Summer 2008 CMSC 351: Homework 7 Clyde Kruskal
Due at the start of class, Friday, June 27.
Problem 1. Consider the bubble-sort algorithm, for an input (a b c) of the three integer
values 1, 2, and 3 in any order, where all input orders are equally likely (uniform
distribution).
(a) Let Mbe the number of exchanges. What are E(M), Var(M), and σ(M)?
(b) Let xbe the number of comparisons. What are E(M), Var(M), and σ(M)?
(c) What p ercentage of the sample space is within a standard deviation of average.
Problem 2. Again consider the bubble-sort algorithm, for an input (a b c) of the three
integer values 1, 2, and 3 in any order. But now suppose the input orders are not
equally likely; instead, there is a 1/5 chance of a= 1 being the first item in the input
(the 1/5 distributed equally among those inputs) and 4/5 of anot being 1 (again the
4/5 shared equally among all these inputs).
(a) Let Mbe the number of exchanges. What are E(M), Var(M), and σ(M)?
(b) Let xbe the number of data comparisons made. What are E(M), Var(M), and
σ(M)?
(c) What p ercentage of the sample space is within a standard deviation of average.
Problem 3. For this problem you may use a calculator for a few calculations.
Consider P100
k=1 k3/2.
(a) Use a non-integral method to show that the sum is between 15,000 and 70,000.
(b) Approximate the sum using integrals. Make sure to get an upper and lower
bound.
Problem 4. Assume you have a list of nelements where the first n/k elements are the
smallest (but not sorted), the next group of n/k elements are the next smallest (but
not sorted), ..., and the last n/k elements are the largest (but not sorted). You may
assume kdivides n.
(a) Give an algorithm that sorts this list with as few comparisons as possible (as a
function of nand k). Just get the high order term right. How many comparisons
does your algorithm use?
(b) Show that your algorithm is optimal using a decision tree argument on the entire
list. (I.e., do not argue that you must solve kindependent sorting problems.)