Search for: All records

In this article, we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. An uncertain curve is a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both problems are NP-hard for the Fréchet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound (discrete [ 5 ] and continuous) Fréchet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) Fréchet distance is #P-hard in some models. On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly δ-separated convex regions. Finally, we study the setting with Sakoe–Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous Fréchet distance for uncertainty modelled as point sets. 

In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters k, d, and ε, that (1+ε)-approximate the minimum expected cost solutions for points in d dimensional Euclidean space. For Faulty k-center we additionally provide a 5-approximation for general metrics. Significantly, all of our algorithms have a small dependence on n. Specifically, our Faulty k-center algorithms have only linear dependence on n, while for our algorithms for Faulty k-median and Faulty k-means the dependence is still only n^(1 + o(1)). 

Given a point set P in the plane, we seek a subset Q ⊆ P, whose convex hull gives a smaller and thus simpler representation of the convex hull of P. Specifically, let cost(Q,P) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q,P) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q,P), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log²n) time algorithm and an O(n log³n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n^2.5302) time algorithm when minimizing k and an O(min{n^2.5302, kn^2.376}) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case. 

In the standard planar k-center clustering problem, one is given a set P of n points in the plane, and the goal is to select k center points, so as to minimize the maximum distance over points in P to their nearest center. Here we initiate the systematic study of the clustering with neighborhoods problem, which generalizes the k-center problem to allow the covered objects to be a set of general disjoint convex objects C rather than just a point set P. For this problem we first show that there is a PTAS for approximating the number of centers. Specifically, if r_opt is the optimal radius for k centers, then in n^O(1/ε²) time we can produce a set of (1+ε)k centers with radius ≤ r_opt. If instead one considers the standard goal of approximating the optimal clustering radius, while keeping k as a hard constraint, we show that the radius cannot be approximated within any factor in polynomial time unless P = NP, even when C is a set of line segments. When C is a set of unit disks we show the problem is hard to approximate within a factor of (√{13}-√3)(2-√3) ≈ 6.99. This hardness result complements our main result, where we show that when the objects are disks, of possibly differing radii, there is a (5+2√3)≈ 8.46 approximation algorithm. Additionally, for unit disks we give an O(n log k)+(k/ε)^O(k) time (1+ε)-approximation to the optimal radius, that is, an FPTAS for constant k whose running time depends only linearly on n. Finally, we show that the one dimensional version of the problem, even when intersections are allowed, can be solved exactly in O(n log n) time.