TY - GEN

T1 - On the locality of distributed sparse spanner construction

AU - Derbel, Bilel

AU - Gavoille, Cyril

AU - Peleg, David

AU - Viennot, Laurent

PY - 2008

Y1 - 2008

N2 - The paper presents a deterministic distributed algorithm that, given k ≥ 1, constructs in k rounds a (2k-1, 0)-spanner of O(kn1+1/k) edges for every n-node unweighted graph, (If n is not available to the nodes, then our algorithm executes in 3k - 2 rounds, and still returns a (2k - 1, 0)-spanner with O(kn1+1/k) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ∈ > 0, constructs a (1 + ∈, 2)-spanner of O(∈-1n3/2) edges in O(∈-1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k - 1, 0)-spanner of o(n1+1/(k-1)) edges for k ∈ {2, 3, 5}. It is also shown that for every k > 1, any (randomized) distributed algorithm that constructs a spanner with fewer than n1+1/k+∈ edges in at most n∈ expected rounds must stretch some distances by an additive factor of n Ω(∈). In other words, while additive stretched spanners with O(n1+1/k) edges may exist, e.g., for k = 2, 3, they cannot be computed distributively in a sub-polynomial number of rounds in expectation.

AB - The paper presents a deterministic distributed algorithm that, given k ≥ 1, constructs in k rounds a (2k-1, 0)-spanner of O(kn1+1/k) edges for every n-node unweighted graph, (If n is not available to the nodes, then our algorithm executes in 3k - 2 rounds, and still returns a (2k - 1, 0)-spanner with O(kn1+1/k) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ∈ > 0, constructs a (1 + ∈, 2)-spanner of O(∈-1n3/2) edges in O(∈-1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k - 1, 0)-spanner of o(n1+1/(k-1)) edges for k ∈ {2, 3, 5}. It is also shown that for every k > 1, any (randomized) distributed algorithm that constructs a spanner with fewer than n1+1/k+∈ edges in at most n∈ expected rounds must stretch some distances by an additive factor of n Ω(∈). In other words, while additive stretched spanners with O(n1+1/k) edges may exist, e.g., for k = 2, 3, they cannot be computed distributively in a sub-polynomial number of rounds in expectation.

KW - Distributed algorithms

KW - Graph spanners

KW - Time complexity

UR - http://www.scopus.com/inward/record.url?scp=57549099437&partnerID=8YFLogxK

U2 - 10.1145/1400751.1400788

DO - 10.1145/1400751.1400788

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AN - SCOPUS:57549099437

SN - 9781595939890

T3 - Proceedings of the Annual ACM Symposium on Principles of Distributed Computing

SP - 273

EP - 282

BT - PODC'08

PB - Association for Computing Machinery (ACM)

T2 - 27th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing

Y2 - 18 August 2008 through 21 August 2008

ER -