# Homework Assignment #7

#### Due on paper in class, Monday 30 November 2009

There are six questions for 50 total points plus 10 extra credit.  All are from the textbook, A Mathematical Foundation for Computer Science. Note that the book has both Exercises and Problems -- make sure you are doing a Problem and not the Exercise with the same number. The number in parentheses following each problem is its individual point value.

Problem 12.3.3 (not 12.2.3) corrected on 23 November 2009 -- see below.

• Problem 11.10.1 (10)
• Problem 12.1.2 (10)
• Problem 12.1.5 (10XC)
• Problem 12.2.2 (10)
• Problem 12.2.3 (10)
• Problem 12.3.3 (10)
• Correction (23 Nov, fixed 24 Nov): As pointed out on the Q and A page, while there is always a steady state of the form described it may not be an attracting steady state. In fact there is an attracting steady state if and only if the undirected graph G has an odd cycle -- a path from some vertex v to itself that never revisits any other vertex and has an odd number of edges. So here is the revised question:
• (a,5) Given any connected undirected graph G, give the steady state distribution for the resulting Markov chain as the problem suggests, and prove that it is a steady state distribution.
• (b,5) Suppose that the connected graph G has the property that for all integers greater than some fixed number t, there is a path of length t from any vertex u to any vertex v. Then prove that the steady state of part (a) is attracting. You may quote the Steady State Theorem.
• (c,10XC) Show that if G has an odd cycle, it has the property in part (b). (Hint: Remember that a path can reuse an edge. Show that there are paths of all sufficiently long odd lengths and of all sufficiently long even lengths.)