Gali Bar-On | Searcharxiv

arXiv2019

Resolving SINR Queries in a Dynamic Setting

We consider a set of transmitters broadcasting simultaneously on the same frequency under the SINR model. Transmission power may vary from one transmitter to another, and a transmitter's signal strength at a given point is modeled by the transmitter's power divided by some constant power $α$ of the distance it traveled. Roughly, a receiver at a given location can hear a specific transmitter only if the transmitter's signal is stronger by a specified ratio than the signals of all other transmitters combined. An SINR query is to determine whether a receiver at a given location can hear any transmitter, and if yes, which one. An approximate answer to an SINR query is such that one gets a definite YES or definite NO, when the ratio between the strongest signal and all other signals combined is well above or well below the reception threshold, while the answer in the intermediate range is allowed to be either YES or NO. We describe compact data structures that support approximate SINR queries in the plane in a dynamic context, i.e., where transmitters may be inserted and deleted over time. We distinguish between two main variants --- uniform power and non-uniform power. In both variants the preprocessing time is $O(n \mathop{\textrm{polylog}} n)$ and the amortized update time is $O(\mathop{\textrm{polylog}} n)$, while the query time is $O(\mathop{\textrm{polylog}} n)$ for uniform power, and randomized time $O(\sqrt{n} \mathop{\textrm{polylog}} n)$ with high probability for non-uniform power. Finally, we observe that in the static context the latter data structure can be implemented differently, so that the query time is also $O(\mathop{\textrm{polylog}} n)$, thus significantly improving all previous results for this problem.

cs.CG↗

arXiv2017

$δ$-Greedy $t$-spanner

We introduce a new geometric spanner, $δ$-Greedy, whose construction is based on a generalization of the known Path-Greedy and Gap-Greedy spanners. The $δ$-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice. More specifically, it has the same theoretical and practical properties as the Path-Greedy spanner: a natural definition, small degree, linear number of edges, low weight, and strong $(1+\varepsilon)$-spanner for every $\varepsilon>0$. The $δ$-Greedy algorithm is an improvement over the Path-Greedy algorithm with respect to the number of shortest path queries and hence with respect to its construction time. We show how to construct such a spanner for a set of $n$ points in the plane in $O(n^2 \log n)$ time. The $δ$-Greedy spanner has an additional parameter, $δ$, which indicates how close it is to the Path-Greedy spanner on the account of the number of shortest path queries. For $δ= t$ the output spanner is identical to the Path-Greedy spanner, while the number of shortest path queries is, in practice, linear. Finally, we show that for a set of $n$ points placed independently at random in a unit square the expected construction time of the $δ$-Greedy algorithm is $O(n \log n)$. Our analysis indicates that the $δ$-Greedy spanner gives the best results among the known spanners of expected $O(n \log n)$ time for random point sets. Moreover, the analysis implies that by setting $δ= t$, the $δ$-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected $O(n \log n)$ time.

cs.CG↗