254 research outputs found

    Output-Sensitive Tools for Range Searching in Higher Dimensions

    Full text link
    Let PP be a set of nn points in Rd{\mathbb R}^{d}. A point pPp \in P is kk\emph{-shallow} if it lies in a halfspace which contains at most kk points of PP (including pp). We show that if all points of PP are kk-shallow, then PP can be partitioned into Θ(n/k)\Theta(n/k) subsets, so that any hyperplane crosses at most O((n/k)11/(d1)log2/(d1)(n/k))O((n/k)^{1-1/(d-1)} \log^{2/(d-1)}(n/k)) subsets. Given such a partition, we can apply the standard construction of a spanning tree with small crossing number within each subset, to obtain a spanning tree for the point set PP, with crossing number O(n11/(d1)k1/d(d1)log2/(d1)(n/k))O(n^{1-1/(d-1)}k^{1/d(d-1)} \log^{2/(d-1)}(n/k)). This allows us to extend the construction of Har-Peled and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set of nn points in Rd{\mathbb R}^{d} (without the shallowness assumption), a spanning tree TT with {\em small relative crossing number}. That is, any hyperplane which contains wn/2w \leq n/2 points of PP on one side, crosses O(n11/(d1)w1/d(d1)log2/(d1)(n/w))O(n^{1-1/(d-1)}w^{1/d(d-1)} \log^{2/(d-1)}(n/w)) edges of TT. Using a similar mechanism, we also obtain a data structure for halfspace range counting, which uses O(nloglogn)O(n \log \log n) space (and somewhat higher preprocessing cost), and answers a query in time O(n11/(d1)k1/d(d1)(log(n/k))O(1))O(n^{1-1/(d-1)}k^{1/d(d-1)} (\log (n/k))^{O(1)}), where kk is the output size

    Independent Range Sampling, Revisited

    Get PDF
    In the independent range sampling (IRS) problem, given an input set P of n points in R^d, the task is to build a data structure, such that given a range R and an integer t >= 1, it returns t points that are uniformly and independently drawn from P cap R. The samples must satisfy inter-query independence, that is, the samples returned by every query must be independent of the samples returned by all the previous queries. This problem was first tackled by Hu, Qiao and Tao in 2014, who proposed optimal structures for one-dimensional dynamic IRS problem in internal memory and one-dimensional static IRS problem in external memory. In this paper, we study two natural extensions of the independent range sampling problem. In the first extension, we consider the static IRS problem in two and three dimensions in internal memory. We obtain data structures with optimal space-query tradeoffs for 3D halfspace, 3D dominance, and 2D three-sided queries. The second extension considers weighted IRS problem. Each point is associated with a real-valued weight, and given a query range R, a sample is drawn independently such that each point in P cap R is selected with probability proportional to its weight. Walker\u27s alias method is a classic solution to this problem when no query range is specified. We obtain optimal data structure for one dimensional weighted range sampling problem, thereby extending the alias method to allow range queries

    Further Results on Colored Range Searching

    Get PDF
    We present a number of new results about range searching for colored (or "categorical") data: 1. For a set of nn colored points in three dimensions, we describe randomized data structures with O(npolylogn)O(n\mathop{\rm polylog}n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(kpolyloglogn)O(k\mathop{\rm polyloglog} n) expected time, where kk is the number of distinct colors in the range, assuming that coordinates are in {1,,n}\{1,\ldots,n\}. Previous data structures require O(lognloglogn+k)O(\frac{\log n}{\log\log n} + k) query time. Our result also implies improvements in higher constant dimensions. 2. Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(klogn)O(k\log n) expected query time. Previous data structures require O(klog2n)O(k\log^2n) query time. 3. For a set of nn colored points in two dimensions, we describe a data structure with O(npolylogn)O(n\mathop{\rm polylog}n) space that can answer colored "type-2" range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O(lognloglogn+kloglogn)O(\frac{\log n}{\log\log n} + k\log\log n), where kk is the number of distinct colors in the range. Naively performing kk uncolored range counting queries would require O(klognloglogn)O(k\frac{\log n}{\log\log n}) time. Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.Comment: full version of a SoCG'20 pape

    Convex Hulls under Uncertainty

    Get PDF
    We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a \emph{null} location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of \some-hull that may be a useful representation of uncertain hulls

    On Range Summary Queries

    Get PDF
    We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter ? and a set P of n points in ?^d where each point is assigned a color from a set C, and the goal is to build a structure such that given any geometric range ?, we can efficiently find a list of approximate heavy hitters in ??P, i.e., colors that appear at least ? |??P| times in ??P, as well as their frequencies with an additive error of ? |??P|. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence S of 1+1/? weights such that the i-th weight in S has approximate rank i?|??P|, meaning, rank i?|??P| up to an additive error of ?|??P|. Previously, optimal results were only known in 1D [Wei and Yi, 2011] but a few sub-optimal methods were available in higher dimensions [Peyman Afshani and Zhewei Wei, 2017; Pankaj K. Agarwal et al., 2012]. We study the problems for two important classes of geometric ranges: 3D halfspace and 3D dominance queries. It is known that many other important queries can be reduced to these two, e.g., 1D interval stabbing or interval containment, 2D three-sided queries, 2D circular as well as 2D k-nearest neighbors queries. We consider the real RAM model of computation where integer registers of size w bits, w = ?(log n), are also available. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time O(log n + 1/?). Note that as the output size is 1/?, after investing the initial O(log n) searching time, our structure takes on average O(1) time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra log_w(1/?) (resp. log log_w(1/?)) factor in 3D (resp. 2D). By spending extra log^O(1)(1/?) factors in both time and space, we can also support quantile queries. We remark that it is hopeless to achieve a similar query bound for dimensions 4 or higher unless significant advances are made in the data structure side of theory of geometric approximations
    corecore