Self-Stabilizing Construction of a Minimal Weakly 𝒮𝒯-Reachable Directed Acyclic Graph
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We propose a self-stabilizing algorithm to construct a minimal weakly 𝒮𝒯-reachable directed acyclic graph (DAG), which is suited for routing messages on wireless networks. Given an arbitrary, simple, connected, and undirected graph G=(V, E) and two sets of nodes, senders 𝒮 (⊂ V) and targets 𝒯 (⊂ V), a directed subgraph G⃗ of G is a weakly 𝒮𝒯-reachable DAG on G, if G⃗ is a DAG and every sender can reach at least one target, and every target is reachable from at least one sender in G⃗. We say that a weakly 𝒮𝒯-reachable DAG G⃗ on G is minimal if any proper subgraph of G⃗ is no longer a weakly 𝒮𝒯-reachable DAG. This DAG is a relaxed version of the original (or strongly) 𝒮𝒯-reachable DAG, where every target is reachable from every sender. This is because a strongly 𝒮𝒯-reachable DAG G does not always exist; some graph has no strongly 𝒮𝒯-reachable DAG even in the case |𝒮|=|𝒯|=2. On the other hand, the proposed algorithm always constructs a weakly 𝒮𝒯-reachable DAG for any |𝒮| and |𝒯|. Furthermore, the proposed algorithm is self-stabilizing; even if the constructed DAG deviates from the reachability requirement by a breakdown or exhausting the battery of a node having an arc in the DAG, this algorithm automatically reconstructs the DAG to satisfy the requirement again. The convergence time of the algorithm is O(D) asynchronous rounds, where D is the diameter of a given graph. We conduct small simulations to evaluate the performance of the proposed algorithm. The simulation result indicates that its execution time decreases when the number of sender nodes or target nodes is large.
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