Assigned:
Thursday, January 29, 2009
Due:
Thursday, February 5, 2009
General Instructions
 Please review the
course information.
 You must write down with whom you worked on the assignment. If this
changes from problem to problem, then you should write down this
information separately with each problem.
 Numbered problems are all from the textbook Network Flows .
Problems
 Problem 13.14. Tree minimax result.
 Problem 13.34. Balanced Spanning Trees
 Suppose I have a graph G with n nodes and 5n edges. For a spanning tree F, we define h(F) to be the number of Fheavy edges in G. Give upper and lower
bounds on h(F) in terms of n. Repeat the problem for a complete graph (with n(n1)/2 edges)
 Problem 4.6. Cluster Analysis.
 Suppose that we order the edge relaxations in each pass of the
BellmanFord algorithm as follows. Before the first pass, we assign
an arbitrary linear order v_{1},v_{2},...,v_{n}
to the vertices of the input graph.
Then, we partition the edge set E into E_{f} and E_{b}, where E_{f} are the forward edges, i.e. the ones of the form (v_{i},v_{j}) with i < j and
E_{b}, the backwards edges, i.e. the ones of the form (v_{i},v_{j}) with i> j.
Define G_{f}=(V,E_{f}) and G_{b}=(V,E_{b}).
 Prove that G_{f} is acyclic with topological sort v_{1},v_{2},...,v_{n} and
that G_{b} is acyclic with
topological sort v_{n},v_{n1},...,v_{1}

Suppose that we implement each pass of the BellmanFord
algorithm in the following way. We visit each vertex in the order
v_{1}, v_{2}, ... , v_{n}, relaxing edges of E_{f} that leave
the vertex. We then visit each vertex in the order v_{n},
v_{n1}, ..., v_{1}, relaxing edges of E_{b} that leave the vertex.
Prove that with this scheme, if G contains no
negativeweight cycles that are reachable from the source vertex s
then after only ceiling(n/2) passes over the edges, d(v) is the correct
shortest path distance for each vertex v.
 Does this scheme improve the asymptotic running time of
the BellmanFord algorithm?
Switch to:
cliff@ieor.columbia.edu