Deterministic Distributed Expander Decomposition and Routing with Applications in Distributed Derandomization

There is a recent exciting line of work in distributed graph algorithms in the CONGEST model that exploit expanders. All these algorithms so far are based on two tools: expander decomposition and expander routing. An ( $\epsilon, \phi$ )-expander decomposition removes $\epsilon$ -fraction of the edges so that the remaining connected components have conductance at least $\phi$ , i.e., they are $\phi$ -expanders, and expander routing allows each vertex $v$ in a $\phi$ -expander to very quickly exchange $\deg(v)$ messages with any other vertices, not just its local neighbors. In this paper, we give the first efficient deterministic distributed algorithms for both tools. We show that an ( $\epsilon, \phi$ ) -expander decomposition can be deterministically computed in $\text{poly} (\epsilon^{-1})n^{o(1)}$ rounds for $\phi= \text{poly} (\epsilon)n^{-o(1)}$ , and that expander routing can be performed deterministically in $\text{poly} (\phi^{-1})n^{o(1)}$ rounds. Both results match previous bounds of randomized algorithms by [Chang and Saranurak, PODC 2019] and [Ghaffari, Kuhn, and Su, PODC 2017] up to subpolynomial factors. Consequently, we derandomize existing distributed algorithms that exploit expanders. We show that a minimum spanning tree on $n^{-o(1)}$ -expanders can be constructed deterministically in $n^{o(1)}$ rounds, and triangle detection and enumeration on general graphs can be solved deterministically in $O(n^{0.58})$ and $n^{2/3+o(1)}$ rounds, respectively. Using similar techniques, we also give the first polylogarithmic-round randomized algorithm for constructing an ( $\epsilon, \phi$ ) -expander decomposition in $\text{poly} (\epsilon^{-1}, \log n)$ rounds for $\phi=1/\text{poly}(\epsilon^{-1}, \log n)$ . This algorithm is faster than the previous algorithm by [Chang and Saranurak, PODC 2019] in all regimes of parameters. The previous algorithm needs $n^{\Omega(1)}$ rounds for any $\phi\geq 1/\text{poly}\log n$ .