Simplex range reporting on a pointer machine

AbstractWe give a lower bound on the following problem, known as simplex range reporting: Given a collection P of n points in d-space and an arbitrary simplex q, find all the points in P ∩ q. It is understood that P is fixed and can be preprocessed ahead of time, while q is a query that must be answered on-line. We consider data structures for this problem that can be modeled on a pointer machine and whose query time is bounded by O(nδ + r), where r is the number of points to be reported and δ is an arbitrary fixed real. We prove that any such data structure of that form must occupy storage Ω(nd(1 − δ)− ε), for any fixed ε > 0. This lower bound is tight within a factor of nε

Data Structure Lower Bounds for Document Indexing Problems

We study data structure problems related to document indexing and pattern matching queries and our main contribution is to show that the pointer machine model of computation can be extremely useful in proving high and unconditional lower bounds that cannot be obtained in any other known model of computation with the current techniques. Often our lower bounds match the known space-query time trade-off curve and in fact for all the problems considered, there is a very good and reasonable match between the our lower bounds and the known upper bounds, at least for some choice of input parameters. The problems that we consider are set intersection queries (both the reporting variant and the semi-group counting variant), indexing a set of documents for two-pattern queries, or forbidden- pattern queries, or queries with wild-cards, and indexing an input set of gapped-patterns (or two-patterns) to find those matching a document given at the query time.Comment: Full version of the conference version that appeared at ICALP 2016, 25 page

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 $O(t^2)$

Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

In the semialgebraic range searching problem, we are to preprocess $n$ points in $\mathbb{R}^d$ s.t. for any query range from a family of constant complexity semialgebraic sets, all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem can be solved using $S(n)$ space and with $Q(n)$ query time with $S(n)Q^d(n) = \tilde{O}(n^d)$ and this trade-off is almost tight. Consequently, there exists low space structures that use $\tilde{O}(n)$ space with $O(n^{1-1/d})$ query time and fast query structures that use $O(n^d)$ space with $O(\log^{d} n)$ query time. However, for the general semialgebraic ranges, only low space solutions are known, but the best solutions match the same trade-off curve as the simplex queries. It has been conjectured that the same could be done for the fast query case but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semilagebraic range searching and related problems. We show that any data structure for reporting the points between two concentric circles with $Q(n)$ query time must use $S(n)=\Omega(n^{3-o(1)}/Q(n)^5)$ space, meaning, for $Q(n)=O(\log^{O(1)}n)$, $\Omega(n^{3-o(1)})$ space must be used. We also study the problem of reporting the points between two polynomials of form $Y=\sum_{i=0}^\Delta a_i X^i$ where $a_0, \cdots, a_\Delta$ are given at the query time. We show $S(n)=\Omega(n^{\Delta+1-o(1)}/Q(n)^{\Delta^2+\Delta})$. So for $Q(n)=O(\log^{O(1)}n)$, we must use $\Omega(n^{\Delta+1-o(1)})$ space. For the dual semialgebraic stabbing problems, we show that in linear space, any data structure that solves 2D ring stabbing must use $\Omega(n^{2/3})$ query time. This almost matches the linearization upper bound. For general semialgebraic slab stabbing problems, again, we show an almost tight lower bounds.Comment: Submitted to SoCG'21; this version: readjust the table and other minor change

Advanced information processing system: Inter-computer communication services

The purpose is to document the functional requirements and detailed specifications for the Inter-Computer Communications Services (ICCS) of the Advanced Information Processing System (AIPS). An introductory section is provided to outline the overall architecture and functional requirements of the AIPS and to present an overview of the ICCS. An overview of the AIPS architecture as well as a brief description of the AIPS software is given. The guarantees of the ICCS are provided, and the ICCS is described as a seven-layered International Standards Organization (ISO) Model. The ICCS functional requirements, functional design, and detailed specifications as well as each layer of the ICCS are also described. A summary of results and suggestions for future work are presented