Token management schemes and random walks yield self-stabilizing mutual exclusion
PODC '90 Proceedings of the ninth annual ACM symposium on Principles of distributed computing
Distributed probabilistic polling and applications to proportionate agreement
Information and Computation
Conductance and congestion in power law graphs
SIGMETRICS '03 Proceedings of the 2003 ACM SIGMETRICS international conference on Measurement and modeling of computer systems
Gossip-Based Computation of Aggregate Information
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Randomized protocols for asynchronous consensus
Distributed Computing - Papers in celebration of the 20th anniversary of PODC
Tight bounds for distributed selection
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and Robustness
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
On Mixing and Edge Expansion Properties in Randomized Broadcasting
Algorithmica - Special Issue: Algorithms and Computation; Guest Editor: Takeshi Tokuyama
Probabilistic local majority voting for the agreement problem on finite graphs
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
Multiple Random Walks in Random Regular Graphs
SIAM Journal on Discrete Mathematics
Stabilizing consensus with the power of two choices
Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures
On the runtime and robustness of randomized broadcasting
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Coalescing-branching random walks on graphs
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
Hi-index | 0.00 |
In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon [14]. Let G=(V,E), be an undirected, connected n vertex graph. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our general result is, that C(n)= O(n/(v(1-λ2))), where v = ⁙u∈Vd2(u)/(d2 n), d(u) is the degree of vertex u, d is the average vertex degree, and λ2 is the second eigenvalue of the transition matrix of the random walk. The parameter v is an indicator of the variability of vertex degrees: 1 ≤ v = O(n), with v = 1 for regular graphs. Our general bound on C(n) holds provided the maximum vertex degree is O(m1-ε), where m is the number of edges in the graph. This result implies, for example, that C(n)=O(n/(1-λ2)) for d-regular graphs with expansion parameterized by the eigenvalue gap 1-λ2. The O(n/(v(1-λ2))) bound is sublinear for some classes of graphs with skewed degree distributions. A system of coalescing particles where initially one particle is located at each vertex, corresponds to the following voter model. Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbour. The voting process can be used for leader election in a distributed context. Let E(Cv) be the expected time for voting to complete, that is, for a unique opinion to emerge. It is known that E(Cv)=C(n ), so our results imply that E(Cv) = O(n/(v(1-λ2))). We also investigate how the voting time improves when a vertex elicits more than one opinion at each step. In a model which we call min-voting, each vertex initially holds a distinct opinion drawn from a linearly ordered domain. At each step each vertex takes the opinions of two random neighbours and keeps the smaller. We show that for regular graphs with very good expansion properties, voting is completed in O(log n) time with high probability. This result can be viewed as an example of the "power of two choices" in distributed voting.