TY - GEN

T1 - Light euclidean spanners with steiner points

AU - Le, Hung

AU - Solomon, Shay

N1 - Publisher Copyright:
© Hung Le and Shay Solomon.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - The FOCS’19 paper of Le and Solomon [59], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1 + )-spanner in Rd is Õ(−d) for any d = O(1) and any = Ω(n− d−1 1 ) (where Õ hides polylogarithmic factors of 1 ), and also shows the existence of point sets in Rd for which any (1 + )-spanner must have lightness Ω(−d).1 Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in R2 with lightness O(−1 log ∆), where ∆ is the spread of the point set.2 In the regime of ∆ 21/, this provides an improvement over the lightness bound of [59]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications often controls the precision, and it sometimes needs to be much smaller than O(1/ log n). Moreover, for spread polynomially bounded in 1/, this upper bound provides a quadratic improvement over the non-Steiner bound of [59], We then demonstrate that such a light spanner can be constructed in O(n) time for polynomially bounded spread, where O hides a factor of poly(1 ). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(−(d+1)/2 + −2 log ∆) for any 3 ≤ d = O(1) and any = Ω(n− d−1).

AB - The FOCS’19 paper of Le and Solomon [59], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1 + )-spanner in Rd is Õ(−d) for any d = O(1) and any = Ω(n− d−1 1 ) (where Õ hides polylogarithmic factors of 1 ), and also shows the existence of point sets in Rd for which any (1 + )-spanner must have lightness Ω(−d).1 Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in R2 with lightness O(−1 log ∆), where ∆ is the spread of the point set.2 In the regime of ∆ 21/, this provides an improvement over the lightness bound of [59]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications often controls the precision, and it sometimes needs to be much smaller than O(1/ log n). Moreover, for spread polynomially bounded in 1/, this upper bound provides a quadratic improvement over the non-Steiner bound of [59], We then demonstrate that such a light spanner can be constructed in O(n) time for polynomially bounded spread, where O hides a factor of poly(1 ). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(−(d+1)/2 + −2 log ∆) for any 3 ≤ d = O(1) and any = Ω(n− d−1).

KW - Euclidean spanners

KW - Light spanners

KW - Steiner spanners

UR - http://www.scopus.com/inward/record.url?scp=85092461464&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2020.67

DO - 10.4230/LIPIcs.ESA.2020.67

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AN - SCOPUS:85092461464

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 28th Annual European Symposium on Algorithms, ESA 2020

A2 - Grandoni, Fabrizio

A2 - Herman, Grzegorz

A2 - Sanders, Peter

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

Y2 - 7 September 2020 through 9 September 2020

ER -