2008990
Dual-Tree Fast Gauss Transforms
In previous work we presented an efficient approach to computing kernel summations which arise in many machine learning methods such as kernel density estimation. This approach, dual-tree recursion with finite-difference approximation, generalized existing methods for similar problems arising in computational physics in two ways appropriate for statistical problems: toward distribution sensitivity and general dimension, partly by avoiding series expansions. While this proved to be the fastest practical method for multivariate kernel density estimation at the optimal bandwidth, it is much less efficient at larger-than-optimal bandwidths. In this work, we explore the extent to which the dual-tree approach can be integrated with multipole-like Hermite expansions in order to achieve reasonable efficiency across all bandwidth scales, though only for low dimensionalities. In the process, we derive and demonstrate the first truly hierarchical fast Gauss transforms, effectively combining the best tools from discrete algorithms and continuous approximation theory.
12896882
Faster Gaussian Summation: Theory and Experiment
We provide faster algorithms for the problem of Gaussian summation, which occurs in many machine learning methods. We develop two new extensions - an O(Dp) Taylor expansion for the Gaussian kernel with rigorous error bounds and a new error control scheme integrating any arbitrary approximation method - within the best discretealgorithmic framework using adaptive hierarchical data structures. We rigorously evaluate these techniques empirically in the context of optimal bandwidth selection in kernel density estimation, revealing the strengths and weaknesses of current state-of-the-art approaches for the first time. Our results demonstrate that the new error control scheme yields improved performance, whereas the series expansion approach is only effective in low dimensions (five or less).
13592174
Critical area computation—a new approach
In this paper we present a new approach for computing the critical area for shorts in a circuit layout. The critical area calculation is the main computational problem in VLSI yield prediction. The method is based on the concept of Voronoi diagrams and computes the critical area for shorts (for all possible defect radii, assuming square defects) accurately in <italic>O</italic>(<italic>n</italic> log <italic>n</italic>) time, where <italic>n</italic> is the size of the input. The method is presented for rectilinear layouts but it is extendible to general layouts. As a byproduct we briefly sketch how to speed up the grid method of Wagner and Koren [16].
5628931
A distributed multicast routing algorithm for real-time applications in wide area networks
In this paper we propose a delay-constrained distributed multicast routing algorithm based on token passing. This, algorithm is fully distributed and generates a multicast routing tree, which not only meets the realtime requirement, but also has a sub-optimal network cost. Simulations have been done and the results have shown that the multicast routing tree generated by our algorithm has better performance than previously known results.
532780
A New Approach for the Geodesic Voronoi Diagram of Points in a Simple Polygon and Other Restricted Polygonal Domains
Abstract. We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divide-and-conquer algorithm. The time complexity is O((n+k) log(n+k)) where n is the number of vertices and k is the number of points, improving upon previously known bounds. Space is O(n+k) . Other polygonal domains where our method is applicable include (among others) a polygonal domain of parallel disjoint line segments and a polygonal domain of rectangles in the L1 metric.
2585986
Fast multiplication of a recursive block Toeplitz matrix by a vector and its application
We present an algorithm for multiplying an N × N recursive block Toeplitz matrix by a vector with cost O (N log N). Its application to optimal surface interpolation is discussed.
6583254
Rectilinear Path Problems among Rectilinear Obstacles Revisited
Efficient algorithms are presented for finding rectilinear collision-free paths between two given points among a set of rectilinear obstacles. The results improve the time complexity of previous results for finding the shortest rectilinear path, the minimum-bend shortest rectilinear path, the shortest minimum-bend rectilinear path and the minimum-cost rectilinear path. For finding the shortest rectilinear path, a graph-theoretic approach is used and an algorithm is obtained with $O(m\log t+ t\log^{3/2} t)$ running time, where $t$ is the number of extreme edges of given obstacles and $m$ is the number of obstacle edges. Based on this result an $O(N\log N+(m+N)\log t + (t+N)\log^2 (t+N))$ running time algorithm for computing the $L_1$ minimum spanning tree of given $N$ terminals among rectilinear obstacles is obtained. For finding the minimum-bend shortest path, the shortest minimum-bend rectilinear path, and the minimum-cost rectilinear path, we devise a new dynamic-searching approach and derive algorithms that run in $O(m\log^2m)$ time using $O(m\log m)$ space or run in $O(m\log^{3/2}m)$ time and space.