CS 2606 Data Structures & OO Devel II Homework 2: Complexity
1
You will submit your solution to this assignment to the Curator System (as HW2). Your solution must be either a plain text file
(e.g., NotePad) or a MS Word document; submissions in other formats will not be graded.
Except as noted, credit will only be given if you show relevant work.
1. [25 points] Using the rules given in the course notes, perform an exact count complexity analysis, for the worst case, of
the body of the following function. (Treat K as a constant. Your answer should be a function of the parameter N.)
int selectKth(int A[], int K, int N) {
int minI, tmp;
for (int i = 0; i < K; i++) {
minI = i;
for (int j = i + 1; j < N; j++) {
if ( A[j] < A[minI] )
minI = j;
}
tmp = A[i];
A[i] = A[minI];
A[minI] = tmp;
}
return A[k-1];
}
2. [15 points] Use Theorem 8 from the course notes on asymptotics to give a formal proof that:
log is ( )
n n n
+ Θ
3. [40 points] For each part, determine the simplest possible function g(n) such that the given function is
( )
g
Θ
. No
justification is necessary.
a)
2
( ) 3 14 47
a n n n
= + +
b)
( ) 14 3 log
b n n n n
= +
Hint: the last three take a little analysis.
c)
0.9
( ) log
c n n n
= +
d)
2
( ) 3 log
d n n n
=
e)
2 2
( ) 3 log 3 log
e n n n n n
= +
4. [20 points] Suppose that executing an algorithm on input of size N requires executing T(N) = N
log N + 8N instructions.
How long would it take to execute this algorithm on hardware capable of carrying out 2
24
instructions per second if N =
2
32
? (Give your answer in hours, minutes and seconds, to the nearest second.)