We present a general technique for dynamizing certain problems posed on point sets in Euclidean space for any fixed dimension d. This technique applies to a large class of structurally similar algorithms, presented previously by the authors, that make use of the well-separated pair decomposition. We prove efficient worst-case complexity for maintaining such computations under point insertions and deletions, and apply the technique to several problems posed on a set P containing n points. In particular, we show how to answer a query for any point x that returns a constant-size set of points, a subset of which consists of all points in P that have x as a nearest neighbor. We then show how to use such queries to maintain the closest pair of points in P . We also show how to dynamize the fast multipole method, a technique for approximating the potential field of a set of point charges. All our algorithms use the algebraic model that is standard in computational geometry, and have worst-ca...

Any connected plane nearest neighbor graph has diameter #(n 1/6 ). This bound generalizes to #(n 1/3d ) in any dimension d. For any set of n points in the plane, we define the nearest neighbor graph by selecting a unique nearest neighbor for each point, and adding an edge between each point and its neighbor. This is a directed graph with outdegree one; thus it is a pseudo-forest. Each component of the pseudo-forest is a tree, with a length-two directed cycle at the root. As with minimum spanning trees, the maximum degree in a nearest neighbor graph is five. Monma and Suri [1] showed that, conversely, any tree with vertex degree at most five is the minimum spanning tree of some point set; thus minimum spanning tree topologies are exactly characterized by their degrees. Paterson and Yao [2] considered the corresponding question for nearest neighbor graphs. They showed that a tree with depth D can have at most O(D 9 ) vertices. Thus unlike minimum spanning trees, nearest neighbor...

For a set S of points in R d,ans-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers. 1

We present algorithms for five interdistance enumeration problems that take as input a set S of n points in IR d (for a fixed but arbitrary dimension d) and as output enumerate pairs of points in S satisfying various conditions. We present: an O(n log n + k) time and O(n) space algorithm that takes as additional input a distance # and outputs all k pairs of points in S separated by a distance of # or less; an O(n log n + k log k) time and O(n+k) space algorithm that enumerates in non-decreasing order the k closest pairs of points in S; an O(n log n + k) time algorithm for the same problem without any order restrictions; an O(nk log n) time and O(n) space algorithm that enumerates in nondecreasing order the nk pairs representing the k nearest neighbors of each point in S; and an O(n log n + kn) time algorithm for the same problem without any order restrictions. The algorithms combine a modification of the planar approach of Dickerson, Drysdale, and Sack [11] with the ...

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Given a database of sites or elements of a metric space, the all-k-nearest-neighbor problem consists in finding, for each element, its k nearest neighbors. Using the brute force approach the problem is solved with O(n²) distance computations. An obvious idea is to index the set somehow so as to reduce this complexity. Yet, it is not obvious what kind of indexing can be helpful and there exist few proposals valid for general metric spaces. In this paper we present a general method to compute the k nearest neighbors of each element which takes advantage of any index able to answer fixed radius queries, a much better known problem. The approach is incremental, starting with a rough guess of the set of neighbors and refining it as more information is available throughout the process. The elements are solved one by one, the cost for the first being high, but dropping as we solve more and more elements. Assuming that the set has n elements and the index examines n (0 1) elements for a fixed radius query that retrieves n elements from the set, our amortized analysis shows that our algorithm takes average time O n 2+ 1+ . Our experimental results confirm the subquadraticity of the algorithm and show that it is very efficient in practice.

Abstract. We present an external memory algorithm to compute a well-separated pair decomposition (WSPD) of a given point set P in £ d in O ¤ sort ¤ N¥¦ ¥ I/Os using O ¤ N § B ¥ blocks of external memory, where N is the number of points in P, and sort ¤ N ¥ denotes the I/O complexity of sorting N items. (Throughout this paper we assume that the dimension d is fixed). We also show how to dynamically maintain the WSPD in O ¤ log B N ¥ I/O’s per insert or delete operation using O ¤ N § B ¥ blocks of external memory. As applications of the WSPD, we show how to compute a linear size t-spanner for P within the same I/O and space bounds and how to solve the K-nearest neighbor and K-closest pair problems in O ¤ sort ¤ KN¥¦¥ and O ¤ sort ¤ N ¨ K¥¦ ¥ I/Os using O ¤ KN § B ¥ and O¤¦ ¤ N ¨ K¥© § B ¥ blocks of external memory, respectively. Using the dynamic WSPD, we show how to dynamically maintain the closest pair of P in O ¤ log B N ¥ I/O’s per insert or delete operation using O ¤ N § B ¥ blocks of external memory. 1

Nearest neighbor graphs are widely used in data mining and machine learning. The brute-force method to compute the exact kNN graph takes Θ(dn 2) time for n data points in the d dimensional Euclidean space. We propose two divide and conquer methods for computing an approximate kNN graph in Θ(dn t) time for high dimensional data (large d). The exponent t depends on an internal parameter and is larger than one. Experiments show that a high quality graph usually requires a small t which is close to one. A few of the practical details of the algorithms are as follows. First, the divide step uses an inexpensive Lanczos procedure to perform recursive spectral bisection. After each conquer step, an additional refinement step is performed to improve the accuracy of the graph. Finally, a hash table is used to avoid repeating distance calculations during the divide and conquer process. The combination of these techniques is shown to yield quite effective algorithms for building kNN graphs. 1

Abstract. Let U be a set of elements and d a distance function defined among them. Let NNk(u) be the k elements in U − {u} which have the smallest distance towards u. The k-nearest neighbor graph (knng) is a weighted directed graph G(U, E) such that E = {(u, v), v ∈ NNk(u)}. We focus on the metric space context, so d is a metric. Several knng construction algorithms are known, but they are not suitable to general metric spaces. We present a general methodology to construct knngs that exploits several features of metric spaces, requiring empirically around O(n 1.27) distance computations for low and medium dimensional spaces, and O(n 1.90) for high dimensional ones. Keywords: Graph Algorithms, Metric Spaces, Nearest Neighbors. 1

Abstract. We extend the classic notion of well-separated pair decomposition [10] to the unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c ≥ 1, there exists a c-wellseparated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where k ≥ 3, there exists a c-wellseparated pair decomposition with O(n 2−2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric. Keywords Well separated pair decomposition, Unit-disk graph, Approximation algorithm