Nevertheless, an approximate polynomial-time algorithm for this problem exists: we apply a fast implementation of the Goemans-Williamson approximate algorithm to the problem of finding a tree representation after an image is transformed by a local symmetry mapping. Examples of extracting tree structures from images illustrate the idea and applicability of the approximate method