IEOR 6614, Spring 2012 : Homework 2

General Instructions

1. Please review the course information.
2. 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.
3. Numbered problems are all from the textbook Network Flows .

Problems

1. Suppose that I have a collection of n items, which I place in single item sets. Each set S has a counter c_S, whose value is initially 0. I execute a series of UNION operations on the sets, where UNION(S,T) creates a new set X which is the union of S and T. I set c_X by the following rule: if c_S = c_T then c_X = c_S +1 ,otherwise c_X = max(c_S , c_T ).
   1. Show that for any sequnce of UNION operations,for all sets S c_S <= lg n.
   2. Give a series of UNION operations for which c_S is lg n (or very close) for some set S.
2. Suppose we have a graph G and the edge weights are not necessarily unique. Prove that a graph has a unique minimum spanning tree if, for every cut of the graph, there is a unique minimum weight edge crossing the cut. Prove that the converse is not true.
3. Problem 13.30. Most vital arc.
4. 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 F-heavy edges in G. Give upper and lower bounds on h(F) in terms of n. Repeat the problem for a complete graph (with n(n-1)/2 edges)
5. 1. Prove that the recurrence relation T(n,m) = T(n/8,m/2) + T(n/8,m/4) + O(n+m) has solution O(n+m).
   2. Let a,b,c,d be positive constants such that a+b < .99 and c+d < .99. Prove that the recurrence relation T(n,m) = T(an,cm) + T(bn,dm) + O(n+m) has solution O(n+m).

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• Homework 1
• Homework 3