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

Complexity of Connected Components in Evolving Graphs 269the arc is present both at times f a (u j ,v j ) − 1 and f a (u j ,v j ), since w(u j ,v j ) < 1,the packet will reach v j in t uj + w(u j ,v j ). If, however, the arc is not present attime f a (u j ,v j ) − 1, then the transmission process itself starts at the f a (u j ,v j ) thstep (i.e. from time f a (u j ,v j ) − 1 to time f a (u j ,v j )), thus reaching v j by timef a (u j ,v j ) − 1+w(u j ,v j ).We remark that a rooted directed tree can also be computed over an o-SCC V G ′. As a modification for that purpose, V G must be replaced by V G ′ andcorrespondingly, Step 3 of Algorithm 1 should be modified to V G ′ ⊂ V f since thefragment can also contain the h-nodes for the vertices in V G ′ and the loop canstop once all the vertices are covered.Algorithm 1 is a greedy algorithm that always chooses the arc that transmitsin minimum time. The proof of its correctness is the same as the proof of thePrim-Dijkstra algorithm [4]. If the maximum outdegree of each vertex is D, theneach step of increasing the fragment will take O(NDlog T ) time and the fragmentwill increase N times adding up to a total execution time of O(N 2 D log T )steps.5 ConclusionThe two important results in this paper are the intractability of the decompositioninto (open) strongly connected components in FSDN’s and the constructionof DMST’s over an already existing strongly connected components.The first result implies that it is possible to lead a non-strongly connectednetwork towards strong connectedness by adding intermediary agents to serveas hops between two nodes that are out of range from each other. An interestingproblem would be to find a way to add such links so as to minimize the numberof intermediary (helping) nodes. Another way for further research is to designapproximation algorithms for (open) strongly connected components in evolvingdigraphs.AcknowledgmentsThe authors are grateful to Aubin Jarry and Stephane Perennes for very fruitfuldiscussions.References1. A. Borodin and R. El-Yaniv. Online computation and competitive analysis. CambridgeUniversity Press, 1998.2. B. Bui-Xuan, A. Ferreira, and A. Jarry. Computing shortest, fastest, and foremostjourneys in dynamic networks. International Journal of Foundations of ComputerScience, 14(2):267–285, April 2003.3. Y. J. Chu and T. H. Liu. On the shortest arborescence of a directed graph. ScienceSinica, 14:1396–1400, 1965.