In this paper, we bring the main tools of the Laplacian paradigm to the Broadcast Congested Clique. We introduce an algorithm to compute spectral sparsifiers in a polylogarithmic number of rounds, which directly leads to an efficient Laplacian solver. Based on this primitive, we consider the linear program solver of Lee and Sidford (FOCS 2014). We show how to solve certain linear programs up to additive error $\epsilon$ with $n$ constraints on an $n$-vertex Broadcast Congested Clique network in $\tilde O(\sqrt{n}\log(1/\epsilon))$ rounds. Using this, we show how to find an exact solution to the minimum cost flow problem in $\tilde O(\sqrt{n})$ rounds.
翻译：在本文中,我们把拉普拉西亚模式的主要工具带到广播中心中心。 我们引入了一种算法, 用来计算多声波数的光谱封闭器, 这直接导致高效拉普拉西亚解答器。 基于这个原始的, 我们考虑李和西德福德的线性程序求解器( FOCS 2014) 。 我们展示了如何解决某些线性程序, 直至添加错误 $\ epsion$, 并限制在$\ tilde O (\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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