TY - JOUR

T1 - Reachability Oracles for Directed Transmission Graphs

AU - Kaplan, Haim

AU - Mulzer, Wolfgang

AU - Roditty, Liam

AU - Seiferth, Paul

N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - Let P⊂ Rd be a set of n points in d dimensions such that each point p∈ P has an associated radiusrp> 0. The transmission graphG for P is the directed graph with vertex set P such that there is an edge from p to q if and only if | pq| ≤ 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 quality of the oracle is measured by the space requirement S(n), the query time Q(n), and the preprocessing time. For transmission graphs of one-dimensional point sets, we can construct in O(nlog n) time an oracle with Q(n) = O(1) and S(n) = O(n). For planar point sets, the ratio Ψ between the largest and the smallest associated radius turns out to be an important parameter. We present three data structures whose quality depends on Ψ : the first works only for Ψ<3 and achieves Q(n) = O(1) with S(n) = O(n) and preprocessing time O(nlog n) ; the second data structure gives Q(n)=O(Ψ3n) and S(n) = O(Ψ 3n3 / 2) ; the third data structure is randomized with Q(n) = O(n2 / 3log 1 / 3Ψ log 2 / 3n) and S(n) = O(n5 / 3log 1 / 3Ψ log 2 / 3n) and answers queries correctly with high probability.

AB - Let P⊂ Rd be a set of n points in d dimensions such that each point p∈ P has an associated radiusrp> 0. The transmission graphG for P is the directed graph with vertex set P such that there is an edge from p to q if and only if | pq| ≤ 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 quality of the oracle is measured by the space requirement S(n), the query time Q(n), and the preprocessing time. For transmission graphs of one-dimensional point sets, we can construct in O(nlog n) time an oracle with Q(n) = O(1) and S(n) = O(n). For planar point sets, the ratio Ψ between the largest and the smallest associated radius turns out to be an important parameter. We present three data structures whose quality depends on Ψ : the first works only for Ψ<3 and achieves Q(n) = O(1) with S(n) = O(n) and preprocessing time O(nlog n) ; the second data structure gives Q(n)=O(Ψ3n) and S(n) = O(Ψ 3n3 / 2) ; the third data structure is randomized with Q(n) = O(n2 / 3log 1 / 3Ψ log 2 / 3n) and S(n) = O(n5 / 3log 1 / 3Ψ log 2 / 3n) and answers queries correctly with high probability.

UR - http://www.scopus.com/inward/record.url?scp=85074684705&partnerID=8YFLogxK

U2 - 10.1007/s00453-019-00641-1

DO - 10.1007/s00453-019-00641-1

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AN - SCOPUS:85074684705

SN - 0178-4617

VL - 82

SP - 1259

EP - 1276

JO - Algorithmica

JF - Algorithmica

IS - 5

ER -