This paper introduces a new proximity graph, called the @math -Semi-Yao graph ( @math -SYG), on a set @math of points in @math , which is a supergraph of the @math -nearest neighbor graph ( @math -NNG) of @math . We provide a kinetic data structure (KDS) to maintain the @math -SYG on moving points, where the trajectory of each point is a polynomial function whose degree is bounded by some constant. Our technique gives the first KDS for the theta graph ( , @math -SYG) in @math . It generalizes and improves on previous work on maintaining the theta graph in @math . As an application, we use the kinetic @math -SYG to provide the first KDS for maintenance of all the @math -nearest neighbors in @math , for any @math . Previous works considered the @math case only. Our KDS for all the @math -nearest neighbors is deterministic. The best previous KDS for all the @math -nearest neighbors in @math is randomized. Our structure and analysis are simpler and improve on this work for the @math case. We also provide a KDS for all the @math -nearest neighbors, which in fact gives better performance than previous KDS's for maintenance of all the exact @math -nearest neighbors. As another application, we present the first KDS for answering reverse @math -nearest neighbor queries on moving points in @math , for any @math .
Using multidimensional range trees, Agarwal, Kaplan, and Sharir (TALG'08) @cite_13 gave KDS's both for maintenance of the closest pair and for all the @math -nearest neighbors in @math . The closest pair KDS by Agarwal al uses @math space and processes @math events, each in amortized time @math ; this KDS is efficient, amortized responsive, local, and compact. Agarwal al gave the first efficient KDS to maintain all the @math -nearest neighbors in @math . For the efficiency of their KDS, they implemented range trees by using randomized search trees (treaps). Their kinetic approach uses @math space and processes @math events; the expected time to process all events is @math . Their all @math -nearest neighbors KDS is efficient, amortized responsive, compact, but in general is not local.
@cite_13
2109064983 2950981509 2115704441 1970995121
A method is presented that uses an approximate nearest neighbor method for determining correspondences within the iterative closest point algorithm. The method is based upon the k-d tree. The standard k-d tree uses a tentative backtracking search to identify nearest neighbors. In contrast, the approximate k-d tree (Ak-d tree) applies a depth-first nontentative search to the k-d tree structure. This search improves runtime efficiency, with the tradeoff of reducing the accuracy of the determined correspondences. This approximate search is applied to early iterations of the iterative closest point algorithm, transitioning to the standard k-d tree for the final iterations after the change in the mean square error of the correspondences becomes sufficiently small. The method benefits both from the improved time performance of the approximate search in early iterations as well as the full accuracy of the complete search in later iterations. Experimental results indicate that the time efficiency of Ak-d tree is superior to the k-d tree and Elias for moderately large point sets. The change in the shape of the minimum potential well space is subtle, and the convergence properties are often identical. In those cases where the global minimum was not achieved, the difference in final mse was very small. In one trial, Ak-d tree converged faster to a better minimum with a smaller mse, which indicates that the use of approximate methods may be beneficial in the presence of outliers. Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure. We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic two-dimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or deletions take place, the structure for the closest pair uses O(n log n) space, and processes O(n2βs+2(n)log n) critical events, each in O(log2n) time. Here s is the maximum number of times where the distances between any two specific pairs of points can become equal, βs(q) e λs(q) q, and λs(q) is the maximum length of Davenport-Schinzel sequences of order s on q symbols. The dynamic version of the problem incurs a slight degradation in performance: If m ≥ n insertions and deletions are performed, the structure still uses O(n log n) space, and processes O(mnβs+2(n)log3 n) events, each in O(log3n) time. Our kinetic data structure for all nearest neighbors uses O(n log2 n) space, and processes O(n2β2s+2(n)log3 n) critical events. The expected time to process all events is O(n2βs+22(n) log4n), though processing a single event may take Θ(n) expected time in the worst case. If m ≥ n insertions and deletions are performed, then the expected number of events is O(mnβ2s+2(n) log3n) and processing them all takes O(mnβ2s+2(n) log4n). An insertion or deletion takes O(n) expected time. For some multimedia applications, it has been found that domain objects cannot be represented as feature vectors in a multidimensional space. Instead, pair-wise distances between data objects are the only input. To support content-based retrieval, one approach maps each object to a k-dimensional (k-d) point and tries to preserve the distances among the points. Then, existing spatial access index methods such as the R-trees and KD-trees can support fast searching on the resulting k-d points. However, information loss is inevitable with such an approach since the distances between data objects can only be preserved to a certain extent. Here we investigate the use of a distance-based indexing method. In particular, we apply the vantage point tree (vp-tree) method. There are two important problems for the vp-tree method that warrant further investigation, the n-nearest neighbors search and the updating mechanisms. We study an n-nearest neighbors search algorithm for the vp-tree, which is shown by experiments to scale up well with the size of the dataset and the desired number of nearest neighbors, n. Experiments also show that the searching in the vp-tree is more efficient than that for the @math -tree and the M-tree. Next, we propose solutions for the update problem for the vp-tree, and show by experiments that the algorithms are efficient and effective. Finally, we investigate the problem of selecting vantage-point, propose a few alternative methods, and study their impact on the number of distance computation.