TY - GEN

T1 - Sparse Euclidean Spanners with Tiny Diameter

T2 - 38th International Symposium on Computational Geometry, SoCG 2022

AU - Le, Hung

AU - Milenkovic, Lazar

AU - Solomon, Shay

N1 - Publisher Copyright:
© Hung Le, Lazar Milenkovi, and Shay Solomon; licensed under Creative Commons License CC-BY 4.0

PY - 2022/6/1

Y1 - 2022/6/1

N2 - In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R admits a (1 + ?)spanner with hop-diameter at most 2 (respectively, 3) and O(nlog n) edges (resp., O(nlog log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(nak(n)) edges, for any k = 2. The function ak is the inverse of a certain Ackermann-style function at the ?k/2?th level of the primitive recursive hierarchy, where a0(n) = ?n/2?, a1(n) = [vn], a2(n) = ?log n?, a3(n) = ?log log n?, a4(n) = log* n, a5(n) = ?12 log* n?,.... Roughly speaking, for k = 2 the function ak is close to ?k-22 ?-iterated log-star function, i.e., log with ?k-22 ? stars. Also, a2a(n)+4(n) = 4, where a(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases k = 2 and k = 3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ? > 0, any (1 + ?)-spanner for the uniform line metric with hop-diameter at most k must have at least ?(nak(n)) edges.

AB - In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R admits a (1 + ?)spanner with hop-diameter at most 2 (respectively, 3) and O(nlog n) edges (resp., O(nlog log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(nak(n)) edges, for any k = 2. The function ak is the inverse of a certain Ackermann-style function at the ?k/2?th level of the primitive recursive hierarchy, where a0(n) = ?n/2?, a1(n) = [vn], a2(n) = ?log n?, a3(n) = ?log log n?, a4(n) = log* n, a5(n) = ?12 log* n?,.... Roughly speaking, for k = 2 the function ak is close to ?k-22 ?-iterated log-star function, i.e., log with ?k-22 ? stars. Also, a2a(n)+4(n) = 4, where a(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases k = 2 and k = 3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ? > 0, any (1 + ?)-spanner for the uniform line metric with hop-diameter at most k must have at least ?(nak(n)) edges.

KW - Euclidean spanners

KW - hop-diameter

KW - inverse-Ackermann

KW - lower bounds

KW - sparse spanners

UR - http://www.scopus.com/inward/record.url?scp=85134298531&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2022.54

DO - 10.4230/LIPIcs.SoCG.2022.54

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AN - SCOPUS:85134298531

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 38th International Symposium on Computational Geometry, SoCG 2022

A2 - Goaoc, Xavier

A2 - Kerber, Michael

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 7 June 2022 through 10 June 2022

ER -