Yeditepe University
Department of Computer Engineering
CSE 311 Algorithm Analisys
FALL 2009
MIDTERM
Student Number : ___________
Student Name : ______________________
1
6
2 3 4 5
16 10 18 8
6 7 8
14 12 16
Total
100
There are 8 questions in 5 pages. The exam duration is two hours.
1. (6 points)
Find the algorithm complexity (Big-Oh) of the following codes. Assume the input to these functions is a
one dimensional array A[n].
(a)
total = 0;
for(i=1;i<n;i++)
for(j=i+1;j≤n;j++)
total++;
n
T(n) = ∑n−1
i=0 ∑j=i+1 1
T(n) = ∑n−1
i=0 (n − (i + 1) + 1)
T(n) = ∑n−1
i=0 (n − i − 1 + 1)
T(n) = ∑n−1
i=0 (n − i)
n−1
T(n) = ∑n−1
i=0 n − ∑i=0 i
T(n) = n2 −
T(n) =
T(n) =
(n−1)n
2
2n2 −(n2 −n)
n2 +n
2
2
T(n) = O(n2)
(b)
i = n;
while (i>=0){
sum = sum + A[n];
i = i+2;
}
T(n)
O(n)
=
∑𝑛/2
𝑘=0 1
(c)
s=0; sum1=0; sum2=0;
for (int i=1; i< (1500*n) ;i++){
sum1 = sum1 +1;
}
for (int i=1; i< n ;i++)
sum2 = sum2 +A[i];
for (int i=1;i<n;i*=2)
{
s= s+A[i];
}
We have three parts in the algorithm:
T(n) = 15000n+n+log n
Dominant term is n, so
O(n)
2. (16 points)
Definitions of Big 0,  and  are given below:
O(g(n))={f(n):there exists a positive constant c and n0 such that 0f(n)cg(n), n, nn0}
(g(n))={f(n):there exists a positive constant c and n0 such that 0cg(n)f(n), n, nn0}
(g(n))={f(n):there exists positive constants c 1,c2 and n0 such that 0c1g(n)f(n)c2g(n), n, nn0}
Using the definitions prove that
a.) n2+4n+4 = O(n2)
We have to prove that:
0n2+4n+4cn2 for a positive constant c and n0 for all nn0
Let’s divide all sides by n 2 (note that n is always positive)
01+4/n+4/n2c
If we pick c as 6 and n0 as 2, the equation holds for all nn0.
01+4/n+4/n2 6 is true for all n2.
b.) n3 ≠ O(n2)
We have to disprove the following:
0n3cn2 for a positive constant c and n0 for all nn0
Let’s divide all sides by n 2 (note that n is always positive)
0nc
It is not possible to pick a constant c to hold this inequality, since n is not a constant.
Therefore
n3 ≠ O(n2)
c.) n2+3n = (n2)
We have to prove that:
0 c1n2n2+3nc2n2 for positive constants c1 and c2 and n0 for all nn0
Let’s divide all sides by n 2 (note that n is always positive)
0 c11+3/nc2
If we pick c1 as 1, c2 as 3 and n0 as 3, the equation holds for all nn0.
0 1 1+3/n 3 is true for all n 3.
d.) n3+3n= (n)
We have to prove that:
0cnn3+3n for a positive constant c and n0 for all nn0
If we pick c as 1 and n0 as 1, the equation holds for all nn0.
0n n3+3n is true for all n1.
3. (10 points)
a. What are the Big-Ohs of the following expressions? (No proof is necessary.)
i. 𝑛log3 5 +5n2
O(n2)
ii. n!+9n
O(n!)
iii. n+log n
O(n)
iv. n2+ nlog n
O(n2)
v. n8+(3/2)n
O((3/2)n)
vi. log log n + log n
O(log n)
vii. n3+2n
O(2n)
viii. n+(n2/log n)
O(n2/log n)
3
ix. ( )n+n2
4
O(n2)
x. n1/2+n3/2+n4/5
O(n3/2)
b. Order the Big-Ohs in part a from smallest to the biggest
O(log n) < O(n) < O(n3/2)< O(n2/log n)< O(n2)< O((3/2)n)< O(2n)< O(n!)
4. (18 points)
The product of a x a x a x a x......x a for n times is denoted by an and called nth power of an
integer a. Example: 2x 2 x 2 x 2 x 2 is written as 25, where 2 is the base and 5 is the index.
a. Write a recursive divide and conquer algorithm that finds n th power of an integer. Explain the
code by making comments on the code. You may use any programming language or pseudocode.
Power (a,n)
if a is 1 or n is 1
return a
else
a 
x =  ;
2
p = power (a,x)
if a is even
return p*p
else
return p*p*a
end if
end if
b. What is the efficiency equation T(n) of your code?
T(n) = T(n/2)+ 1
c. What is the growth rate of your code? (You can use any method to prove it.)
T(n) = Θ(lg n)
5. 8 points
Suppose we have the following cities and the distances for 5 cities. We would like to find the minimum
length tour for Travelling Salesman Problem using exhaustive search. Suppose tour starts at city A.
B
2
4
2
A
E
5
3
8
6
5
7
5
C
D
a. How many different paths do exist? Show at least four possible paths and their lengths.
24
A-B-E-D-C-A = 24
A-E-B-C-D-A = 20
A-C-B-D-E-A = 26
A-B-D-C-E-A = 21
b. How many different paths do exist if we add one more city to our graph?
120
c. What is the algorithmic complexity of examining every possible path using exhaustive search?
O((n-1)!)
6. (15 points)
a. Find the runtime T(n) function for the following function and then using Master Theorem
calculate the time complexity.
function funcA(ArrayA,leftIndex,rightIndex,increment){
if (leftIndex==rightIndex) return;
for (i=leftIndex;i<rightIndex;i++)
ArrayA[i]=Array[i]+increment;
 rightIndex - leftIndex 
;
2


mid = leftindex + 
funcA(ArrayA,leftIndex,mid,increment);
funcA(ArrayA,mid+1, rightIndex,increment);
funcA(ArrayA,leftIndex,mid,increment+1);
}
T(n) = 3 T(n/2) + n
Master Theorem Case 1:
a = 3 b = 2 f(n)=n
T(n) = Θ(n𝑛log2 3 )
b. Find the runtime T(n) function for the following function and then using Substitution Method
calculate the time complexity. Use induction to prove your result.
function fact (n) {
if n=0 return 1
else return n * fact(n-1)
}
T(n)= T(n-1)+1
T(n) = Θ(n)
For proving with substitution method, you need to use induction.
7. (12 points)
The Master Theorem applies to recurrences of the following form:
T (n) = aT (n/b) + f(n)
where a ≥ 1 and b > 1 are constants and f(n) is an asymptotically positive function.
There are 3 cases:
1. If f(n) = O(𝑛log𝑏 𝑎−𝜖 ) for some constant 𝜖 > 0, then T (n) = Θ (nlogba).
2. If f(n) = Θ (𝑛log𝑏 𝑎 log k n) with1 k ≥0, then T (n) = Θ (𝑛log𝑏 𝑎 log k+1 n).
3. If f(n) =Ω (𝑛log𝑏 𝑎+𝜖 ) with 𝜖 > 0, and f(n) satisfies the regularity condition, then T (n) =
Θ(f(n)).
Regularity condition: af(n/b) ≤ cf(n) for some constant c < 1 and all sufficiently large n.
Apply master theorem to the following equations. Show your work, and calculate the complexities.
a. T (n) = 16T (n/4)+ n
Master Theorem Case 1:
a = 3 b = 2 f(n)=n
T(n) = Θ(nlg3)
b. T(n) = 3T (n/4)+ n log n
Master Theorem Case 3:
a = 3 b = 4 f(n)=n log n
T(n) = Θ(nlogn)
c. T (n) = 0.5T (n/4)+ 1
Master Theorem can’t be applied:
a = 0.5 b = 2 f(n)=1
a<1
d. T(n) =3T (n/3)+ √𝑛
Master Theorem Case 1:
a = 3 b = 3 f(n)= √𝑛
T(n) = Θ(n)
8. (16 points)
In the following table, you are to examine 5 different sorting algorithms. The algorithms are randomly
distributed within the table. Suppose there is an unsorted string array, we are going to sort it. The
unsorted array is displayed in the first column of the table, and the sorting transform the array into the
one in the second column.
We have applied 5 sorting algorithms to the same array with the same original positioning. Then at some
point of the execution of the algorithms, we have stopped the execution and printed the arrays as they
are at that specific time. So we have 5 semi-sorted arrays which can give you a clue about the algorithms
used. Based on the layout of array in columns I to V, figure out which algorithm is used in which
column. The used algorithms are Bubble Sort, Insertion Sort, Selection Sort, Quicksort and Merge
Sort.
ORIGINAL SORTED
I
II
III
IV
V
bath
acid
acid
acid
acid
azur
acid
tile
ajar
bath
anti
ajar
anti
anti
nine
anti
game
amid
anti
acid
amid
mode
amid
lost
bath
amid
away
away
lost
away
mode
cash
away
amid
bath
game
azur
nine
game
azur
ajar
ajar
acid
bath
tile
lost
bath
bath
cash
anti
bold
anti
mode
nine
cool
cool
cash
cash
cash
nine
cash
bold
doom
amid
cool
amid
tile
mode
cash
fame
away
doom
away
ajar
lost
doom
bold
fame
etik
fame
away
fame
last
etik
ajar
fame
ajar
azur
tile
nine
azur
zoom
game
zoom
bold
zoom
zoom
game
cool
last
cool
cool
cool
fame
last
doom
lost
doom
doom
doom
mode
lost
last
mode
last
etik
last
tile
mode
bold
nine
bold
fame
bold
game
nine
etik
tile
etik
last
etik
etik
tile
azur
zoom
azur
zoom
game
lost
zoom
a. Please write which algorithm is used in the following columns :
Column I
Column II
Column III
Column IV
Column V
:…………..…Insertion……………………………..
:………………Merge……………………………..
:………….…Selection……………………………….
:………….…Quick……………………………….
:………………Bubble……………………………..
b. Give average time complexities of each algorithm
Bubble
:…………..…n2……………………………..
Selection
:………………n2……………………………..
Quicksort
:………….…n lg n ……………………………….
Insertion
:………….……n 2…………………………….
Merge :…………………………n lg n…………………..