Local Mixing Time: Distributed Computation and Applications

The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We introduce a new notion of mixing of a random walk on a (undirected) graph, called local mixing. Informally, the local mixing with respect to a given node $s$, is the mixing of a random walk probability distribution restricted to a large enough subset of nodes --- say, a subset of size at least $n/\beta$ for a given parameter $\beta$ --- containing $s$. The time to mix over such a subset by a random walk starting from a source node $s$ is called the local mixing time with respect to $s$. The local mixing time captures the local connectivity and expansion properties around a given source node and is a useful parameter that determines the running time of algorithms for partial information spreading, gossip etc. Our first contribution is formally defining the notion of local mixing time in an undirected graph. We then present an efficient distributed algorithm which computes a constant factor approximation to the local mixing time with respect to a source node $s$ in $\tilde{O}(\tau_s)$ rounds, where $\tau_s$ is the local mixing time w.r.t $s$ in an $n$-node regular graph. This bound holds when $\tau_s$ is significantly smaller than the conductance of the local mixing set (i.e., the set where the walk mixes locally); this is typically the interesting case where the local mixing time is significantly smaller than the mixing time (with respect to $s$). We also present a distributed algorithm that computes the exact local mixing time in $\tilde{O}(\tau_s \mathcal{D})$ rounds, where $\mathcal{D} =\min\{\tau_s, D\}$ and $D$ is the diameter of the graph. We further show that local mixing time tightly characterizes the complexity of partial information spreading.Comment: 16 page

Locality of Distributed Graph Problems

Locality is one of the central themes in distributed computing. Suppose in a network each node only has direct communication with its local neighbors, how efficiently can a global task be solved? We aim to investigate the locality of fundamental distributed graph problems. Toward this goal, we consider the following three basic abstract models of distributed computing. • LOCAL: each device has direct communication links with its neighbors, there is no message size constraint. • CONGEST: each device has direct communication links with its neighbors, the size of each message is at most O(log n) bits. • CONGESTED-CLIQUE: each device has direct communication links with all other devices, the size of each message is at most O(log n) bits. A brief summary of our results is as follows. 1. Complexity Theory for the LOCAL Model: We study the spectrum of natural problem complexities that can exist in the LOCAL model. We provide answers to the following fundamental questions regarding the nature of the LOCAL model: (i) How to classify the distributed problems according to their complexities? (ii) How much does randomness help? (iii) Can we solve more problems given more time? 2. Complexity of Distributed Coloring: The coloring problem is a classical and well-studied problem in distributed computing. We devise distributed algorithms for the edge-coloring problem and the vertex-coloring problem in the LOCAL model that improve upon the previous state of the art. 3. Bandwidth Constraint: We develop a new framework for algorithm design based on expander decompositions that allows us to apply CONGESTED-CLIQUE techniques to the CONGEST model. Using this approach, we provide improved algorithms for the triangle detection and enumeration problem in CONGEST.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149872/1/cyijun_1.pd