Parallel rotor walks on finite graphs and applications in discrete load balancing

Abstract

We study the parallel rotor walk process, which works as follows: Consider a graph along with an arbitrary distribution of tokens over its nodes. Every node is equipped with a rotor that points to its neighbours in a fixed circular order. In each round, every node distributes all of its tokens using the rotor. One token is allocated to the neighbour pointed at by the rotor, then the rotor moves to the subsequent neighbour, and so on, until no token remains.The process can be considered as a deterministic analogue of a process in which tokens perform one independent random walk step in each round. We compare the distribution of tokens in the rotor walk process with expected distribution in the random walk model. The similarity between the two processes is measured by their discrepancy, which is the maximum difference between the corresponding distribution entries over all rounds and nodes. We analyze a lazy variation of rotor walks that simulates a random walk with loop probability of 1/2 on each node, and each node sends not all its tokens, but every other token in each round.Viewing the rotor walk as a load balancing process, we prove that the rotor walk falls in the class of bounded-error diffusion processes introduced in [11]. This gives us discrepancy bounds of O(log 3/2 n) and O(1) for hypercube and r-dimensional torus with r = O(1), respectively, which improve over the best existing bounds of O(log 2 n) and O(n 1/r ). Also, as a result of switching to the load balancing view, we observe that the existing load balancing results can be translated to rotor walk discrepancy bounds not previously noticed in the rotor walk literature.We also use the idea of rotor walks to propose and analyze a randomized rounding discrete load balancing process that achieves the same balancing quality as similar protocols [11,3], but uses fewer number of random bits compared to [3], and avoids the negative load problem of [11].

Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains

“…Friedrich et al [15] proposed the BED algorithm for load balancing, which uses some extra information in the previous time, and they gave O(d 1.5 ) for hypercube and O(1) for constantdimensional tori. Akbari and Berenbrink [1] discussed the relation between the BED algorithm and the rotorrouter model, and gave the same bounds for a rotorrouter model. Berenbrink et al [6] investigated about cumulatively fair balancers algorithms, which includes the rotor-router model, and gave an upper bound…”

“…Deterministic random walks [10,9,13,8,23,22,27] may be used as a substitute for Markov chains, for the purpose. Deterministic random walk Deterministic random walk is a deterministic process analogous to a (multiple) random walk 1 . A configuration χ (t) ∈ Z V ≥0 of M tokens distributed over a (finite) vertex set V is deterministically updated from time t to t + 1 by routers equipped on vertices.…”

The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

“…As the number of agents k increases, the speed-up of the rotor-router with Table 1 The cover time of the multi-agent roter-router on the ring compared to multiple random walks (for k < n 1 …”

Limit Behavior of the Multi-agent Rotor-Router System

“…The structural properties of the distribution of tokens for a rotor-router system on the 2-dimensional grid were considered by Doerr and Friedrich [9]. Akbari and Berenbrink [2] proved an upper bound of O(log 3/2 n) on the load-balancing discrepancy for hypercubes, and for tori of constant dimensions, they showed that the discrepancy is bounded by a constant. For general d-regular graphs, a bound of O(d log n/µ) on the discrepancy of the rotor-router mechanism with respect to continuous diffusion follows from the general framework of [18], where µ is the eigenvalue gap of the graph, under the assumption that a sufficient number of self-loops are present at each node of the graph.…”