TY - GEN

T1 - Efficient algorithms for constructing very sparse spanners and emulators

AU - Elkin, Michael

AU - Neiman, Ofer

N1 - Funding Information:
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: elkinm@cs.bgu.ac.il. This research was supported by the ISF grant 724/15. yDepartment of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: neimano@cs.bgu.ac.il. Supported in part by ISF grant No. (523/12), by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n303809, and by BSF Grant No. 2015813.
Publisher Copyright:
Copyright © by SIAM.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Miller et al. [43] 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 expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k - 1)- spanner with edges, with probability, for any . Moreover, when, our algorithm produces (still in k rounds) ultra-sparse 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 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 spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any there exists a emulator with O(n) edges. All previous constructions of 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. [43] 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 expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k - 1)- spanner with edges, with probability, for any . Moreover, when, our algorithm produces (still in k rounds) ultra-sparse 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 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 spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any there exists a emulator with O(n) edges. All previous constructions of 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.

UR - http://www.scopus.com/inward/record.url?scp=85016174746&partnerID=8YFLogxK

U2 - 10.1137/1.9781611974782.41

DO - 10.1137/1.9781611974782.41

M3 - Conference contribution

AN - SCOPUS:85016174746

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 652

EP - 669

BT - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

A2 - Klein, Philip N.

PB - Association for Computing Machinery

T2 - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

Y2 - 16 January 2017 through 19 January 2017

ER -