 
Summary: CMPSCI 611: Advanced Algorithms
Micah Adler
Problem Set 3 Out: November 2, 2000
Due: November 9, 2000
1. CLR Problem 271 (page 625).
2. Recall from Lecture 13, Karger's mincut algorithm:
While number of vertices > 2:
Select edge e = (u; v) uniformly at random.
Merge u and v into a single vertex.
Return remaining edges.
We proved that this returns a minimum cut in the original graph with probability at least 1
n 2
, where
n = jV j.
(a) Describe a data structure that allows each iteration of the while loop to be performed in O(n)
time, and justify this bound. Describe how this allows us to nd a minimum cut with probability
at least 1 in time O(n 4 log 1 ).
We next consider how to improve on this run time. The idea behind this improvement is that the
probability of the mincut algorithm hitting an edge in the minimum cut is fairly small for the rst few
edges that are chosen, but increases with each merge that is performed. Thus, we should be repeating
