Short correctness proofs for two self-stabilizing algorithms under the distributed daemon model

Concurrency and Computation

Masuzawa

2020

A ( , )-directed acyclic mixed graph (DAMG) is a mixed graph, which allows both arcs (or directed edges) and (undirected) edges such that there exist exactly source nodes and sink nodes, but there exists no directed cycle (consisting of only arcs). Each source (resp. sink) node has at least one outgoing (resp. incoming) arc, but no incoming (resp. outgoing) arc. Moreover any other node is neither a source nor a sink node; it has no incident arc or both outgoing and incoming arcs. This article considers maximal ( , )-DAMG constructions: when an arbitrary undirected connected graph G = (V, E) and two distinct subsets S and T of node set V, where |S|= and |T|= , are given, construct a maximal ( , )-DAMG with source node set S and sink node set T by assigning directions to as many edges as possible (ie, by changing edges into arcs). The maximality implies that changing any more edges to arcs violates the conditions of a ( , )-DAMG (eg, a sink node has an outgoing arc or a directed cycle is created). As a previous work, a self-stabilizing algorithm for constructing a maximal (1,1)-DAMG in an arbitrary undirected connected graph is proposed for the case of = = 1. In this article, we consider construction of a maximal ( , )-DAMG for any and . First, we introduce a self-stabilizing algorithm for a maximal (1,2)-DAMG construction in any connected graph (with few constraints), which is based on the previous work. Concerning generalization of and to arbitrary values, we first clarify the necessary and sufficient condition under which a ( , )-DAMG can be constructed in which a source and a sink node sets are given. Then, we propose a generalized self-stabilizing algorithm that constructs a ( , )-DAMG when a given graph with a source and a sink node sets satisfies the above condition.

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Section: Schedulementioning

confidence: 99%

Section: Preliminaries: Model and Definitionsmentioning

confidence: 99%

A self‐stabilizing algorithm for constructing a maximal (σ,τ)‐directed acyclic mixed graph

Concurrency and Computation

Masuzawa

2020

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“…There are some scheduler models depending on the number of concurrently operating processes and granularity of an atomic action. In this paper, we consider the distributed daemon [4,10] which assumes the followings:…”

Section: Definition 4 (Fair Schedule) If Every Process P I ∈ P Appeamentioning

confidence: 99%

A Self-Stabilizing Algorithm for Constructing a Maximal (1,1)-Directed Acyclic Mixed Graph

Ohno²,

et al. 2018

IJNC

We introduce a new network structure named a maximal (σ, τ )-directed acyclic mixed graph (DAMG). A maximal (σ, τ )-DAMG allows both arcs (directed edges) and (undirected) edges which is constructed, for any given connected undirected graph with a set of σ nodes specified as source nodes and a set of τ nodes specified as sink nodes, by assigning directions to as many (undirected) edges as possible (i.e., by changing edges into arcs) so that the following conditions are satisfied: (i) each node specified as a source node has at least one outgoing arc but no incoming arc, (ii) each node specified as a sink node has at least one incoming arc but no outgoing arc, (iii) each other node has no arc or has both outgoing and incoming arcs, and (iv) no directed cycle (consisting only of arcs) exists. This maximality implies that changing any more edges to arcs violates these conditions, for example, a source node has an incoming arc, a node which is specified as neither a source node nor a sink node has only outgoing or incoming arcs other than edges, or a directed cycle is created in the network.In this paper, we propose a self-stabilizing algorithm which constructs a (1,1)-maximal DAMG in any connected graph with a specified source node and a specified sink node by assigning directions to as many edges as possible.

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A Self-Stabilizing Algorithm for Constructing a Maximal (2,2)-Directed Acyclic Mixed Graph

Aono

2018 Sixth International Symposium on Computing and Networking (CANDAR)

et al. 2018