Yuichi Sudo

Loosely-stabilizing leader election in a population protocol model

A self-stabilizing protocol guarantees that starting from any arbitrary initial configuration, a system eventually comes to satisfy its specification and keeps the specification forever. Although self-stabilizing protocols show excellent fault-tolerance against any transient faults (e.g. memory crash), designing self-stabilizing protocols is difficult and, what is worse, might be impossible due to the severe requirements. To circumvent the difficulty and impossibility, we introduce a novel notion of loose-stabilization, that relaxes the closure requirement of self-stabilization; starting from any arbitrary configuration, a system comes to satisfy its specification in a relatively short time, and it keeps the specification not forever but for a long time. To show the effectiveness and feasibility of this new concept, we present a probabilistic loosely-stabilizing leader election protocol in the Probabilistic Population Protocol (PPP) model of complete networks. Starting from any configuration, the protocol elects a unique leader within O(nNlogn) expected steps and keeps the unique leader for @W(Ne^N) expected steps, where n is the network size (not known to the protocol) and N is a known upper bound of n. This result proves that introduction of the loose-stabilization circumvents the already-known impossibility result; the self-stabilizing leader election problem in the PPP model of complete networks cannot be solved without the knowledge of the exact network size.

An agent exploration in unknown undirected graphs with whiteboards

We consider the exploration problem with a single agent in undirected graphs. Starting from an arbitrary node, the agent has to explore all the nodes and edges in the graph and return to the starting node. Our goal is to minimize both the number of agent moves and the memory space of the agent, which dominate the amount of communication during the exploration. In our setting, the agent is allowed to use the local memory called the whiteboard on each node (the whiteboard model), while most of existing exploration algorithms do not use the whiteboard (the no-whiteboard model). In the no-whiteboard model, the agent must memorize in its memory all information needed to explore the graph, and thus designing an exploration algorithm of small agent memory is difficult. In this paper, by allowing the agent to use whiteboards, we present four exploration algorithms such that both the number of agent moves and the agent memory space are small.

Loosely-Stabilizing leader election in population protocol model

A self-stabilizing protocol guarantees that starting from an arbitrary initial configuration, a system eventually comes to satisfy its specification and keeps the specification forever. Although self-stabilizing protocols show excellent fault-tolerance against any transient faults (e.g. memory crash), designing self-stabilizing protocols is difficult and, what is worse, might be impossible due to the severe requirements. To circumvent the difficulty and impossibility, we introduce a novel notion of loose-stabilization, that relaxes the closure requirement of self-stabilization; starting from an arbitrary configuration, a system comes to satisfy its specification in a relatively short time, and it keeps the specification for a long time, though not forever. To show effectiveness and feasibility of this new concept, we present a probabilistic loosely-stabilizing leader election protocol in the Probabilistic Population Protocol (PPP) model of complete networks. Starting from any configuration, the protocol elects a unique leader within O(nNlogn) expected steps and keeps the unique leader for Ω(NeN) expected steps, where n is the network size (not known to the protocol) and N is a known upper bound of n. This result proves that introduction of the loose-stabilization circumvents the already-known impossibility result; the self-stabilizing leader election problem in the PPP model of complete networks cannot be solved without the knowledge of the exact network size.

Loosely-Stabilizing Leader Election on Arbitrary Graphs in Population Protocols

In the population protocol model Angluin et al. proposed in 2004, there exists no self-stabilizing protocol that solves leader election on complete graphs without knowing the exact number of nodes. To circumvent the impossibility, we previously introduced the concept of loose-stabilization, which relaxes the closure requirement of self-stabilization. A loosely-stabilizing protocol guarantees that starting from any initial configuration a system reaches a loosely-safe configuration, and after that, the system keeps its specification (e.g. the unique leader) not forever, but for a sufficiently long time. Our previous work presented a loosely-stabilizing protocol that solves the leader election on complete graphs using only the upper bound N of n, not the exact value of n. We take this work one step further in this paper: We propose two loosely-stabilizing protocols that solve leader election for arbitrary graphs. One is a deterministic protocol that uses the identifiers of nodes while the other is a probabilistic protocol that works on anonymous networks. Given the upper bounds N and Δ of the number of nodes and the maximum degree of nodes respectively, both protocols keep a unique leader for Ω(Ne N ) expected steps after entering a loosely-safe configuration. The former enters a loosely-safe configuration within O(mΔN logn) expected steps while the latter does within O(mΔ2 N 3logN) expected steps where m is the number of edges of the graph.

Loosely-Stabilizing Leader Election on Arbitrary Graphs in Population Protocols Without Identifiers nor Random Numbers

In the population protocol model Angluin et al. proposed in 2004, there exists no self-stabilizing leader election protocol for complete graphs, arbitrary graphs, trees, lines, degree-bounded graphs and so on unless the protocol knows the exact number of nodes. To circumvent the impossibility, we introduced the concept of loose-stabilization in 2009, which relaxes the closure requirement of self-stabilization. A loosely-stabilizing protocol guarantees that starting from any initial configuration a system reaches a safe configuration, and after that, the system keeps its specification (e.g. the unique leader) not forever, but for a sufficiently long time (e.g. exponentially large time with respect to the number of nodes). Our previous works presented two loosely-stabilizing leader election protocols for arbitrary graphs; One uses agent identifiers and the other uses random numbers to elect a unique leader. In this paper, we present a loosely-stabilizing protocol that solves leader election on arbitrary graphs without agent identifiers nor random numbers. By the combination of virus-propagation and token-circulation, the proposed protocol achieves polynomial convergence time and exponential holding time without such external entities. Specifically, given upper bounds N and Delta of the number of nodes n and the maximum degree of nodes delta respectively, it reaches a safe configuration within O(m*n^3*d + m*N*Delta^2*log(N)) expected steps, and keeps the unique leader for Omega(N*e^N) expected steps where m is the number of edges and d is the diameter of the graph. To measure the time complexity of the protocol, we assume the uniformly random scheduler which is widely used in the field of the population protocols.

Brief Announcement: Reduced Space Self-stabilizing Center Finding Algorithms in Chains and Trees

In this work, we consider the problem of finding the center, or centers, of a chain network and a tree network.

A Self-Stabilizing Minimal k-Grouping Algorithm

We consider the minimal k-grouping problem: given a graph G = (V, E) and a constant k, partition G into subgraphs of diameter no greater than k, such that the union of any two subgraphs has diameter greater than k. We give a silent self-stabilizing asynchronous distributed algorithm for this problem in the composite atomicity model of computation, assuming the network has unique process identifiers. Our algorithm works under the weakly-fair daemon. The time complexity (i.e. the number of rounds to reach a legitimate configuration) of our algorithm is O (nD/k) where n is the number of processes in the network and D is the diameter of the network. The space complexity of each process is O((n + nfalse) log n) where nfalse is the number of false identifiers, i.e., identifiers that do not match the identifier of any process, but which are stored in the local memory of at least one process at the initial configuration. Our algorithm guarantees that the number of groups is at most 2n/k + 1 after convergence. We also give a novel composition technique to concatenate a silent algorithm repeatedly, which we call loop composition.

The Same Speed Timer in Population Protocols

A novel concept of the same speed timer is presented, and is applied in the population protocol (PP) model to improve the convergence time of existing loosely-stabilizing leader election protocols. Loosely-stabilizing leader election guarantees that, starting from any configuration, the system reaches a safe configuration within a short time (convergence), and after that, the system keeps the unique leader for a long time (closure). Two loosely-stabilizing leader election protocols for arbitrary graphs exist in the literature; one uses identifiers of nodes and the other uses random numbers to elect a unique leader. Both protocols guarantee that the expected convergence time is polynomial and the expected holding time (the time the leader is kept) is exponential. In this paper, convergence time of these protocols is dramatically improved by the same speed timer without impairing the exponential holding time. Specifically, a fast deterministic loosely-stabilizing leader election protocol that uses identifiers of nodes and a fast randomized looselystabilizing leader election protocol are given. The expected convergence time and expected holding time of the former protocol are O(mN log N) and Ω(Ne2N), respectively, where m is the number of edges in the graph and N is a given upper bound on the number of nodes n. The expected convergence time and expected holding time of the latter protocol are O(mN2 log n) and Ω(Ne2N), respectively. A self-stabilizing two-hop coloring protocol that uses only O(log n) memory space of each agent is given as a tool of the latter protocol. A lower bound is also given: any loosely-stabilizing leader election protocol with expected exponential holding time requires Ω(mN) expected convergence time.