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We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $tilde{O}(n^{1/2})$ rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were $tilde{O}(n^{2/3})$ and $tilde{O}(n^{3/4})$, respectively, due to Izumi and LeGall (PODC 2017). The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in $tilde{O}(n^{1-delta})$ rounds we can partition the edge set of the network $G=(V,E)$ into three parts $E=E_mcup E_scup E_r$ such that (a) Each connected component induced by $E_m$ has minimum degree $Omega(n^delta)$ and conductance $Omega(1/text{poly} log(n))$. As a consequence the mixing time of a random walk within the component is $O(text{poly} log(n))$. (b) The subgraph induced by $E_s$ has arboricity at most $n^{delta}$. (c) $|E_r| leq |E|/6$. All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set $E_s$ by an algorithm that is efficient for low-arboricity graphs, and deal with the set $E_r$ using recursive calls. For each connected component induced by $E_m$, we are able to simulate congested clique algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.
An $(epsilon,phi)$-expander decomposition of a graph $G=(V,E)$ is a clustering of the vertices $V=V_{1}cupcdotscup V_{x}$ such that (1) each cluster $V_{i}$ induces subgraph with conductance at least $phi$, and (2) the number of inter-cluster edges i
We design an algorithm for computing connectivity in hypergraphs which runs in time $hat O_r(p + min{lambda^{frac{r-3}{r-1}} n^2, n^r/lambda^{r/(r-1)}})$ (the $hat O_r(cdot)$ hides the terms subpolynomial in the main parameter and terms that depend o
In this paper, we propose a new construction of constantdegree expanders motivated by their application in P2P overlay networks and in particular in the design of robust trees overlay. Our key result can be stated as follows. Consider a complete bina
We present a simple deterministic distributed algorithm that computes a $(Delta+1)$-vertex coloring in $O(log^2 Delta cdot log n)$ rounds. The algorithm can be implemented with $O(log n)$-bit messages. The algorithm can also be extended to the more g
In this note, we study the expander decomposition problem in a more general setting where the input graph has positively weighted edges and nonnegative demands on its vertices. We show how to extend the techniques of Chuzhoy et al. (FOCS 2020) to thi