arXiv subjects

Liam Roditty

Publications and source records attributed to Liam Roditty.

40 records · Page 3Linked to original sources

arXiv2010

Fast, precise and dynamic distance queries

We present an approximate distance oracle for a point set S with n points and doubling dimension λ. For every ε>0, the oracle supports (1+ε)-approximate distance queries in (universal) constant time, occupies space [ε^{-O(λ)} + 2^{O(λ log λ)}]n, and can be constructed in [2^{O(λ)} log3 n + ε^{-O(λ)} + 2^{O(λ log λ)}]n expected time. This improves upon the best previously known constructions, presented by Har-Peled and Mendel. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2^{O(λ)} log n + ε^{-O(λ)} + 2^{O(λ log λ)} update time. This is the first fully dynamic (1+ε)-distance oracle.

cs.DS↗

arXiv2010

Relaxed spanners for directed disk graphs

Let $(V,δ)$ be a finite metric space, where $V$ is a set of $n$ points and $δ$ is a distance function defined for these points. Assume that $(V,δ)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $δ(p,q)\leq r(p)$. In \cite{PeRo08} we presented an algorithm for constructing a $(1+\eps)$-spanner of size $O(n/\eps^d \log M)$, where $M$ is the maximal radius $r(p)$. The current paper presents two results. The first shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of $M$. The second result shows that by slightly relaxing the requirements and allowing a small perturbation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph $I(V,E,r_{1+\eps})$, where $r_{1+\eps}(p) = (1+\eps)\cdot r(p)$ for every $p\in V$, then it is possible to get a $(1+\eps)$-spanner of size $O(n/\eps^d)$ for $I(V,E,r)$. Our algorithm is simple and can be implemented efficiently.

cs.DS↗

arXiv2008

SINR Diagrams: Towards Algorithmically Usable SINR Models of Wireless Networks

The rules governing the availability and quality of connections in a wireless network are described by physical models such as the signal-to-interference & noise ratio (SINR) model. For a collection of simultaneously transmitting stations in the plane, it is possible to identify a reception zone for each station, consisting of the points where its transmission is received correctly. The resulting SINR diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard. SINR diagrams appear to be fundamental to understanding the behavior of wireless networks, and may play a key role in the development of suitable algorithms for such networks, analogous perhaps to the role played by Voronoi diagrams in the study of proximity queries and related issues in computational geometry. So far, however, the properties of SINR diagrams have not been studied systematically, and most algorithmic studies in wireless networking rely on simplified graph-based models such as the unit disk graph (UDG) model, which conveniently abstract away interference-related complications, and make it easier to handle algorithmic issues, but consequently fail to capture accurately some important aspects of wireless networks. The current paper focuses on obtaining some basic understanding of SINR diagrams, their properties and their usability in algorithmic applications. Specifically, based on some algebraic properties of the polynomials defining the reception zones we show that assuming uniform power transmissions, the reception zones are convex and relatively well-rounded. These results are then used to develop an efficient approximation algorithm for a fundamental point location problem in wireless networks.

cs.NI↗

arXiv2008

Dynamic Connectivity: Connecting to Networks and Geometry

Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems. Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by "on" vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in O (m^{2/3}) amortized time, where m denotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC'02], which required fast matrix multiplication and had an update time of O(m^0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We provide similarly improved update times, O (n^{2/3}), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries, such as arbitrary 2D line segments, d-dimensional simplices, and d-dimensional balls.

cs.DS↗