exam #1 analysis of algorithms page 1 - baylor...

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Exam #1 Analysis of Algorithms Page 1Open Book June 13, 1995

NAME:_________________________________________________(10 Points)

1. For the following program, what is the order of the running time? For an input valueof N, what value does func1 return? (10 Points)

int func1(int N){

int A,I,J;

A=0;for (I=0 ; I<N ; I++){

for (J=I ; J>=0 ; J--){

A++;}

}return A;

}



int func2(int N){

int A,I,J;

A=0;for (I=0 ; I<N/2 ; I++){

A++;}for (J=I ; J<N ; J++){

A++;}return A;

}



int func3(int N){

int A;

if (N<=0){

return 0;}else{

A = func3(N-1);A += N;return A;

}}


4. What is the worst case order of each of the following sortingalgorithms? (20 Points)

Insertion Sort______________________

Bubble Sort _______________________

Selection Sort _____________________

Merge Sort________________________

Quick Sort ________________________

Heap Sort_________________________

Radix Sort_________________________

5. How many comparisons does Insertion Sort do to sort the followinglist? (10 Points)

1, 3, 5, 7, 0, 2, 4, 6

6. How many comparisons does Selection Sort do to sort the following list? (10 Points)

1, 3, 5, 7, 0, 2, 4, 6


7. How many times must Heapify (DownHeap) be called to convert the following listinto a heap? Show the array after it is converted to a heap. (10 Points)

1, 2, 3, 4, 5, 6, 7, 8

8. Show the lists and the merging process (as I did in class), for MergeSort on thefollowing list. (10 Points)

6, 3, 8, 5, 2, 1, 4, 7

Exam #2 Analysis of Algorithms Page 1Open Book July 20, 1995

NAME:__________________________________________________________

1. What important historical event occured on this date in 1969? (10 points.)


int func1(int N){

int A,I,J,K;

A=0;for (I=0 ; I<N ; I++){

for (J=0 ; J<N ; J++){

for (K=0 ; K<J ; K++){

A++;}

}}return A;

}

Sorry, son. I said “No Extra Books!”


3. For each of the following Algorithms, write the name of the algorithm on the line thatcorresponds to its worst-case running time. Insertion Sort, Bubble Sort, SelectionSort, Merge Sort, Quicksort, Heap Sort, Radix sort.(10 Points)

O(lg n) _______________________________________________________

O(n) _________________________________________________________

O(n lg n) _____________________________________________________

O(n2)_________________________________________________________

O(n3) ________________________________________________________

4. Suppose you have a hash table with 10 entries, each of which is the head of a linkedlist. Suppose you add 100 keys to this table. In the worst case how manycomparisons will it take to determine that a key is NOT in the table? Answer thequestion assuming both a maximum and a minimum number of collisions.(10 Points)

In hashing, as in life,the fewer the collisions, the better.


5. The following is a red-black tree. The fat edges represent RED edges, the thin edgesrepresent BLACK edges. The numbers in the vertices are the keys. Show how thistree will look after adding the key 14. (10 Points)

20

10

15

12 17

8

40

Yet another use for trees.


6. Find the shortest path from A to B. Which vertices are in the tree when the algorithmterminates? (10 Points)

A

B

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

x13x14

x15

x16

x17

x18

x19

x20

x21

x22

x23

x24

x25

1

1

1

1

11

1

1

1

11

1

13

14

20

22

1010

1 1

1

1

11

10

1010

10

1010

10

10

10

10

10

10

10

10

10

1010

10

10

10

While it may be necessary to explore thewrong path, it’s important not to stray

too far in the wrong direction


7. Find the minimum spanning tree of the following graph. Circle the weights that belongto the minimum spanning tree in the following graph. (10 Points)

1

2

3

45

6

7

8

9

10

12

13

1415

1617

1819

2011

21

22

23

24

25

8. Find biconnected components of the following graph. (10 Points)

A

B

C

D

E

F

G

H

J

K

BiDirectional Connections


9. Find the strongly connected components of the following graph. (10 Points)

A B

C

D

E

F

G

H

JK L

M

N

P

10. What does it mean to say that a problem, such as Satisfiability, is NP-Complete? (10Points)

Miller Time

Analysis of Algorithms Test #1 Oct 3, 1995

100 Points Open Book, Open Notes Full Period

Page 1

NAME ______________________________________________________________

1. For an input N, how many times does the following program print “Hello World”?What is the order of the algorithm? (10 Points)

Prog1(N:Integer)begin

I := 0;While (I<N)

Print “Hello World”;I := I + 1;

EndWhileend


Prog1(N:Integer)Begin

For I := 1 to N doFor J := 1 to I do

Print “Hello World”;EndFor

EndForEnd



Page 2

3. For an input N, how many times does the following program print “Hello World”?What is the order of the algorithm? (Remember the summations that appear inChapter 1.) (10 Points)



For K := 1 to I doPrint “Hello World”;

EndForEndFor

EndForEnd



if N > 0 ThenFor I := 1 to N do

Print “Hello World”;EndForProg1(N/2);Prog1(N/2);

EndIfEnd



Page 3

5. Place the following functions in ascending order by growth rate. If two functions havethe same growth rate, say so. (20 Points)

1. n2

2. log10 n

3. 2n

4. n

5. n3

6. nn

7. ln n

8. 2n + n5

9. n3 + 2n2 + n + 1

10. 3n

11. lg n

12. 5n

6. Professor Gooms of the University of Florida has invented a new algorithm forsearchng an ordered list. The algorithm has no loops in it. There is somethingsuspicious about this. What is it? (10 Points)

My growth rate has always exceededlg n!



Page 4

7. In Quicksort, all comparisons are done in the Split algorithm. If you make the splitalgorithm twice as fast, what happens to the order of the algorithm? What happens tothe actual running time? (10 Points)

8. Fill in the following table. (20 Points.)

Algorithm Worst Case Order Average Case Order Extra Space Required

InsertionSort

QuickSort

MergeSort

HeapSort

(Well, Almost Nothing!)

There’s nothing more repulsive than an Ugly Test!

Analysis of Algorithms Test #1 Nov 7, 1995


Page 1

NAME ______________________________________________________________

1. Find the biconnected components in the following graph. (20 points.)

1

2

3

4

5

6 7

8

9

10

11

12

13

(You may not need to use all lines.)

Component Number Vertices

1

2

3

4

5

6

7

8

9



Page 2

2. Find the strongly connected components of the following graph. (20 points)

1

2

3

4

5

6

7 8

9 10

11

12

13

14

15

16

17

18



1

2

3

4

5

6

7

8

9

10



Page 3

11

3. Find the minimum spanning tree for the following graph. Indicate which edges arepart of the minimum spanning tree. (10 points)

x1

x2

x3

x4

x5

x6

x7x8

x9

x10

x11

x12

x13 x14

x15

x16

x17 x18

x19

x20

x21

1 2

3

4

5

6

7

8

9

10

11

12

13

14

1516

17

1819

20

21

22 23

24

25

2627 28

29

3031

32

4. In the following graph, find the shortest path from A to B. Indicate which edges arepart of the shortest path. (10 points)

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

x13

x14

x15

x16

x17

x18 x19

x20

x21

x22

x23

x24

x25

x26

x27 x28

x29

x30

A

B

1

2

1

2

3

12

3

1

2

2

3

1

1 1

1

22

3

12

1

14 18

8

1214

3 7

10

12

2410

10

15

7

22 1

1

11

1

1

32

27

4522 26

31

34

OK, what happened to all the bugs?



Page 4

5. Suppose the MST algorithm has been run on the following graph, starting with vertexx1. Suppose that vertex x5 has just been added to the graph. At this point, whichvertices are in the tree, which vertices are Fringe vertices, and which vertices areunseen? (10 points)

x1

x2

x3

x4

x5x6

x7

x8 x9 x10

x11

x121

2

34

5

6

7

8

9

1011

12

13

14

15

16

17

1819

Vertex Tree/Fringe/Unseen

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

Yeah! What happened to all the bugs?



Page 5

x12

6. Suppose the Shortest Path algorithm is being used to find the shortest path from A toB. Suppose that vertex x5 has just been added to the tree. At this point, whichvertices are Tree vertices, which vertices are Fringe vertices, and which vertices areUnseen? For the Tree vertices and the Fringe vertices, what is the current distancefrom vertex A when x5 is added to the tree? (10 points)

x2

x3x4

x5

x6x7 x8

x9

x10 x11

AB

11

1

1

1

10

10

10

10

10

1010

10

10

10

10

1010

1010

You ate them all yourself, didn’t you.?



Page 6

Vertex Tree/Fringe/Unseen Distance

A

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

B

7. For the following graph, list the vertices in the order they would be visited during adepth-first-search, and during a breadth-first-search. Start with vertex x1. (20 points)

x1

x2

x3 x4

x5

x6

x7

x8

x9

x12

x11

x10



Page 7

Order Depth FirstSearch

Breadth First Search

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

11th

12th

And now for something

completely different!

Analysis of Algorithms Test #3 Dec 7, 1995


Page 1

NAME ______________________________________________________________

1. What is the order of the following algorithm? How many times does the followingalgorithm print the word “Hello”? Express both results as a function of N. (15points.)

MyAlg(long N){

for (I=0 ; I<N ; I++){

for (J=0 ; J<N ; J++){

for (K=0 ; K<N ; K++){

for (L=0 ; L<K ; L++){

Print "Hello!";}

}}

}}



Page 2

2. Suppose someone figured out that it was possible to compute the value of a 2x2matrix using only 4 multiplications and 10 additions. Suppose that it was notnecessary to use the commutative law anywhere in this method. How fast wouldStrassen’s Matrix Multiply be if you used this method. (Use recurrence relations tojustify your answer.) (15 Points)



Page 3

3. The following 16 numbers are being used to compute a 16 point discrete FFT (as inthe book.) First 8 2-point FFT’s are computed, then 4 4 point FFTs then 2 8 pointFFTs and finally 1 16 point FFT, as in the diagram below. Write the numbers in thecorrect order for computing the 2-point FFTs. (15 Points)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

OriginalNumbers

Write thenew numbershere.



Page 4

4. What does it mean to say that a problem P is NP-Complete? (20 Points)

5. Give a diagram similar to that on P. 214 for the string ABABBAABA. (10 Points.)



Page 5

6. The following is an example of a straight-line program for the polynomial x2+2x+1.temp1 = x*x;temp2 = 2*xFinal = temp1+temp2+1

Create a straight-line program for 3x4+5x3+7x2+4x+2, but make sure that yourprogram is optimal. (10 Points)



Page 6

7. Show the following red-black tree after adding the element 45. The fat edges are red,the black edges are skinny. (15 Points)

60

30

35

4033

25

12

82

9677



Page 1

NAME ______________________________________________________________



For I := 1 to N+1 DoPrint “Hello World”;

EndForI := 0;While (I<N)

Print “Hello World”;I := I + 1;

EndWhileend



For I := 1 to N doProg2(I);

EndForEnd


For I := 1 to N doPrint “Hello World”;

EndForEnd



Page 2

3. For an input N, how many times does the following program print “Hello World”?What is the order of the algorithm? (Remember the summations that appear inChapter 1.) (10 Points)



Print “Hello World”;EndForFor K := N/2 to N do


EndForEnd

4. For an input N, how many times does the following program print “Hello World”?(Order is sufficient.) What is the order of the algorithm? (10 Points)

Prog1(N:Integer)



Page 3

Beginif N > 0 Then

For I := 1 to N doPrint “Hello World”;

EndForProg1(N/2);

EndIfEnd



Page 4


1. n2

2. ln n

3. 2n

4. n/10000

5. n5.1

6. n!

7. ln2 n

8. n5

9. n5lg n

10. (2.1)n

11. lg n

12. n2+2n+1

6. Suppose a list of numbers, is sorted into reverse order. Is this list a heap? Inparticular, is the following list a heap? Justify your answer.(10 Points)

10 9 8 7 6 5 4 3 2 1



Page 5

7. Show How Merge Sort will sort the keys in this list. (15 Points)

8 7 6 5 4 3 2 1

8. Show how the Quicksort Split Algorithm will split the following list. Show theposition of the pivot point. (15 Points.)

7 5 14 12 16 2 13 1 6 4 11 10 8 3 15 9



Page 1

NAME ______________________________________________________________


1 2

3

45

6

78 9 10

1112

1314

15 1617

1819

20


Component Number VerticesVertices

11

22

33

44

55

66

77

88

99



Page 2

1010


1 23 4

56

7 8 9

10 11

1213

1415

16 17

18

19 20(You may not need to use all lines.)


1

2

3

4

5

6

7

8

9

10



Page 3

11

3. Find the minimum spanning tree for the following graph. Indicate which edges arepart of the minimum spanning tree. (15 points)

1 2 3

4 5 6 7

8 9 10 11 12

13 14 15 16

17 18 19 20

1

2

345

67

8

9

10

11

12

13

1415

16

17 18

19

20

21

22

23

24 25

26

27

28

29

30 31

32

3334

35

3637

38 3940

41

42

43 4445

Draw the edges of the MST into the following diagram.

1 2 3

4 5 6 7

8 9 10 11 12

13 14 15 16

17 18 19 20



Page 4

4. In the following graph, find the shortest path from A to B. Indicate which edges arepart of the shortest path. Fill in the table with minimum distances. (15 points)

AB

12

3

45

6 7 89

10 11 1213

14 1516

17182

7

2

6

12

14

12

15

1 1

1

11

1

2 2 2

2

2

119

18

17

19

252119

28

Distance from A to: Minimum Distance

B

11

12

13

15

1



Page 5

5. Show the Binary Search Tree that you get when you add the following integers in thegiven order. (10 points)

10, 9, 30, 55, 12, 17, 15, 31, 22, 81, 6, 1, 5

6. In the following binary search trees, the fat edges represent red edges, while the thinedges represent black edges. This tree is illegal. Show how to fix it. (10 points)

40

51

67

8259

63

65

61

38

12

9 20

47

55

57



Page 6

7. For the following graph, list the vertices in the order they would be visited during adepth-first-search, and during a breadth-first-search. Start with vertex 1. (10 points)

12 3

4 56

7 8 910

1112

1314

151617

1819

2021



1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16th17th18th19th20th



Page 7

21st



Page 1

NAME ______________________________________________________________

1. What is the order of the following algorithm? (15 points.)

MyAlg(long N){

if (N>0){

Print “HELLO”MyAlg(N/2);MyAlg(N/2);

}}

2. Professor Gooms of the University of Florida has invented a new Matrix MultiplyAlgorithm, similar to Strassen’s Matrix Multiply, except that Professor Gooms’salgorithm requires 16 multiplications and 2 additions to multiply a 2x2 matrix. IsProfessor Gooms going to win the Turing Award this year because of this? Why orwhy not, and PLEASE BE SPECIFIC!(15 Points)



Page 2

3. The following 16 numbers are being used to compute a 16 point discrete FFT (as inthe book.) First 8 2-point FFT’s are computed, then 4 4 point FFTs then 2 8 pointFFTs and finally 1 16 point FFT, as in the diagram below. Write the numbers in thecorrect order for computing the 2-point FFTs. (15 Points)



Page 3

4. What does it mean to say that a problem Q is NP-Complete? (20 Points)

5. Give a diagram similar to that on P. 214 for the string AAAABBAAAABA. (10Points.)



Page 4

6. Part 1: The following is an example of a straight-line program for the polynomialwrite an optimal program for computing the value of 5th degree polynomials. Thecoefficients will be supplied in a global array A, which is indexed from 0 through 5.Part 2: While you’re at it, write an optimal program for computing x13, assumingthat x is another global variable. (10 Points)



Page 5

7. Show the following red-black tree after adding the element 45. The fat edges are red,the black edges are skinny. (15 Points)



Page 1

NAME ______________________________________________________________



If N > 1 ThenFor I := 1 to N do

Print “Hello World”EndforProg1(N/2);

Endifend



For I := 1 to N doFor J := I to N do


EndForEnd



Page 2



For I := 1 to N doFor J := 1 to N do

For K := 1 to N doPrint “Hello World”;

EndForEndFor

EndForEnd

4. Professor Gooms from the University of Florida claims to have invented a new Mergealgorithm that can merge two lists in-place doing only 1 comparsion. If you used thisalgorithm in Merge Sort, what would the order of this new algorithm? ProfessorGooms claims that it would be Θ(lg2 n). Is he correct? Is it likely that his algorithmreally works? (20 Points)



Page 3


1. n3

2. lg n

3. 1.00001n

4. n/10000

5. n1024

6. n!

7. lg2 n

8. n3.0001

9. n3lg n

10. nn

11. log30 n

12. n

6. Is the following list a heap? Justify your answer. If it isn’t a heap, show how the heapconstruction portion of the Heapsort algorithm will convert this to a heap. (10 Points)

3 8 4 2 7 9 1 10 5 6



Page 4

7. Sort the following binary numbers using radix sort. Show each step. (10 Points)

Are all seven steps necessary? If not, say why not, and skip the extra steps.

Number Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 701010

10101

11111

10100

10111

01011

01111

01100

01101

11101

00011

00101

00001

01110

11110

11100



Page 5

8. Show how the Quicksort Split Algorithm will split the following list. Show theposition of the pivot point. (10 Points.)

8 3 13 9 2 7 10 1 14 6 12 4 15 5 11



Page 1

NAME ______________________________________________________________



Component Number VerticesVertices

11

22

33

44

55

66

77

88

99

1010

1 2

3

4 5

6

7

89

1011

12

13

14

15

16

17 19

18

20



Page 2




11

22

33

44

55

66

77

88

99

1010

1111

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

1920

21



Page 3

3. Find the minimum spanning tree for the following graph. Indicate which edges are

part of the minimum spanning tree. (15 points)

Draw the edges of the MST into the following diagram.

1 23

4

56

7 89

1011 12 13

14

15

16 17

1819

20

21

1 23

4

56

7 89

1011 12 13

14

15

16 17

1819

20

21

1

1

2

1

1

11

1

1

1

1

2

2

2

2

2

2

2

2

3

3

3

4

56

7 8

9

4

7

8

9

67

89

6



Page 4

4. In the following graph, find the shortest path from A to B. Indicate which edgesare part of the shortest path. Fill in the table with minimum distances. (15 points)

Distance from A to: Minimum Distance

BB

1111

1212

33

66

1616

1

1

1

1

1

1

11

12

2

2

2

2

3

43

3

4 4

4

5

55

6

7

6

7

8

7

9

AB

1

2

3

4

56

789

10

11

12 13 14

15

16

17

18



Page 5

5. Assume that you have run the Biconnected Component algorithm on the followinggraph, and have obtained the following start times. Fill in the table by providing theback value for each vertex. (15 points)

Vertex Start Time Back Value

1 1

2 3

3 4

4 9

5 5

6 8

7 13

8 10

9 6

10 2

11 11

12 7

13 12

1

2

3 5

6

4

7

8

910

11

1213



Page 6

6. Show the shortest-path TREE that will be computed by Dijkstra’s shortest pathalgorithm for the following graph. All edges have the weight 1. The shortest pathcomputation starts at Vertex A, and ends at Vertex B. (15 points)

A

B



Page 7

7. For the following graph, list the vertices in the order they would be visited during adepth-first-search, and during a breadth-first-search. Start with vertex 1. (10 points)



1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th

1

2

3

4

5

6

7

8

9

10 1112 1314 15



Page 1

NAME ______________________________________________________________

1. Show the binary search tree that will be created if the following values are added inthe given order. (15 points.)

5, 4, 2, 1, 3, 9, 7, 6, 10, 13 11, 12, 17, 32, 14



Page 2

2. Given the following two matrices, how many multiplications will be done byStrassen's Matrix Multiply to compute the final result? Show the sub-matrices thatwill be used at the first level of recurrsion.(15 Points)

7 9 20 13 122 176 124 149

63 32 11 27 113 191 179 164

22 55 89 37 143 183 132 199

21 28 91 66 121 101 155 133

Number of Multiplies:

Sub Matrices:



Page 3

3. What does it mean to say that a problem R is NP-Complete? (20 Points)

4. Write an optimal straight-line program for computing x11. (15 Points.)



Page 4

5. Using only addition and multiplication, write an optimal program for computing the

polynomial: Ax5+Bx4+Cx3+Dx2+Ex+F (15 Points)



Page 5

6. Show the following red-black tree after adding the elements 76 and 67. The fat edgesare red, the skinny edges are black. (20 Points)

40

25 60

72

81

12

195

32 50

42 68

65

exam #1 analysis of algorithms page 1 - baylor...

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