Spanners for directed transmission graphs

Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Let P ⊂ R2 be a planar n-point set 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 for any p, q ∈ P, there is an edge from p to q if and only if d(p, q) ≤ rp. Let t > 1 be a constant. A t-spanner for G is a subgraph H ⊆ G with vertex set P so that for any two vertices p, q ∈ P, we have dH(p, q) ≤ tdG(p, q), where dH and dG denote the shortest path distance in H and G, respectively (with Euclidean edge lengths). We show how to compute a t-spanner for G with O(n) edges in O(n(log n + log Ψ)) time, where Ψ is the ratio of the largest and smallest radius of a point in P. Using more advanced data structures, we obtain a construction that runs in O(n log5 n) time, independent of Ψ. We give two applications for our spanners. First, we show how to use our spanner to find a BFS tree in G from any given start vertex in O(n log n) time (in addition to the time it takes to build the spanner). Second, we show how to use our spanner to extend a reachability oracle to answer geometric reachability queries. In a geometric reachability query we ask whether a vertex p in G can “reach” a target q which is an arbitrary point in the plane (rather than restricted to be another vertex q of G in a standard reachability query). Our spanner allows the reachability oracle to answer geometric reachability queries with an additive overhead of O(log n log Ψ) to the query time and O(n log Ψ) to the space.

Original languageEnglish
Pages (from-to)1585-1609
Number of pages25
JournalSIAM Journal on Computing
Issue number4
StatePublished - 2018

Bibliographical note

Publisher Copyright:
© 2018 Haim Kaplan, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth.


  • Quadtrees
  • Reachability oracles
  • Spanners
  • Transmission graphs


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