Near-optimal Distributed Triangle Enumeration via Expander Decompositions

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