Exercise 1: Maximum element

for the following code a- what is the basic operation? b- what is C(n)?

d-what is the running time?

2

Exercise 2 : Element uniqueness problem

for the following code a- what is the basic operation? b- what is C(n)?

d-what is the running time?

3 Exercise 3: For each of the following pairs of functions, indicate whether the first function of each of the following pairs has a lower, same, or higher order of growth (to within a constant multiple) than the second function.

a. n(n + 1) and 2000n

2 b. 100n 2 and 0.01n

3

c. log 2

n and ln n d. log 22

n and log 2

e. 2n−1 and 2n

f. (n − 1)! and n!

n 2

4

homwork

1 . Consider the definition-based algorithm for adding two n × n matrices.

What is its basic operation? How many times is it performed as a function of the matrix order n? As a function of the total

number of elements in the

input matrices?

b. Answer the same questions for the definition-based algorithm for matrix

multiplication .

2 . Consider the following algorithm

.

ALGORITHM Mystery(n)

//Input: A nonnegative integer n

S ←0

for i ←1 to n do

S ←S + i

∗

i

return S a. What does this algorithm compute?

b. What is its basic operation?

c. How many times is the basic operation executed?

d. What is the efficiency class of this algorithm?

e. Suggest an improvement, or a better algorithm

altogether, and indicate its efficiency class. If you cannot do 5 it, try to prove that, in fact, it cannot be