Output-Sensitive Tools for Range Searching in Higher Dimensions

Let $P$ be a set of $n$ points in ${\mathbb R}^{d}$. A point $p \in P$ is $k$\emph{-shallow} if it lies in a halfspace which contains at most $k$ points of $P$ (including $p$). We show that if all points of $P$ are $k$-shallow, then $P$ can be partitioned into $\Theta(n/k)$ subsets, so that any hyperplane crosses at most $O((n/k)^{1-1/(d-1)} \log^{2/(d-1)}(n/k))$ 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 $P$, with crossing number $O(n^{1-1/(d-1)}k^{1/d(d-1)} \log^{2/(d-1)}(n/k))$. 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 $n$ points in ${\mathbb R}^{d}$ (without the shallowness assumption), a spanning tree $T$ with {\em small relative crossing number}. That is, any hyperplane which contains $w \leq n/2$ points of $P$ on one side, crosses $O(n^{1-1/(d-1)}w^{1/d(d-1)} \log^{2/(d-1)}(n/w))$ edges of $T$. Using a similar mechanism, we also obtain a data structure for halfspace range counting, which uses $O(n \log \log n)$ space (and somewhat higher preprocessing cost), and answers a query in time $O(n^{1-1/(d-1)}k^{1/d(d-1)} (\log (n/k))^{O(1)})$, where $k$ is the output size

Further Results on Colored Range Searching

We present a number of new results about range searching for colored (or "categorical") data: 1. For a set of $n$ colored points in three dimensions, we describe randomized data structures with $O(n\mathop{\rm polylog}n)$ space that can report the distinct colors in any query orthogonal range (axis-aligned box) in $O(k\mathop{\rm polyloglog} n)$ expected time, where $k$ is the number of distinct colors in the range, assuming that coordinates are in $\{1,\ldots,n\}$. Previous data structures require $O(\frac{\log n}{\log\log n} + k)$ 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 $O(k\log n)$ expected query time. Previous data structures require $O(k\log^2n)$ query time. 3. For a set of $n$ colored points in two dimensions, we describe a data structure with $O(n\mathop{\rm polylog}n)$ 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 $O(\frac{\log n}{\log\log n} + k\log\log n)$, where $k$ is the number of distinct colors in the range. Naively performing $k$ uncolored range counting queries would require $O(k\frac{\log n}{\log\log n})$ 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

Independent Range Sampling, Revisited Again

We revisit the range sampling problem: the input is a set of points where each point is associated with a real-valued weight. The goal is to store them in a structure such that given a query range and an integer k, we can extract k independent random samples from the points inside the query range, where the probability of sampling a point is proportional to its weight. This line of work was initiated in 2014 by Hu, Qiao, and Tao and it was later followed up by Afshani and Wei. The first line of work mostly studied unweighted but dynamic version of the problem in one dimension whereas the second result considered the static weighted problem in one dimension as well as the unweighted problem in 3D for halfspace queries. We offer three main results and some interesting insights that were missed by the previous work: We show that it is possible to build efficient data structures for range sampling queries if we allow the query time to hold in expectation (the first result), or obtain efficient worst-case query bounds by allowing the sampling probability to be approximately proportional to the weight (the second result). The third result is a conditional lower bound that shows essentially one of the previous two concessions is needed. For instance, for the 3D range sampling queries, the first two results give efficient data structures with near-linear space and polylogarithmic query time whereas the lower bound shows with near-linear space the worst-case query time must be close to n^{2/3}, ignoring polylogarithmic factors. Up to our knowledge, this is the first such major gap between the expected and worst-case query time of a range searching problem

On Geometric Range Searching, Approximate Counting and Depth Problems

In this thesis we deal with problems connected to range searching, which is one of the central areas of computational geometry. The dominant problems in this area are halfspace range searching, simplex range searching and orthogonal range searching and research into these problems has spanned decades. For many range searching problems, the best possible data structures cannot offer fast (i.e., polylogarithmic) query times if we limit ourselves to near linear storage. Even worse, it is conjectured (and proved in some cases) that only very small improvements to these might be possible. This inefficiency has encouraged many researchers to seek alternatives through approximations. In this thesis we continue this line of research and focus on relative approximation of range counting problems. One important problem where it is possible to achieve significant speedup through approximation is halfspace range counting in 3D. Here we continue the previous research done and obtain the first optimal data structure for approximate halfspace range counting in 3D. Our data structure has the slight advantage of being Las Vegas (the result is always correct) in contrast to the previous methods that were Monte Carlo (the correctness holds with high probability). Another series of problems where approximation can provide us with substantial speedup comes from robust statistics. We recognize three problems here: approximate Tukey depth, regression depth and simplicial depth queries. In 2D, we obtain an optimal data structure capable of approximating the regression depth of a query hyperplane. We also offer a linear space data structure which can answer approximate Tukey depth queries efficiently in 3D. These data structures are obtained by applying our ideas for the approximate halfspace counting problem. Approximating the simplicial depth turns out to be much more difficult, however. Computing the simplicial depth of a given point is more computationally challenging than most other definitions of data depth. In 2D we obtain the first data structure which uses near linear space and can answer approximate simplicial depth queries in polylogarithmic time. As applications of this result, we provide two non-trivial methods to approximate the simplicial depth of a given point in higher dimension. Along the way, we establish a tight combinatorial relationship between the Tukey depth of any given point and its simplicial depth. Another problem investigated in this thesis is the dominance reporting problem, an important special case of orthogonal range reporting. In three dimensions, we solve this problem in the pointer machine model and the external memory model by offering the first optimal data structures in these models of computation. Also, in the RAM model and for points from an integer grid we reduce the space complexity of the fastest known data structure to optimal. Using known techniques in the literature, we can use our results to obtain solutions for the orthogonal range searching problem as well. The query complexity offered by our orthogonal range reporting data structures match the most efficient query complexities known in the literature but our space bounds are lower than the previous methods in the external memory model and RAM model where the input is a subset of an integer grid. The results also yield improved orthogonal range searching in higher dimensions (which shows the significance of the dominance reporting problem). Intersection searching is a generalization of range searching where we deal with more complicated geometric objects instead of points. We investigate the rectilinear disjoint polygon counting problem which is a specialized intersection counting problem. We provide a linear-size data structure capable of counting the number of disjoint rectilinear polygons intersecting any rectilinear polygon of constant size. The query time (as well as some other properties of our data structure) resembles the classical simplex range searching data structures