Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

Complexity of Connected Components in Evolving Graphs 261FSDN’s. Concisely, an evolving graph is an indexed sequence of T subgraphs ofa given graph, where the subgraph at a given index point corresponds to thenetwork connectivity at the time interval indicated by the index number. Thetime domain is further incorporated into the model by restricting journeys (i.e.,the equivalent of paths in usual graphs) to never move into edges which existedonly in past subgraphs (cf. Figure 2 below, and Section 2).0,1,3C0,1A0,1,2,332,31,2,3B0,1,20,1,310,2,300,1,21,2,30,1,2,30FD0,3E0,1,2,3Fig. 2. Evolving digraph corresponding to FSDN in Figure 1. Edges are labeled withcorresponding time-steps. Observe that CBF is not a valid journey since BF existsonly in the past with respect to CB.Notice that this model allows for arbitrary changes between two consecutivetime steps, with the possible creation and/or deletion of any number ofvertices and edges. Evolving graph edges can also be associated with traversaltimes. In [2], algorithms were proposed for finding foremost, shortest, and fastestjourneys in dynamic mobile networks modeled by evolving graphs. Other pathproblems in evolving graphs can be found under the merit approach [7]. Resultsproven include finding a sequence of paths that connect a given pair of nodesthroughout the system, such that the global routing plus re-routing costs areminimized.Our work. We focus on the analysis of connectivity properties in FSDN’s andthe design of algorithms for building directed minimal spanning trees (DMST’s)to generate multicast routes in FSDN’s. The DMST problem in wireless networkswas defined in [12] as finding N minimum weight trees, or arborescences, in anetwork modeled by a strongly connected digraph with N vertices. A centralizedalgorithm for finding DMST’s in static wireless networks is presented by Chuand Liu [3], and Tarjan [18] provides an efficient implementation of the same.Humblet [12] provides a distributed algorithm for finding DMST’s in stronglyconnected networks. Furthermore, minimum energy multicast trees for wirelessnetworks have also been studied for the static case in [20,19]. In contrast, ourapproach differs from these in that our algorithm builds DMST’s over dynamicmobile networks modeled by evolving digraphs, which can be seen as dynamicallychanging digraphs.