Reachability Oracles for Disk Transmission Graphs
نویسندگان
چکیده
Let P ⊆ R be a set of n points, each with an associated radius rp > 0. This induces a directed graph on P with an edge from p to q if and only if q lies in the ball with radius rp around p. We show that for d = 1 there is a data structure that answers reachability queries (given two vertices, is there a directed path between them?) in time O(1) using O(n) space and O(n log n) preprocessing time. With di erent techniques we can get a similar result for d = 2 as long as the radii are between 1 and √ 3.
منابع مشابه
Disk Intersection Graphs: Models, Data Structures, and Algorithms
Let P ⊂ R2 be a set of n point sites. The unit disk graph UD(P) on P has vertex set P and an edge between two sites p,q ∈ P if and only if p and q have Euclidean distance |pq| 6 1. If we interpret P as centers of disks with diameter 1, then UD(P) is the intersection graph of these disks, i.e., two sites p and q form an edge if and only if their corresponding unit disks intersect. Two natural ge...
متن کاملReachability Oracles for Directed Transmission Graphs
Let P ⊂ R be a set of n points in the d dimensions such that each point p ∈ P has an associated radius rp > 0. The transmission graph G for P is the directed graph with vertex set P such that there is an edge from p to q if and only if d(p, q) ≤ rp, for any p, q ∈ P . A reachability oracle is a data structure that decides for any two vertices p, q ∈ G whether G has a path from p to q. The quali...
متن کاملSimple, Fast, and Scalable Reachability Oracle
A reachability oracle (or hop labeling) assigns each vertex v two sets of vertices: Lout(v) and Lin(v), such that u reaches v iff Lout(u) ∩ Lin(v) 6= ∅. Despite their simplicity and elegance, reachability oracles have failed to achieve efficiency in more than ten years since their introduction: The main problem is high construction cost, which stems from a set-cover framework and the need to ma...
متن کاملFinding the Girth in Disk Graphs and a Directed Triangle in Transmission Graphs∗
Suppose we are given a set S ⊂ R2 of n point sites in the plane, each with an associated radius rs > 0, for s ∈ S. The disk graph D(S) for S is the undirected graph with vertex set S and an edge between s and t in S if and only if |st| ≤ rs + rt, i.e., if the disks with radius rs around s and with radius rt around t intersect. The transmission graph T (S) for S is the directed graph with vertex...
متن کاملConnectivity in time-graphs
Dynamic networks are characterized by topologies that varywith time and are represented by time-graphs. The notion of connectivity in time-graphs is fundamentally different from that in static graphs. End-to-end connectivity is achieved opportunistically by the store–carry-forward paradigm if the network is so sparse that source–destination pairs are usually not connected by complete paths. In ...
متن کامل