Near-optimal Distributed Triangle Enumeration via Expander Decompositions

We present improved distributed algorithms for variants of the triangle finding problem in model. show that detection, counting, and enumeration can be solved $\tilde{O}(n^{1/3})$?> rounds using expander decompositions . This matches lower bound $\tilde{\Omega }(n^{1/3})$?> by Izumi Le Gall [PODC’17] Pandurangan, Robinson, Scquizzato [SPAA’18], which holds even {CONGESTED}\text{-}\mathsf {CLIQUE}$?> The previous upper bounds detection were $\tilde{O}(n^{2/3})$?> $\tilde{O}(n^{3/4})$?> , respectively, due to [PODC’17]. An $(\epsilon ,\phi)$?> -expander decomposition a graph $G=(V,E)$?> is clustering vertices $V=V_{1}\cup \cdots \cup V_{x}$?> such (i) each cluster $V_{i}$?> induces subgraph with conductance at least $\phi$?> (ii) number inter-cluster edges most $\epsilon |E|$?> an $\phi =(\epsilon /\log n)^{2^{O(k)}}$?> constructed $O(n^{2/k}\cdot {\operatorname{poly}}(1/\phi ,\log n))$?> any \in (0,1)$?> positive integer $k$?> For example, $(1/n^{o(1)},1/n^{o(1)})$?> only requires $n^{o(1)}$?> compute, optimal up subpolynomial factors, $\left(0.1, 1/{\operatorname{poly}}\log n\right)$?> computed $O\left(n^{\gamma }\right)$?> rounds, arbitrarily small constant $\gamma \gt 0$?> Our are based on following generic framework decompositions, independent interest. first construct decomposition. cluster, we simulate overhead applying routing algorithm Ghaffari, Kuhn, Su Finally, deal recursive calls.