TY - JOUR
AU - de Berg, Mark
PY - 2023/05/11
Y2 - 2024/05/20
TI - A Note on Reachability and Distance Oracles for Transmission Graphs
JF - Computing in Geometry and Topology
JA - CompGeomTop
VL - 2
IS - 1
SE - Original Research Articles
DO - 10.57717/cgt.v2i1.25
UR - https://cgt-journal.org/index.php/cgt/article/view/25
SP - 4:1-4:15
AB - <p>Let P be a set of n points in the plane, where each point p in P has a transmission radius r(p) > 0. The transmission graph defined by P and the given radii, denoted by G_{tr}(P), is the directed graph whose nodes are the points in P and that contains the arcs (p,q) such that |pq|<strong>≤</strong> r(p).</p><p>An and Oh [Algorithmica 2022] presented a reachability oracle for transmission graphs. Their oracle uses O(n^<sup>5/3</sup>) storage and, given two query points s,t in P, can decide in O(n^<sup>2/3</sup>) time if there is a path from s to t in G_{tr}(P). We show that the clique-based separators introduced by De Berg et al. [SICOMP 2020] can be used to improve the storage of the oracle to O(n\sqrt{n}) and the query time to O(\sqrt{n}). Our oracle can be extended to approximate distance queries: we can construct, for a given parameter eps > 0, an oracle that uses O((n/\eps)\sqrt{n}\log n) storage and that can report in O((\sqrt{n}/\eps)\log n) time a value d*_{hop}(s,t) satisfying d_{hop}(s,t) \leq d*_{hop}(s,t) < (1+eps) d_{hop}(s,t) + 1, where d_{hop}(s,t) is the hop-distance from s to t. We also show how to extend the oracle to so-called continuous queries, where the target point t can be any point in the plane.</p><p>To obtain an efficient preprocessing algorithm, we show that a clique-based separator of a set F of convex fat objects in R<sup>d</sup> can be constructed in O(n log n) time.</p>
ER -