TY - JOUR

T1 - Efficient algorithms for constructing very sparse spanners and emulators

AU - Elkin, Michael

AU - Neiman, Ofer

N1 - Funding Information:
Michael Elkin this research was supported by the ISF grant No. (724/15). Ofer Neiman supported in part by ISF grant No. (1718/18) and by BSF grant No. 2015813.
Funding Information:
A preliminary version [38] of this article appeared in SODA’17. Michael Elkin this research was supported by the ISF grant No. (724/15). Ofer Neiman supported in part by ISF grant No. (1718/18) and by BSF grant No. 2015813. Authors’ addresses: M. Elkin and O. Neiman, Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel; emails: {elkinm, neimano}@cs.bgu.ac.il. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2018 Association for Computing Machinery. 1549-6325/2018/11-ART4 $15.00 https://doi.org/10.1145/3274651

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Miller et al. [48] devised a distributed1 algorithm in the CONGEST model that, given a parameter k = 1, 2, . . . , constructs an O(k)-spanner of an input unweighted n-vertex graph with O(n1+1/k ) expected edges in O(k) rounds of communication. In this article, we improve the result of Reference [48] by showing a k-round distributed algorithm in the same model that constructs a (2k - 1)-spanner with O(n1+1/k /∈ ) edges, with probability 1 - ∈ for any ∈ > 0. Moreover, when k = ω(logn), our algorithm produces (still in k rounds) ultrasparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 - o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that are sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm. We also devise improved bounds (and algorithms realizing these bounds) for (1 + ∈, β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any ∈ > 0, there exists a (1 + ∈, ( log log n/∈)log log n )-emulator withO(n) edges. All previous constructions of (1 + ∈, β)-spanners and emulators employ a superlinear number of edges for all choices of parameters. Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs.

AB - Miller et al. [48] devised a distributed1 algorithm in the CONGEST model that, given a parameter k = 1, 2, . . . , constructs an O(k)-spanner of an input unweighted n-vertex graph with O(n1+1/k ) expected edges in O(k) rounds of communication. In this article, we improve the result of Reference [48] by showing a k-round distributed algorithm in the same model that constructs a (2k - 1)-spanner with O(n1+1/k /∈ ) edges, with probability 1 - ∈ for any ∈ > 0. Moreover, when k = ω(logn), our algorithm produces (still in k rounds) ultrasparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 - o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that are sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm. We also devise improved bounds (and algorithms realizing these bounds) for (1 + ∈, β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any ∈ > 0, there exists a (1 + ∈, ( log log n/∈)log log n )-emulator withO(n) edges. All previous constructions of (1 + ∈, β)-spanners and emulators employ a superlinear number of edges for all choices of parameters. Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs.

KW - CONGEST

KW - Shortest path

KW - Spanners

UR - http://www.scopus.com/inward/record.url?scp=85058350417&partnerID=8YFLogxK

U2 - 10.1145/3274651

DO - 10.1145/3274651

M3 - Article

AN - SCOPUS:85058350417

VL - 15

SP - 4:1-4:29

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 1

ER -