Deterministic Data Structure Research Articles

On approximate near-neighbors search under the (continuous) Fréchet distance in higher dimensions

Previous studies on Approximate Near-Neighbors Search (ANNS) among curves are either focused on curves in R1 or under the discrete Fréchet distance. In this paper, we propose the first data structure for curves under the (continuous) Fréchet distance in higher dimensions. Given a set P of n curves each with number of vertices at most m in Rd, and a fixed δ>0, we aim to preprocess P into a data structure so that for any given query curve Q with k vertices, we can efficiently report all curves in P whose Fréchet distances to Q are at most δ. In the case that k is given in the preprocessing stage, for any ε>0 we propose a deterministic data structure whose space is n⋅O(max⁡{(dε)kd,(Ddε2)kd}) that can answer (1+ε)δ-ANNS queries in O(kd) query time, where D is the diameter of P. Considering k as part of the query slightly changes the space to n⋅O(1/ε)md with O(kd) query time within an approximation factor of 5+ε. Moreover, we show that our generic data structure for ANNS can give an alternative treatment of the approximate subtrajectory range searching problem studied by de Berg et al. [1].

Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph $G = (V,E)$, with $|V| = n$ and $|E| =m$, in $o(\sqrt{m})$ time per update. In particular, for minimum vertex cover, we provide deterministic data structures for maintaining a $(2+\epsilon)$ approximation in $O(\log n/\epsilon^2)$ amortized time per update. For maximum matching, we show how to maintain a $(3+\epsilon)$ approximation in $O(\min(\sqrt{n}/\epsilon, m^{1/3}/\epsilon^2)$ amortized time per update and a $(4+\epsilon)$ approximation in $O(m^{1/3}/\epsilon^2)$ worst-case time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld [in 42nd ACM Symposium on Theory of Computing, Cambridge, MA, ACM, 2010, pp. 457--464].

A note on data structures for maintaining bipartitions

We consider the following problem: given a ground set U of elements { 1 , 2 , … , n } , and a set S of bipartitions of U, design a data structure to support the following three operations: Report ( S ) —report the partition of U induced by S , Insert ( P , S ) —add a new bipartition P to S , and Delete ( P , S ) —delete the existing partition P from S , where the partition of U induced by S is given by two elements of U being in the same class if and only if they are in the same class for every bipartition of S . We describe a straightforward deterministic data structure with an amortized bound of O ( n ) per update, which is optimal.

Data structures for maintaining set partitions

AbstractEfficiently maintaining the partition induced by a set of features is an important problem in building decision‐tree classifiers. In order to identify a small set of discriminating features, we need the capability of efficiently adding and removing specific features and determining the effect of these changes on the induced classification or partition. In this paper we introduce a variety of randomized and deterministic data structures to support these operations on both general and geometrically induced set partitions. We give both Monte Carlo and Las Vegas data structures that realize near‐optimal time bounds and are practical to implement. We then provide a faster solution to this problem in the geometric setting. Finally, we present a data structure that efficiently estimates the number of partitions separating elements. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

Maintaining dynamic sequences under equality tests in polylogarithmic time

We present a randomized and a deterministic data structure for maintaining a dynamic family of sequences under equality tests of pairs of sequences and creations of new sequences by joining or splitting existing sequences. Both data structures support equality tests inO(1) time. The randomized version supports new sequence creations inO(log2 n) expected time wheren is the length of the sequence created. The deterministic solution supports sequence creations inO(logn(logmlog* m+logn)) time for themth operation.

Queries on Voronoi diagrams of moving points

Suppose we are given n moving postmen described by their motion equations p i ( t) = s i + v i t, i = 1,…, n, where s i ∈ R 2 is the position of the ith postman at time t = 0, and v i ∈ R 2 is his velocity. The pr we address is how to preprocess the postmen data so as to be able to efficiently answer two types of nearest-neighbor queries. The first one asks “who is the nearest postman at time t q to a dog located at point s q . In the second type a query dog is located at point s q at time t q , its speed is v q > | v i | (for all i = 1,…, n), and we know which postman the dog can catch first. The first type of query is relatively simple to address, the second type at first seems much more complicated. We show that the problems are very closely related, with efficient solutions to the first type of query leading to efficient solutions to the second. We then present two solutions to these problems, with tradeoff between preprocessing time and query time. Both solutions use deterministic data structures.