The Best Mixing Time for Random Walks on Trees

We characterize the extremal structures for mixing walks on trees that start from the most advantageous vertex. Let $G=(V,E)$ be a tree with stationary distribution $\pi$. For a vertex $v \in V$, let $H(v,\pi)$ denote the expected length of an optimal stopping rule from $v$ to $\pi$. The \emph{best mixing time} for $G$ is $\min_{v \in V} H(v,\pi)$. We show that among all trees with $|V|=n$, the best mixing time is minimized uniquely by the star. For even $n$, the best mixing time is maximized by the uniquely path. Surprising, for odd $n$, the best mixing time is maximized uniquely by a path of length $n-1$ with a single leaf adjacent to one central vertex.Comment: 25 pages, 7 figures, 3 table

Location-Aided Fast Distributed Consensus in Wireless Networks

Existing works on distributed consensus explore linear iterations based on reversible Markov chains, which contribute to the slow convergence of the algorithms. It has been observed that by overcoming the diffusive behavior of reversible chains, certain nonreversible chains lifted from reversible ones mix substantially faster than the original chains. In this paper, we investigate the idea of accelerating distributed consensus via lifting Markov chains, and propose a class of Location-Aided Distributed Averaging (LADA) algorithms for wireless networks, where nodes' coarse location information is used to construct nonreversible chains that facilitate distributed computing and cooperative processing. First, two general pseudo-algorithms are presented to illustrate the notion of distributed averaging through chain-lifting. These pseudo-algorithms are then respectively instantiated through one LADA algorithm on grid networks, and one on general wireless networks. For a $k\times k$ grid network, the proposed LADA algorithm achieves an $\epsilon$-averaging time of $O(k\log(\epsilon^{-1}))$. Based on this algorithm, in a wireless network with transmission range $r$, an $\epsilon$-averaging time of $O(r^{-1}\log(\epsilon^{-1}))$ can be attained through a centralized algorithm. Subsequently, we present a fully-distributed LADA algorithm for wireless networks, which utilizes only the direction information of neighbors to construct nonreversible chains. It is shown that this distributed LADA algorithm achieves the same scaling law in averaging time as the centralized scheme. Finally, we propose a cluster-based LADA (C-LADA) algorithm, which, requiring no central coordination, provides the additional benefit of reduced message complexity compared with the distributed LADA algorithm.Comment: 44 pages, 14 figures. Submitted to IEEE Transactions on Information Theor

Dynamic Size Counting in Population Protocols

The population protocol model describes a network of anonymous agents that interact asynchronously in pairs chosen at random. Each agent starts in the same initial state $s$. We introduce the *dynamic size counting* problem: approximately counting the number of agents in the presence of an adversary who at any time can remove any number of agents or add any number of new agents in state $s$. A valid solution requires that after each addition/removal event, resulting in population size $n$, with high probability each agent "quickly" computes the same constant-factor estimate of the value $\log_2 n$ (how quickly is called the *convergence* time), which remains the output of every agent for as long as possible (the *holding* time). Since the adversary can remove agents, the holding time is necessarily finite: even after the adversary stops altering the population, it is impossible to *stabilize* to an output that never again changes. We first show that a protocol solves the dynamic size counting problem if and only if it solves the *loosely-stabilizing counting* problem: that of estimating $\log n$ in a *fixed-size* population, but where the adversary can initialize each agent in an arbitrary state, with the same convergence time and holding time. We then show a protocol solving the loosely-stabilizing counting problem with the following guarantees: if the population size is $n$, $M$ is the largest initial estimate of $\log n$, and s is the maximum integer initially stored in any field of the agents' memory, we have expected convergence time $O(\log n + \log M)$, expected polynomial holding time, and expected memory usage of $O(\log^2 (s) + (\log \log n)^2)$ bits. Interpreted as a dynamic size counting protocol, when changing from population size $n_{prev}$ to $n_{next}$, the convergence time is $O(\log n_{next} + \log \log n_{prev})$

Reversal of Markov Chains and the Forget Time

We study three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to "forget" where it started. One of these is the mixing time, the minimum mean length of a stopping rule that yields the stationary distribution from the worst starting state. A second is the forget time, the minimum mean length of any stopping rule that yields the same distribution from any starting state. The third is the reset time, the minimum expected time between independent samples from the stationary distribution. Our main results state that the mixing time of a chain is equal to the mixing time of the time-reversed chain, while the forget time of a chain is equal to the reset time of the reverse chain. In particular, the forget time and the reset time of a time-reversible chain are equal. Moreover, the mixing time lies between absolute constant multiples of the sum of the forget time and the reset time. We also derive an explicit formula for the forget time, in terms o..