Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Abstract

Miller et al. [43] devised a distributed 1 algorithm in the CONGEST model, that given a parameter k = 1, 2, . . ., constructs an O(k)-spanner of an input unweighted nvertex graph with O(n 1+1/k ) 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 O(n 1+1/k / ) edges, with probability 1 − , for any > 0. Moreover, when k = ω(log n), 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 (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 with O(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. * Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: elkinm@cs.bgu.ac.il. 1 They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in [43].

Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

“…superclustering-and-interconnection approach (see [EP01,EN16,EN17] for an elaborate discussion of this technique). Its randomized distributed implementation by Elkin and Neiman [EN17] relies on random sampling of clusters. Those sampled clusters join nearby unsampled clusters to create superclusters, and this is iteratively repeated.…”

“…(1), and |H| = O(βn 1+1/κ ), in deterministic time O ,κ,ρ (n ρ ) in the CONGEST model. The overall structure of our algorithm is reminiscent of that in [EN17]. However, unlike the algorithm of [EN17], the current algorithm does not use any randomization.…”

“…Also, spanners are a fundamental combinatorial construct, and their existential properties are being extensively explored [ADD + 93, PS89, EP01, BKMP10, TZ06, Pet09, AB15, ENS15, ABP17]. from a superlinear in n running time, O(n 1+ 1 2κ ), and from additive term β E = κ log κ · (ρ −1 ) ρ −1 , which is significantly larger than β EN (the additive term of [EN17]). Also, the number of edges in the resulting spanner is by a polylog(n) factor larger than the desired bound of O ,κ,ρ (n 1+1/κ ) (see [EP01,Pet10,EN17,ABP17]).…”

New (α, β) Spanners and Hopsets

“…(11). As in [EN19a], we introduce an efficiency parameter ρ ∈ (0, 1] that determines the trade-off between the β value, the number of edges in the spanner, and the construction time. For a given parameter ρ, define: i 0 = log (k · ρ) , i 1 = 2/ρ − 1 , and T := i 0 + i 1 .…”

“…By computing efficiently the (3 + , β) spanners, we also get the following fast computation of the S × V distances. The next is Corollary 19 of [EN19a] while enjoying a better tradeoff in the expense of increasing the multiplicative stretch from (1 + ) to (3 + ): Corollary 1. There exists an algorithm that computes for any graph G = (V, E), integer k ≥ 1, any parameters > 0, ρ ∈ (0, 1) and any vertex set S ⊆ V, a (3 + , β) approximate shortest paths for S × V, for β = O((5 + 16/ ) log ρ+2/ρ · k log(5+16/ ) ) in O((log (k · ρ) + 1/ρ) · |E| · n ρ + |S| · (n 1+1/k + (5 + 16/ ) log ρ+2/ρ · k log(5+16/ ) · n) time.…”