TY - JOUR

T1 - δ-Greedy t-spanner

AU - Abu-Affash, A. Karim

AU - Bar-On, Gali

AU - Carmi, Paz

N1 - Funding Information:
This work was partially supported by Grant 2016116 from the United States ? Israel Binational Science Foundation.
Funding Information:
This work was partially supported by Grant 2016116 from the United States – Israel Binational Science Foundation .
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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+ε)-spanner, for every ε>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(n2logn) 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(nlogn). Our analysis indicates that the δ-Greedy spanner gives the best results among the known spanners of expected O(nlogn) time for random point sets. Moreover, analysis implies that by setting δ=t, the δ-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected O(nlogn) time.

AB - 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+ε)-spanner, for every ε>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(n2logn) 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(nlogn). Our analysis indicates that the δ-Greedy spanner gives the best results among the known spanners of expected O(nlogn) time for random point sets. Moreover, analysis implies that by setting δ=t, the δ-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected O(nlogn) time.

KW - Computational geometry

KW - Geometric spanners

KW - Path-greedy

UR - http://www.scopus.com/inward/record.url?scp=85112308283&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2021.101807

DO - 10.1016/j.comgeo.2021.101807

M3 - Article

AN - SCOPUS:85112308283

SN - 0925-7721

VL - 100

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

M1 - 101807

ER -