Summer 2009 CMSC 351: Homework 2 Clyde Kruskal
Due at the start of class Wednesday, June 10, 2009.
Problem 1. Professor Hilbert invents a program that uses exactly 1000n3operations to
solve the borogove problem of size n. Professor Poincare invents a program that uses
1.5n/105operations to solve the same problem.
(a) Assume that you are willing to run Hilbertās program for some specific amount
of time. Assume that your computer speed increases by a factor of 10. How
much larger a problem can you solve in the same amount of time? Show your
work.
(b) Assume that your computer executes one billion operations per second. Approx-
imately how large a problem can you solve in a year? Show your work (but not
the arithmetic).
(c) Assume that you are willing to run Poincareās program for some specific amount
of time. Assume that your computer speed increases by a factor of 10. How
much larger a problem can you solve in the same amount of time? Show your
work.
(d) Assume that your computer still executes one billion operations per second. Ap-
proximately how large a problem can you now solve in a year? Show your work
(but not the arithmetic).
(e) Whose program should you use if you are going to run it for a year?
(f) For approximately what values of nis Professor Hilbertās program faster than
Professor Poincareās? Show your work.
Problem 2. Show the following. In each of (a), (b), and (c) state specific values of the
constants (e.g. c1,c2,n0) you used to satisfy the conditions, and show how you arrived
at the values.
(a) 2n2+ 2n+ 6 = O(n2ā2n+ 4)
(b) 4n3ā3n2+ 8 = Ī(n3)
(c) 2n2+ 5n(lg n)3=O(n2) [Hint: Find n0such that (lg n)3ā¤n, for all nā„n0.]
(d) nlog n=o(n1.5)
Problem 3. For each pair of expressions (A, B ) below, indicate whether Ais O,o, ā¦, Ļ,
or Ī of B. Note that zero, one or more of these relations may hold for a given pair;
list all correct ones.
A B
(a)n100 2n
(b) (log n)12 ān
(c)ān ncos(Ļn/8)
(d) 10n100n
(e)nlog n(log n)n
(f) log (n!) nlog n
