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Efficient algorithms for constructing (1+∊,β)-spanners in the distributed and streaming models

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Abstract

For an unweighted undirected graph G = (V,E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G′ = (V,H), H ⊂eqE, is called an (α,β)-spanner of G if for every pair of vertices u,v ∊ V, dist G′(u,v) ≤ α ⋅ dist G (u,v) + β.

It was shown in [21] that for any ∊ > 0, κ = 1,2,…, there exists an integer β = β(∊,κ) such that for every n-vertex graph G there exists a (1+∊,β)-spanner G′ with O(n 1+1/κ) edges. An efficient distributed protocol for constructing (1+∊,β)-spanners was devised in [19]. The running time and the communication complexity of that protocol are O(n 1+ρ) and O(|E|n^ρ), respectively, where ρ is an additional control parameter of the protocol that affects only the additive term β.

In this paper we devise a protocol with a drastically improved running time (O(n^ρ) as opposed to O(n 1+ρ)) for constructing (1+∊,β)-spanners. Our protocol has the same communication complexity as the protocol of [19], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [19]. The protocol can be easily extended to a parallel implementation which runs in O(log n + (|E|⋅ n ρlog n)/p) time using p processors in the EREW PRAM model. In particular, when the number of processors, p, is at least |E|⋅ n ρ, the running time of the algorithm is O(log n). We also show that our protocol for constructing (1+∊,β)-spanners can be adapted to the streaming model, and devise a streaming algorithm that uses a constant number of passes and O(n 1+1/κ⋅ {log} n) bits of space for computing all-pairs-almost-shortest-paths of length at most by a multiplicative factor (1+∊) and an additive term of β greater than the shortest paths. Our algorithm processes each edge in time O(n^ρ), for an arbitrarily small ρ > 0. The only previously known algorithm for the problem [23] constructs paths of length κ times greater than the shortest paths, has the same space requirements as our algorithm, but requires O(n 1+1/κ) time for processing each edge of the input graph. However, the algorithm of [23] uses just one pass over the input, as opposed to the constant number of passes in our algorithm.

We also show that any streaming algorithm for o(n)-approximate distance computation requires Ω(n) bits of space.

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Authors and Affiliations

  1. Department of Computer Science, Yale University, P.O. Box 208285, New Haven, CT, 06520-8285, USA

    Michael Elkin & Jian Zhang

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  1. Michael Elkin
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  2. Jian Zhang
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Correspondence to Michael Elkin.

Additional information

This work was Supported by the DoD University Research Initiative (URI) administered by the Office of Naval Research under Grant N00014-01-1-0795.

Michael Elkin was supported by ONR grant N00014-01-1-0795.

Jian Zhang was supported by ONR grant N00014-01-1-0795 and NSF grants CCR-0105337 and ITR-0331548.

Preliminary version of this paper was published in PODC’04, see [22].

After the preliminary version of our paper [22] appeared on PODC’04, Feigenbaum et al. [24] came up with a new streaming algorithm for the problem that is far more efficient than [23] in terms of time-per-edge processing. However, our algorithm is still the only existing streaming algorithm that provides an almost additive approximation of distances.

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Elkin, M., Zhang, J. Efficient algorithms for constructing (1+∊,β)-spanners in the distributed and streaming models. Distrib. Comput. 18, 375–385 (2006). https://doi.org/10.1007/s00446-005-0147-2

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