Simple Parallel and Distributed Algorithms for Spectral Graph Sparsification

Abstract

We describe a simple algorithm for spectral graph sparsification, based on iterative computations of weighted spanners and uniform sampling. Leveraging the algorithms of Baswana and Sen for computing spanners, we obtain the first distributed spectral sparsification algorithm. We also obtain a parallel algorithm with improved work and time guarantees. Combining this algorithm with the parallel framework of Peng and Spielman for solving symmetric diagonally dominant linear systems, we get a parallel solver which is much closer to being practical and significantly more efficient in terms of the total work. IntroductionThe efficient transformation of dense instances of graph problems to nearly equivalent sparse instances is a powerful tool in algorithm design. Spectral sparsifiers are sparse graphs that preserve within an 1 + ǫ factor the quadratic form x T L G x, where L G is the Laplacian of G and ǫ is a parameter of choice. They were introduced by Spielman and Teng [24] as a basic component of the first nearly-linear time solvers for linear systems on symmetric diagonally dominant (SDD) matrices 1 . Such linear system solvers are a key algorithmic primitive with numerous applications [17,25].The Spielman and Teng sparsification algorithm produces sparsifiers with O(n log c n/ǫ 2 ) edges for some fairly large constant c, where n is the number of vertices in the graph. At a high level their algorithm is based on graph decompositions into edge-disjoint sets that get sparsified independently via uniform sampling. As noted in [22] the algorithm can be parallelized if the original partitioning subroutine is substituted by a more recent one due to Orecchia and Vishnoi [20]. Peng and Spielman [22] recently presented a novel algebraic framework for solving SDD systems. It enables the use of parallel sparsification algorithms for constructing parallel solvers. Combined with the parallelized Spielman and Teng sparsification algorithm, or a more recent approach due to Peng (Section 3.4,[21]), this algebraic framework yields the first 'truly' parallel SDD solver that does near-linear work and runs in polylogarithmic time.The new parallel solver leaves something to be desired: its work is by several logarithmic factors larger than that of the fastest known sequential algorithm that runs inÕ(m log n) time 2 ; here m is the number * This work is supported by NSF CAREER award CCF-1149048.1 A symmetric matrix A is SDD if for all i, Aii ≥ j =i |Aij|. 2 We useÕ() to hide a poly(log log n) factor.

Deterministic Distributed Algorithms and Lower Bounds in the Hybrid Model

“…To this end, we will employ the sparsification algorithm developed by Koutis [40]. We should remark that the algorithm of Koutis actually returns a spectral sparsifier, which is a strictly stronger notion than a cut sparsifier [7], but we will not use this property here.…”

“…Proof. First, we apply the sparsification algorithm of Koutis [40], employing only the local network for O(1/ϵ 2 ) rounds in order to identify a subgraph H. Theorem 13 implies that with high probability (1…”

Constructing light spanners deterministically in near-linear time

“…This special case is of particular interest in the literature (see e.g. [4,22]). Furthermore, all of our constructions are deterministic, giving the first subquadratic deterministic construction without the additional dependence on k in the size of the spanner.…”

Scaling industrial applications for the Big Data era

“…Spielman and Teng [17], and Spielman and Srivastava [20] inspired further research in finding a distributed approach. Koutis and Xu [21] introduce a theoretical algorithm for spectral sparsification. Their work focuses on the use of weighted spanners.…”

“…That is why we choose to build upon the original algorithm [17], which carries an arguably more intuitive set theory mindset. Unfortunately, Koutis and Xu [21] didn't provide any benchmark, so it is hard to compare our solution to their approach. Similarly, Sun and Zanetti [22] approach the sparsification problem from a clustering perspective avoiding spectral methods.…”