Homework 1
Problem 1: (5 points)
Both Dijkstras algorihm and Bellmanford Algorithm
generates shortest paths to all destinations. Modify the
algorithm to compute shortest paths to a single specified
destination and analyze its complexity.
Problem 2: (5 points)
Can you modify Dijkstras algorithm to generate shortest
paths to all vertices from a given source vertex, if all these
vertices are connected to the source vertex, and detect that
the graph is not connected if the source is not connected to
some vertices?
Problem 3: (5 points)
Generate a counter example where Dijkstras
algorithm fails to generate shortest paths in graphs
with negative edge weights, but no non-positive
weight cycle.
Problem 4: (5 points)
Give an algorithm for computing the maximum
weight spanning tree in a graph and analyze its
complexity.
Problem 5: (10 points)
Consider two graphs G and G’ which are identical except the
weights of the edges. Weight in edge e of graph G is w(e) >=0,
and weight in edge e of Graph G’ is aw(e) + b, where a and b
are constants. Are the minimum spanning trees of G and G’
identical? Are the shortest paths between any two vertices in
the two graphs identical? Justify your answer. (If you answer
positive, then prove your answer. If you answer negative, then
give counter examples). If you answer negative for any of
these, can you think of conditions on a and b, for which your
answer would be positive?
Suppose the weight of an edge e in graph G’ is 1/w(e). Is the
minimum weight spanning tree in G the maximum weight
spanning tree in G’ and vice-versa? Justify.
Problem 6: (5 points)
Consider the minimum steiner tree problem.
Suppose someone tells you the intermediate
vertices in a minimum steiner tree (intermediate
vertices are the ones which are not the vertices in
the multicast group). Can you present a
polynomial complexity algorithm to compute the
minimum steiner tree? Analyze its complexity.
Problem 7: (5 points)
Consider a vertex v. Does the mnimum spanning tree
always consist of shortest paths from v to all other
vertices?
Problem 8: (10 points)
Prove that the markov chain representing the wireless network
taught in lectures 2 and 3 has one closed set, and possibly a few
other open sets, under the routing and scheduling policy
discussed in class. Also prove that the closed set has period 1,
under certain assumptions on the arrival probabilities. You may
assume that every packet has duration 1 slot.
Problem 9: (10 points)
Consider a network with a reward associated with every
edge. The reward associated with every path is the
minimum reward in any link in the path. Present an
algorithm which computes the highest reward path
between a source and a destination. Analyze its
complexity.
Problem 10: (15 points)
Consider a unirate, multicast network with N sessions. Assume that the
traffic for every session traverses a predetermined route. Construct a
unicast network with N sessions with predetermined routes for each
session such that a feasible rate vector in the unirate, multicast network is
a feasible rate vector in the unicast network and vice versa. (The size of
the unicast network will be larger than the size of the multicast network).
Now consider a multirate, multicast network with M virtual sessions.
Again assume that the traffic for every session traverses a predetermined
route. Construct a unicast network with M sessions with predetermined
routes for each session such that a feasible rate vector in the multirate,
multicast network is a feasible rate vector in the unicast network and vice
versa. (The size of the unicast network will be much larger than the size of
the multicast network).