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

268 S. Bhadra and A. FerreiraTheorem 5. o-COMPONENT is NP-complete.Proof: We know that CLIQUE is NP-Complete. So from Theorem 3 and Theorem4, o-COMPONENT is NP-Complete.4 Computing the Directed Minimum Spanning TreesConsidering a strongly connected evolving digraph G, the object is to find N =|V G | rooted directed minimum spanning trees rooted at each of the nodes r ∈ V G .Our algorithm is a modification of the Prim-Dijkstra algorithm [4] for findingMST’s in undirected standard graphs. The algorithm proceeds by building afragment which is a subset of the DMST starting from the root r. The propertyof the fragment f(r) is that it consists of those edges by which informationtransmitted at the beginning of the time interval from the root r will travel inthe shortest time to the vertices included already in the fragment. Having defineda fragment as such, it is easy to see how the algorithm for the DMST proceeds.In the following algorithm we choose from among the set of arcs outgoing fromthe fragment f(r), the arc with the smallest arc schedule time such that it canform a valid journey starting from the root. A number t v is associated with eachvertex v ∈ V G denoting the minimum time required for that vertex to receivethe information given that the root r originates the information.Since each node can transmit information only after it has received it, theinformation cannot pass simultaneously through two edges. Recall that thetime required for transmission over one arc is denoted as an arbitrary weight,w(u, v) < 1.Algorithm 11. Start with f(r) =∅ and a set V f containing vertices already considered infragment f(r).2. V f = {r}, t r =13. while V f ≠ V G do(a) Let Γ f be the set of all arcs (u i ,v i ) such that u i ∈ V f , v i /∈ V f . For each(u i ,v i ) ∈ Γ f , choose the smallest arc schedule time f a (u i ,v i ), such thatf a (u i ,v i ) ≥ t ui + w(u i ,v i ).(b) Choose arc (u j ,v j ) where j = min −1i (f a (u i ,v i )+w(u i ,v i )).(c) if f a (u j ,v j )=t uj + w(u j ,v j ), then t vj ← f a (u j ,v j ),(d) else if f a (u j ,v j ) − 1 ∈ {arc schedule of (u j ,v j )}, then t vj ← t uj +w(u j ,v j ),(e) else, t vj ← f a (u j ,v j ) − 1+w(u j ,v j )(f) add v j to V f and (u j ,v j ) to f(r).In the above algorithm, an arc schedule time i indicates the presence of thelink from time i − 1toi. Note that two cases might arise depending on whetherf a (u j ,v j )=t uj + w(u j ,v j )orf a (u j ,v j ) >t uj + w(u j ,v j ). For the first case, theinformation reaches the node exactly at the time f a (u j ,v j ). For the other case, if