For $n\in\mathbb N$, let $\mu_n$ be the uniform measure on $2^{[n]}$. A family $\mathcal F \subseteq2^{[n]}$ is called $k$-AP-intersecting if $F_1\cap F_2$ contains a nonconstant $k$-term arithmetic progression for every $F_1, F_2\in\mathcal F$. Writing $m_k$ for the supremum of $\mu_n(\mathcal F)$ over all $n$ and all...
A. Kupavskii, F. Noskov, Georgy Sokolov et al.· 0 citations
We study a random graph process that connects the Erd\H{o}s--R\'enyi random graph with the semi-random process. In each round, a uniformly random vertex is revealed together with a uniformly random candidate set of $s$ possible partners, and Builder chooses one of them. We study how this limited adaptivity $s$ changes...
Allreduce is a fundamental communication primitive in large-scale distributed training, and torus interconnects remain widely deployed in high-performance systems. We introduce SP-allreduce, a new allreduce algorithm for $d$-dimensional tori that exploits the symmetry of the topology and routes only along shortest path...
M. Cherniavskii, A. Kupavskii, Г. Г. Миронов et al.· 0 citations
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