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Linear-Size hopsets with small hopbound, and constant-hopbound hopsets in RNC

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Abstract

Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with \(\Omega (n \log n)\) edges, or with a hopbound \(n^{\Omega (1)}\). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on \(\epsilon \)) \((\log \log n)^{\log \log n + O(1)}\). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [19] for computing hopsets with a constant (i.e., independent of n) hopbound requires \(n^{\Omega (1)}\) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [19]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.

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Notes

  1. We may use \(G\cup H\) instead of \(G\cup G_H\) for notational convenience.

  2. The class RNC (Randomized Nick’s Class) stands for the class of problems that can be solved on a randomized parallel computer with polynomially many processors in polylogarithmic time.

  3. This is not an error. Indeed, the lower bound is meaningless for \(\epsilon >1/\log \kappa \).

  4. The aspect ratio \(\Lambda \) of a graph G is given by \(\Lambda = {{\max _{u,v \in V} d_G(u,v)} \over {\min _{u,v\in V, u\ne v} d_G(u,v)}}\).

  5. To the best of our understanding, the algorithm of [5] scales linearly with s for the s-SSP problem, and so, for large s, the state-of-the-art bounds are based on hopsets.

  6. A graph \(G' = (V,E',\omega ')\) is called a sublinear-error emulator of an unweighted graph \(G = (V,E)\), if for every pair of vertices \(u,v \in V\), we have \(d_{G}(u,v) \le d_{G'}(u,v) \le d_G(u,v) + \alpha (d_G(u,v))\) for some sub-linear stretch function \(\alpha \). If \(G'\) is a subgraph of G, it is called a sublinear-error spanner of G.

  7. Our construction is simpler than that of [4] in two aspects: First, [4] use truncated TZ-bunches, while we use the standard TZ bunches. Second, [4] applies the construction for all possible distances scales, while we apply it just once. The distributed implementations of [26, 27, 38] are even more complicated, since the hop-reduction technique requires a recursive application of the construction.

  8. Our definition is slightly different than that of [43], which used \(p_i=|A_i|/n^{1+\nu }\), but it gives rise to the same expected size of \(A_i\). We use our version since it allows efficient implementation in various models of computation.

  9. We stress that we define bunches in the same way as [43]. As mentioned above, the only difference is the probabilities for the sets \(A_i\).

  10. Typically, in the CONGEST model only messages of size \(O(\log n)\) bits are allowed, but edge weights are restricted to be at most polynomial in n. Our definition is geared to capture a more general situation, when there is no restriction on the aspect ratio. Hence results achieved in our more general model are more general than previous ones.

  11. The cluster C(v) is defined as follows: each point \(u\in C(v)\) iff \(v\in B(u)\).

  12. In the CONGEST model one needs to multiply the bound by an \(O(\log _n\log \Lambda )\) factor.

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  1. Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Michael Elkin & Ofer Neiman

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A preliminary version of this paper was published in SPAA’19 [20]. M. Elkin: This research was supported by the ISF grant No. (724/15). O. Neiman: Supported in part by ISF grant No. (1817/17) and by BSF grant No. 2015813.

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Elkin, M., Neiman, O. Linear-Size hopsets with small hopbound, and constant-hopbound hopsets in RNC. Distrib. Comput. 35, 419–437 (2022). https://doi.org/10.1007/s00446-022-00431-z

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