A Self-Stabilizing Minimal k-Grouping Algorithm

Abstract

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 + n false ) log n) where n false 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. * This is a revised version of the conference paper [6], which appears in the proceedings of the 18th International Conference on Distributed Computing and Networking (ICDCN), ACM, 2017. This revised version slightly generalize Theorem 1. PreliminariesA connected undirected network G = (V, E) of n = |V | processes is given where n ≥ 2. Each process v has a unique identifier v.id chosen from a set ID of non-negative integers. We assume that |ID | ≤ O(n c ) holds for some constant c, thus, a process can store an identifier in O(log n) space. By an abuse of notation, we will identify each process with its identifier, and vice versa, whenever convenient. We call a member of ID a false identifier if it does not match the identifier of any process in V .We use the locally shared memory model [8]. A process is modeled by a state machine and its state is defined by the values of its variables. A process can read the values of its own and its neighbors' variables simultaneously, but can update only its own variables. An algorithm of each process v is

A Self-stabilizing 1-maximal Independent Set Algorithm

“…Its convergence time is O(nD) rounds, where D is the diameter of the network, while the space complexity is O(log n) bits per process. We use a composition technique called loop composition, which Datta et al [3] introduced recently. This technique enables the processes to execute the same subalgorithm repeatedly in a consistent way until a 1-MIS is constructed, which results in a smaller space complexity, O(log n) bits per process.…”

“…This technique enables the processes to execute the same subalgorithm repeatedly in a consistent way until a 1-MIS is constructed, which results in a smaller space complexity, O(log n) bits per process. To the best of our knowledge, the loop composition technique is utilized only for the k-grouping problem [3] although it seems applicable to many problems. Thus, our result shows the applicability by providing the second success case of the loop composition.…”

“…We use the loop composition [3] to design a silent selfstabilizing 1-MIS algorithm. The loop composition is a technique to execute a given algorithm repeatedly in a consistent way.…”

“…Otherwise, we say that γ is non-erroneous for E. Algorithm P is a silent self-stabilizing algorithm that brings the system to a non-erroneous configuration for E, starting from any configuration. Then, we obtain a composite algorithm Loop(A, E, P) [3], which is a silent and self-stabilizing algorithm for L. Very roughly speaking, Loop(A, E, P) shows the following behavior (See Fig. 1).…”

“…If we design A, E, and P that satisfy the above requirements, the composited algorithm Loop(A, E, P) presented in Ref. [3] has the property described in the following theorem. We briefly describe the behavior of Loop(A, E, P) in the rest of this section.…”