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تسجيل مستخدم جديدSparse Hopsets in Congested Clique
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تأليف
Yasamin Nazari
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We give the first Congested Clique algorithm that computes a sparse hopset with polylogarithmic hopbound in polylogarithmic time. Given a graph $G=(V,E)$, a $(beta,epsilon)$-hopset $H$ with hopbound $beta$, is a set of edges added to $G$ such that for any pair of nodes $u$ and $v$ in $G$ there is a path with at most $beta$ hops in $G cup H$ with length within $(1+epsilon)$ of the shortest path between $u$ and $v$ in $G$. Our hopsets are significantly sparser than the recent construction of Censor-Hillel et al. [6], that constructs a hopset of size $tilde{O}(n^{3/2})$, but with a smaller polylogarithmic hopbound. On the other hand, the previously known constructions of sparse hopsets with polylogarithmic hopbound in the Congested Clique model, proposed by Elkin and Neiman [10],[11],[12], all require polynomial rounds. One tool that we use is an efficient algorithm that constructs an $ell$-limited neighborhood cover, that may be of independent interest. Finally, as a side result, we also give a hopset construction in a variant of the low-memory Massively Parallel Computation model, with improved running time over existing algorithms.
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