Let G = (V, E) be an undirected, acyclic, connected graph (that is, a tree). For...

   

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Let G = (V, E) be an undirected, acyclic, connected graph (that is, a tree). For any vertex v EV, the eccentricity of u is the length of a longest path from u to any other vertex of G. A vertex of G with minimum eccentricity is called an omphalos of G. 1. Design an efficient algorithm that computes an omphalos of G. (O(n + m)-time is possible and that should be your goal.) Provide some arguments behind the correctness of your algorithm and give the analysis of its asymptotic (worst-case) running time. 2. Is the omphalos unique? If not, how many different omphaloi can a tree have? Let G = (V, E) be an undirected, acyclic, connected graph (that is, a tree). For any vertex v EV, the eccentricity of u is the length of a longest path from u to any other vertex of G. A vertex of G with minimum eccentricity is called an omphalos of G. 1. Design an efficient algorithm that computes an omphalos of G. (O(n + m)-time is possible and that should be your goal.) Provide some arguments behind the correctness of your algorithm and give the analysis of its asymptotic (worst-case) running time. 2. Is the omphalos unique? If not, how many different omphaloi can a tree have?

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To find an omphalos of a tree GV E we can use the following algorithm 1 Choose an arbitrary vertex u ... View the full answer