def-dec - CS648 Randomized Algorithms Lecture...

CS648 Randomized Algorithms
Lecture notes :Principle of deferred decision II
Expected distance between two vertices in a complete graph with
random edge weights
Principle of deferred decision : Let f be a function of n random variables and there is an underlying
algorithm that computes the value of f . These n random variables may be initially chosen to be independent
and assumed to have certain probability distribution in the beginning. Sometimes there is an algorithm
which computes f . In such scenario, the naive and usual way of calculating the expected value of f will be
to compute value of f for each possible input (assigning all possible values to the random variables), and
then taking an average of these values. This approach is a very cumbersome approach to compute expected
value of f . On the contrary, there is a lazy way of exposing the random variables during the execution where
we defer revealing the exact value of a random variable unless it is absolutely needed. Taking this approach
turns out to be very effective in calculating the expected value of f . The reason why it is sometimes helpful
is that the this approach imposes the least dependency on the values of the unexposed random variables in
terms of the values taken by the already exposed random variables. As a result, it is much easier to analyze
a particular step of the algorithm for calculating the final value taken by f . We shall highlight the usefulness
of this approach with the following problem :
Problem : Given a complete graph G = ( V, E ) on n