CSCI203 ASSIGNMENT 1

CSCI203 ASSIGNMENT 1
1. Prove that x log y = y log x
2. Let g(n) be the number of ways to write a string of n zeros and ones so that
there are never two successive zeros. Show that g(n) = fib(n+2). (Hint: try induction.)
3. Which of the following statements are true? Prove your answers.
a. n2
 O(n3
)
b. n2
 Ω(n3
)
c. 2n  Θ(2n+1)
d. n!  Θ((n+1)!)
4. An alternative proof that there are an infinite number of primes begins as
follows. Assume that the set of prime numbers is finite. Let P be the largest
prime. Consider X=P! and Y=X+1. Complete the proof and construct an
algorithm bigprime(P) based on the proof. It should be clear from the
algorithm that it terminates with a new prime larger than P. (Note: this
algorithm should not involve brute-force testing of candidate primes by
exhaustive division.)
5. Suppose A and B are two finite sets. Show that
 where denotes the number of
elements in set A. What is the equivalent equation for three sets, A, B and C?
6. Two algorithms take n^2 days and n^3 seconds to solve an instance of size n.
Which is the better algorithm? What is the break-even value for n? How long
will the algorithms take at this value of n?
7. A function f (x) such that f (a) and f (b) have different signs must have at least
one root (value where the function is zero) between a and b. Devise an
algorithm based on finding the value of x at which the line joining the points
(a, f(a)) and (b, f(b)) crosses the x-axis and then determining the sign of the
function at this point.
8. Use the secant method of question 7 to perform three iterations with f(x) = x3
-
4x + 1, a = 1 and b = 2. Tabulate the values of all variables at each stage.
9. Give two functions f(n) and g(n) such that neither f(n) nor g(n) tends to a limit
as n tends to infinity, but both f(n)+g(n) and f(n)/g(n) do tend to a limit.
10. A certain algorithm takes 0.0001 x 2n
seconds to solve an instance of size n.
Show that it just solves an instance of size 38 in a year. What size problem
could be solved in a year on a machine 100 times as fast? A second algorithm
takes 0.01 x n3
seconds to solve an instance of size n. What size problem could
it solve in a year? With a machine 100 times as fast? For what range of n is the
first algorithm better?
Assignments should be typed into a text file called ass1.txt and submitted via the
submit program in Unix.
submit -u user -c csci203 -a 1 ass1.txt
where your unix userid should appear instead of user.
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