CAIRO UNIVERSITY
FACULTY OF ENGINEERING
COMPUTER ENG. DEPT.
CMP 302, CMPN 302
Algorithms
Assignment 2
Due Date: Wednesday November 30th
Q1: Design an algorithm for computing  n  for any positive integer n.
Besides assignment and comparison, your algorithm may only use the four
basic arithmetic operations.
look at one of these algorithms:
http://en.wikipedia.org/wiki/Methods_of_computing_square_roots
Q2: Locker Doors: there are n lockers in a hallway, numbered sequentially from 1
to n. Initially all the doors are closed. You make n passes by the lockers, each
time starting from locker #1. On the ith pass, i=1,2,…,n, you toggle the door of
every ith locker: if the door is closed, you open it; if it is open, you close it. For
example, after the first pass, every door is open; on the second pass, you toggle the
even-numbered lockers (#2, #4,….) so that after the second pass, the even doors
are closed and the odd ones are open; the third time through, you close the door of
locker #3 (open from the first pass), open the door #6 (closed from the second
pass), and so on. After the last pass, which locker doors are open and which are
closed? How many of them open?
Q3: Solve the following problems:
1. 1,000,000 n2  O(n2) why /why not?
2. (n - 1)n / 2  O(n2) why /why not?
3. n / 2  O(n2) why /why not?
4. lg (n2)  O ( lg n ) why /why not?
5. n2  O(n) why /why not?
6. Does
1 2
n  3n  (n 2 ) ?
2
7. Does
1 2
n  3n  O(n 2 ) ?
2
8. Does
1 2
n  3n  (n 2 ) ?
2
Q3: There are 22 gloves in a drawer: 5 pairs of red gloves, 4 pairs of yellow, and 2 pairs of
green. You select the gloves in the dark and can check them only after a selection has been
made. What is the smallest number of gloves you need to select to have at least one
matching pair in the best case? in the worst case? Explain?
Q4: Question 9 exercise 2.1 in the textbook
Indicate whether the first function of each of the following pairs has smaller, the same, or
larger order of growth than the second function:
a.
b.
c.
d.
e.
n(n-1)
100 n2
Log n
(n-1)!
2n-1
and 2000n2
and 0.01 n3
and ln n
and n!
and
2n
Q5: Give a careful proof that ∑ni=1 i log(i) = Θ(n2 log n)?
Q6: What is the complexity of the Selection Problem algorithm?
Q 7: Solve Q3 on page 187 of the textbook?