On Range Searching with Semialgebraic Sets II

Let $P$ be a set of $n$ points in $\R^d$. We present a linear-size data structure for answering range queries on $P$ with constant-complexity semialgebraic sets as ranges, in time close to $O(n^{1-1/d})$. It essentially matches the performance of similar structures for simplex range searching, and, for $d\ge 5$, significantly improves earlier solutions by the first two authors obtained in~1994. This almost settles a long-standing open problem in range searching. The data structure is based on the polynomial-partitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter $r$, $1 < r \le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $\R^d\setminus Z(f)$ contains at most $n/r$ points of $P$, where $Z(f)$ is the zero set of $f$. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications

Approximate Range Counting Revisited

We study range-searching for colored objects, where one has to count (approximately) the number of colors present in a query range. The problems studied mostly involve orthogonal range-searching in two and three dimensions, and the dual setting of rectangle stabbing by points. We present optimal and near-optimal solutions for these problems. Most of the results are obtained via reductions to the approximate uncolored version, and improved data-structures for them. An additional contribution of this work is the introduction of nested shallow cuttings

A New Lower Bound for Semigroup Orthogonal Range Searching

We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle's result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao's influential result had shown that the problem is already non-trivial in one dimension~\cite{Yao-1Dlb}: using $m$ units of space, the query time $Q(n)$ must be $\Omega( \alpha(m,n) + \frac{n}{m-n+1})$ where $\alpha(\cdot,\cdot)$ is the inverse Ackermann's function, a very slowly growing function. In $d$ dimensions, Bernard Chazelle~\cite{Chazelle.LB.II} proved that the query time must be $Q(n) = \Omega( (\log_\beta n)^{d-1})$ where $\beta = 2m/n$. Chazelle's lower bound is known to be tight for when space consumption is `high' i.e., $m = \Omega(n \log^{d+\varepsilon}n)$. We have two main results. The first is a lower bound that shows Chazelle's lower bound was not tight for `low space': we prove that we must have $m (n) = \Omega(n (\log n \log\log n)^{d-1})$. Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions

Approximate Range Queries for Clustering

We study the approximate range searching for three variants of the clustering problem with a set P of n points in d-dimensional Euclidean space and axis-parallel rectangular range queries: the k-median, k-means, and k-center range-clustering query problems. We present data structures and query algorithms that compute (1+epsilon)-approximations to the optimal clusterings of P cap Q efficiently for a query consisting of an orthogonal range Q, an integer k, and a value epsilon>0

Effizient algorithms for generalized intersection searching on non-iso-oriented objects

In a generalized intersection searching problem, a set $S$ of colored geometric objects is to be preprocessed so that, given a query object $q$, the distinct colors of the objects of $S$ that are intersected by $q$ can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented, i.e., axes-parallel, or where the color classes satisfy additional properties, e.g., connectedness. In this paper, efficient algorithms are given for several generalized problems involving non-iso-oriented objects. These problems include: generalized halfspace range searching in ${\cal R}^d$, for any fixed $d \geq 2$, segment intersection searching, triangle stabbing, and triangle range searching in ${\cal R}^2$. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search

On the complexity of range searching among curves

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 $\mathbb{R}^d$, preprocess $S$ into a data structure that answers queries with a query curve $q$ and radius $\rho$ for the curves of $S$ that have \Frechet distance at most $\rho$ 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 $\Omega\left((n/Q(n))^2\right)$ space in the worst case, even if queries are mere points (for the discrete \Frechet distance) or line segments (for the continuous \Frechet 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 in the range searching literature: 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 \Frechet distance by using semialgebraic range searching. Our solution for the discrete \Frechet 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 \Frechet distance, the number of levels increases to $O(t^2)$

Idempotent permutations

Together with a characteristic function, idempotent permutations uniquely determine idempotent maps, as well as their linearly ordered arrangement simultaneously. Furthermore, in-place linear time transformations are possible between them. Hence, they may be important for succinct data structures, information storing, sorting and searching. In this study, their combinatorial interpretation is given and their application on sorting is examined. Given an array of n integer keys each in [1,n], if it is allowed to modify the keys in the range [-n,n], idempotent permutations make it possible to obtain linearly ordered arrangement of the keys in O(n) time using only 4log(n) bits, setting the theoretical lower bound of time and space complexity of sorting. If it is not allowed to modify the keys out of the range [1,n], then n+4log(n) bits are required where n of them is used to tag some of the keys.Comment: 32 page

Electromagnetic form factors in the J/\psi mass region: The case in favor of additional resonances

Using the results of our recent analysis of e^+e^- annihilation, we plot the curves for the diagonal and transition form factors of light hadrons in the time-like region up to the production threshold of an open charm quantum number. The comparison with existing data on the decays of J/\psi into such hadrons shows that some new resonance structures may be present in the mass range between 2 GeVand the J/\psi mass. Searching them may help in a better understanding of the mass spectrum in both the simple and a more sophisticated quark models, and in revealing the details of the three-gluon mechanism of the OZI rule breaking in K\bar K channel.Comment: Formulas are added, typo is corrected, the text is rearranged. Replaced to match the version accepted in Phys Rev