A Distributed Algorithm for Minimum Distance-k Domination in Trees

Abstract

While efficient algorithms for finding minimal distance-k dominating sets exist, finding minimum such sets is NP-hard even for bipartite graphs. This paper presents a distributed algorithm to determine a minimum (connected) distance-k dominating set and a maximum distance-2k independent set of a tree T. It terminates in O(height(T)) rounds and uses O(log k) space. To the best of our knowledge this is the first distributed algorithm that computes a minimum (as opposed to a minimal) distancek dominating set for trees. The algorithm can also be applied to general graphs, albeit the distance-k dominating sets are not necessarily minimal.

Acyclic Strategy for Silent Self-stabilization in Spanning Forests

“…We have shown, using a toy example, that our method allow to easily obtain tight complexity bounds, precisely a stabilization time which is asymptotically optimal in rounds and polynomial in moves. Finally, we reviewed a number of existing silent self-stabilizing solutions from the literature [9,35,8,36,37,38,40] where our method applies. In some of them, we were able to provide more general results than those proven in the original papers.…”

“…This algorithm is made of two families of actions A 1 and A 2 : for every node p, We already know that A 1 is bottom-up and correct-alone. Then, from the definitions given in [9], we can easily deduce that A 2 is top-down and correct-alone since f ading(p) depends on p.parent.f ading and q.L with q ∈ p.children, which is not written by the family A 2 . Hence, the graph of actions' causality is…”

“…A Distributed Algorithm for Minimum Distance-k Domination in Trees [9]. This paper proposes three algorithms for directed trees.…”

“…To illustrate the versatility of our method, we review several existing works where our results apply.After the seminal work of Dijkstra, many self-stabilizing algorithms have been proposed to solve various tasks such as spanning tree constructions [2], token circulations [3], clock synchronization [4], propagation of information with feedbacks [5]. Those works consider a large taxonomy of topologies: rings [6,7], (directed) trees [5,8,9], planar graphs [10,11], arbitrary connected graphs [12,13], etc. Among those topologies, the class of directed (in-) trees (i.e., trees where one process is distinguished as the root and edges are oriented toward the root) is of particular interest.…”

Distance domination and generalized eccentricity in graphs with given minimum degree

“…k A survey of distance domination can be found in [15]. For more recent papers see, for example [3,7,9,13,16,18,25].…”