Peyman Afshani

Orthogonal Range Reporting in Three and Higher Dimensions

In orthogonal range reporting we are to preprocess N points in d-dimensional space so that the points inside a d-dimensional axis-aligned query box can be reported efficiently. This is a fundamental problem in various fields, including spatial databases and computational geometry. In this paper we provide a number of improvements for three and higher dimensional orthogonal range reporting: In the pointer machine model, we improve all the best previous results, some of which have not seen any improvements in almost two decades. In the I/O-model, we improve the previously known three-dimensional structures and provide the first (non-trivial) structures for four and higher dimensions.

Orthogonal range reporting: query lower bounds, optimal structures in 3-d, and higher-dimensional improvements

Orthogonal range reporting is the problem of storing a set of <i>n</i> points in <i>d</i>-dimensional space, such that the <i>k</i> points in an axis-orthogonal query box can be reported efficiently. While the 2-d version of the problem was completely characterized in the pointer machine model more than two decades ago, this is not the case in higher dimensions. In this paper we provide a space optimal pointer machine data structure for 3-d orthogonal range reporting that answers queries in <i>O</i>(log <i>n</i> + <i>k</i>) time. Thus we settle the complexity of the problem in 3-d. We use this result to obtain improved structures in higher dimensions, namely structures with a log <i>n</i>/ log log n factor increase in space and query time per dimension. Thus for <i>d</i> e 3 we obtain a structure that both uses optimal <i>O</i>(<i>n</i>(log <i>n</i>/ log log <i>n</i>)^<i>d</i>--1) space and answers queries in the best known query bound <i>O</i>(log <i>n</i>(log <i>n</i>/ log log <i>n</i>)^<i>d</i>--3 + <i>k</i>). Furthermore, we show that any data structure for the <i>d</i>-dimensional orthogonal range reporting problem in the pointer machine model of computation that uses <i>S</i>(<i>n</i>) space must spend Ω((log <i>n</i>/ log(<i>S</i>(<i>n</i>)/<i>n</i>))^{⌊<i>d</i>/2⌋--1}) time to answer queries. Thus, if <i>S</i>(<i>n</i>)/<i>n</i> is poly-logarithmic, then the query time is at least Ω((log <i>n</i>/ log log <i>n</i>)^{⌊<i>d</i>/2⌋--1}). This is the first known non-trivial higher dimensional orthogonal range reporting query lower bound and it has two important implications. First, it shows that the query bound increases with dimension. Second, in combination with our upper bounds it shows that the optimal query bound increases from Θ(log <i>n</i> + <i>k</i>) to Ω((log <i>n</i>/ log log <i>n</i>)² + <i>k</i>) somewhere between three and six dimensions. Finally, we show that our techniques also lead to improved structures for point location in rectilinear subdivisions, that is, the problem of storing a set of <i>n</i> disjoint <i>d</i>-dimensional axis-orthogonal rectangles, such that the rectangle containing a query point <i>q</i> can be found efficiently.

Approximate) uncertain skylines

Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each points uncertainty is described as a probability distribution over a discrete set of locations, we improve the best known exact solution. We also suggest why we believe our solution might be optimal. Next, we describe simple, near-linear time approximation algorithms for computing the probability of each point lying on the skyline. In addition, some of our methods can be adapted to construct data structures that can efficiently determine the probability of a query point lying on the skyline.

Cache-oblivious range reporting with optimal queries requires superlinear space

We consider a number of range reporting problems in two and three dimensions and prove lower bounds on the amount of space required by any cache-oblivious data structure for these problems that achieves an optimal query bound of O(logBN + K/B) block transfers in the worst case, where K is the size of the query output. The problems we study are three-sided range reporting, 3-d dominance reporting, and 3-d halfspace range reporting. We prove that, in order to achieve the above query bound or even a bound of O((logB N)c (1 + K/B)), for any constant c > 0, the structure has to use Ohmega(N (log log N)ε) space, where ε > 0 is a constant that depends on c and on the constant hidden in the big-Oh notation of the query bound. Our result has a number of interesting consequences. The first one is a new type of separation between the I/O model and the cache-oblivious model, as I/O-efficient data structures with the optimal query bound and using linear or O(N logAST N) space are known for the above problems. The second consequence is the non-existence of a linear-space cache-oblivious persistent B-tree with worst-case optimal 1-d range reporting queries.

The Query Complexity of Finding a Hidden Permutation

We study the query complexity of determining a hidden permutation. More specifically, we study the problem of learning a secret (z,π) consisting of a binary string z of length n and a permutation π of [n]. The secret must be unveiled by asking queries x ∈ {0,1} n , and for each query asked, we are returned the score f z,π (x) defined as

Deterministic rectangle enclosure and offline dominance reporting on the RAM

f_{z,\pi}(x):= \max \{ i \in[0..n]\mid \forall j \leq i: z_{\pi(j)} = x_{\pi(j)}\}\,;

On the complexity of finding an unknown cut via vertex queries

i.e., the length of the longest common prefix of x and z with respect to π. The goal is to minimize the number of queries asked. We prove matching upper and lower bounds for the deterministic and randomized query complexity of Θ(n logn) and Θ(n loglogn), respectively.

Optimal deterministic shallow cuttings for 3D dominance ranges

We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex range reporting as well as some other geometric problems. Simplex range reporting is the problem of storing n points in the d-dimensional space in a data structure such that the k points that lie inside a query simplex can be found efficiently. This is one of the fundamental and extensively studied problems in computational geometry. Currently, the best data structures for the problem achieve Q(n) + O(k) query time using space in which the notation either hides a polylogarithmic or an ne factor for any constant e > 0, (depending on the data structure and Q(n)). The best lower bound on this problem is due to Chazelle and Rosenberg who showed any pointer machine data structure that can answer queries in O(nγ + k) time must use Ω(nd-e-dγ) space. Observe that this bound is a polynomial factor away from the best known data structures. In this article, we improve the space lower bound to . Not only this bridges the gap from polynomial to sub-polynomial, it also offers a smooth trade-off curve. For instance, for polylogarithmic values of Q(n), our space lower bound almost equals Ω((n/Q(n))d); the latter is generally believed to be the “right” bound. By a simple geometric transformation, we also improve the best lower bounds for the halfspace range reporting problem. Furthermore, we study the external memory model and offer a new simple framework for proving lower bounds in this model. We show that answering simplex range reporting queries with Q(n)+O(k/B) I/Os requires ) space or blocks, in which B is the block size.