No abstract

In this paper, we study the 3D dominance reporting problem in different models of computations and offer optimal results in the pointer machine and the external memory models and a near optimal result in the RAM model; all our results consume linear space. We can answer queries in O(log n + k) time on a pointer machine, with O(log B n + k/B) I/Os in the external memory model and in O((log log n) 2 + log log U + k) time in the RAM model and in a U ×U ×U integer grid. These improve the results of various papers, such as Makris and Tsakalidis (IPL'98), Vengroff and Vitter (STOC'96) and Nekrich (SOCG'07). Here, n, k and B are the input, output and block size respectively. With a log 3 n fold increase in the space complexity these can be turned into orthogonal range reporting algorithms with matching query times, improving the previous orthogonal range searching results in the pointer machine and RAM models. Using our 3D results as base cases, we can provide improved orthogonal range reporting algorithms in R d , d ≥ 4. We use randomization only in the preprocessing part and our query bounds are all worst case.

We prove the existence of an algorithm A for computing 2D or 3D convex hulls that is optimal for every point set in the following sense: for every sequence σ of n points and for every algorithm A ′ in a certain class A , the running time of A on input σ is at most a constant factor times the running time of A ′ on the worst possible permutation of σ for A ′. In fact, we can establish a stronger property: for every sequence σ of points and every algorithm A ′, the running time of A on σ is at most a constant factor times the average running time of A ′ over all permutations of σ. We call algorithms satisfying these properties instance optimal in the order-oblivious and random-order setting. Such instance-optimal algorithms simultaneously subsume output-sensitive algorithms and distribution-dependent average-case algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input are given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instance-specific lower bound, we deviate from traditional Ben-Or-style proofs and adopt a new adversary argument. For 2D convex hulls, we prove that a version of the well-known algorithm by Kirkpatrick and Seidel [1986] or Chan, Snoeyink, and Yap [1995] already attains this lower bound. For 3D convex hulls, we propose a new algorithm. We further obtain instance-optimal results for a few other standard problems in computational geometry, such as maxima in 2D and 3D, orthogonal line segment intersection in 2D, finding bichromatic L ∞ -close pairs in 2D, offline orthogonal range searching in 2D, offline dominance reporting in 2D and 3D, offline half-space range reporting in 2D and 3D, and offline point location in 2D. Our framework also reveals a connection to distribution-sensitive data structures and yields new results as a byproduct, for example, on online orthogonal range searching in 2D and online half-space range reporting in 2D and 3D.

No abstract

In orthogonal range reporting we are to preprocess

In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in ddimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported efficiently. Rectangle stabbing is the "dual" problem where a set of n axis-aligned rectangles should be stored in a data structure, such that the t rectangles that contain a query point can be reported efficiently. Very recently an optimal O(log n + t) query time pointer machine data structure was developed for the three-dimensional version of the orthogonal range reporting problem. However, in four dimensions the best known query bound of O(log 2 n/ log log n + t) has not been improved for decades.We describe an orthogonal range reporting data structure that is the first structure to achieve significantly less than O(log 2 n + t) query time in four dimensions. More precisely, we develop a structure that uses O(n(log n/ log log n) d ) space and can answer d-dimensional orthogonal range reporting queries (for d ≥ 4) in O(log n(log n/ log log n) d−4+1/(d−2) +t) time. Ignoring log log n factors, this speeds up the best previous query time by a log 1−1/(d−2) n factor. For the rectangle stabbing problem, we show that any data structure that uses nh space must use Ω(log n(log n/ log h) d−2 + t) time * This research was done while the author was a postdoctoral researcher at Dalhousie University, and was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) through the Postdoctoral Fellowships Program (PDF). † Is supported in part by MADALGO and in part by a Google Fellowship in Search and Information Retrieval ‡ Center for Massive Data Algorithmics, a center of the Danish National Research Foundation.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. to answer a query. This improves the previous results by a log h factor, and is the first lower bound that is optimal for a large range of h, namely for h ≥ log d−2+ε n where ε > 0 is an arbitrarily small constant. By a simple geometric transformation, our result also implies an improved query lower bound for orthogonal range reporting.

No abstract

Modern tracking technology has made the collection of large numbers of densely sampled trajectories of moving objects widely available. We consider a fundamental problem encountered when analysing such data: Given n polygonal curves S in R d , preprocess S into a data structure that answers queries with a query curve q and radius ρ for the curves of S that have Fréchet distance at most ρ to q.We initiate a comprehensive analysis of the space/query-time trade-off for this data structuring problem. Our lower bounds imply that any data structure in the pointer model model that achieves Q(n) + O(k) query time, where k is the output size, has to use roughly Ω (n/Q(n)) 2 space in the worst case, even if queries are mere points (for the discrete Fréchet distance) or line segments (for the continuous Fréchet distance). More importantly, we show that more complex queries and input curves lead to additional logarithmic factors in the lower bound. Roughly speaking, the number of logarithmic factors added is linear in the number of edges added to the query and input curve complexity. This means that the space/query time trade-off worsens by an exponential factor of input and query complexity. This behaviour addresses an open question (see [1,9]) in the range searching literature concerning multilevel partition trees which may be of independent interest, namely, whether it is possible to avoid the additional logarithmic factors in the space and query time of a multilevel partition tree. We answer this question negatively.On the positive side, we show we can build data structures for the Fréchet distance by using semialgebraic range searching. The space/query-time trade-off of our data structure for the discrete Fréchet distance is in line with the lower bound, as the number of levels in the data structure is O(t), where t denotes the maximal number of vertices of a curve. For the continuous Fréchet distance, the number of levels increases to O(t 2 ).