Spring 2009 CMSC 451: Homework 3 Samir Khuller
Feb 26
(1) Describe an efficient algorithm that given an undirected graph G, determines a span-
ning tree of Gwhose largest edge weight is minimum, over all spanning trees of G.
Give an argument justifying your algorithm.
(2) The diameter of a tree T= (V, E ) is given by
max
u,v∈Vδ(u, v)
where δ(u, v) is the distance between uand vin the tree T. Give an O(|V|) algorithm
for computing the diameter of the tree. Write a proof of correctness for your algorithm.
(3) In a directed graph, a get-stuck vertex is one that has in-degree |V| − 1 and out-
degree 0. Assume that the adjacency matrix representation is used. Design an O(|V|)
algorithm to determine if a given graph has a get-stuck vertex. (Yes, this problem
can be solved without even looking at the entire input matrix.) Write a proof of
correctness for your algorithm.
(4) Assume that we have a network (a connected undirected graph) in which each edge ei
has an associated bandwidth bi. If we have a path P, from sto v, then the capacity of
the path is defined to be the minimum bandwidth of all the edges that belong to the
path P. We define capacity(s, v) = maxP(s,v )capacity(P). (Essentially, capacity(s, v)
is equal to the maximum capacity path from sto v.) Give an efficient algorithm to
compute capacity(s, v), for each vertex v; where sis some fixed source vertex. Show
that your algorithm is “correct”, and analyze its running time.
(Design something that is no more than O(|V|2), and with the right data structures
takes O(|E|log |V|) time.)
(5) Let Gbe a directed graph. The vertices of Ghave been numbered 1 . . . n (where nis
the number of vertices in G). Let small(i) = min{j|jis reachable from i}. In other
words, for a vertex numbered i,small(i) is the smallest numbered vertex reachable
from it. Design an O(V+E) algorithm to compute small(i) for al l vertices in the
graph.
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