CS 1501 Algorithm Implementation

Exam 2

 

 

SOLUTIONS

 

 

 

 

Problem

Possible Pts

Pts Received

1

20

 

2

10

 

3

12

 

4

12

 

5

10

 

6

10

 

7

8

 

8

10

 

9

8

 

Total

100

 

 

For all questions, be sure to show your work.  Answers without work will not receive full credit.

1)      (20 points – 2 points each) Fill in the Blanks. Complete the statements with the MOST APPROPRIATE word(s) and/or phrase(s).

a)      The sum 1 + 2 + 3 + … + N evaluates to _________N(N+1)/2___________________.

b)      If a program's run-time can be modeled by the function   8N5/2 + 2N2(3lgN + 6N) , the program's Theta runtime growth rate is _________Theta(N3)_____.

c)      The Boyer-Moore string matching algorithm can do as few as Theta (______N/M_________________) comparisons for an unsuccessful search of a pattern of length M within a string of length N.

d)      Collisions can be avoided in a hash table as long as the table size, M, is greater than or equal to ___the size of the key space___________.

e)      A block code containing K bits can represent up to ________2K_______________ distinct characters.

f)       The Unix compress implementation of LZW tries to use as few bits as possible in the codewords that it outputs, so as to maximize compression.  However, it still allows for a large number of distinct codewords.  It accomplishes both of these goals by _____starting with 9 bits and increasing the number of bits as needed_________________________________________.

g)      Run-time analysis of recursive algorithms is typically done via ____recurrence relations________.

h)      An undirected graph with V vertices has a minimum of _____0_________________ edges and a maximum of ____(V-1)(V)/2__________ edges.

i)        PFS on a graph with V vertices and E edges runs in time ______Theta([V+E]lgV)_____ with an adjacency list and in time ____Theta(V2)_____________ with an adjacency matrix.

j)        A sequence of N Inserts followed by N DeleteMins on a sorted array will have a total run-time of Theta(________N2___________________), but the same sequence on a min-heap will have a total run-time of Theta(_________NlgN_________________).

 

2)      (10 points – 2 points each) Indicate if each of the following statements is TRUE or FALSE.  For FALSE statements, INDICATE WHY THEY ARE FALSE.

a)      The "branch and bound" technique reduces worst case run-times for search algorithms from exponential to polynomial run-times. False – in the worst case they are still exponential

b)      DLBs are an improvement over regular multiway tries because they have faster search times.  False – it improves in terms of memory useage

c)      If someone comes up with a factoring algorithm that is 1,000,000 times faster than current factoring algorithms, RSA will no longer be a useful encryption scheme.   False – the improvement is not asymptotic, so RSA could still be used with larger key sizes.

d)      NP-Complete problems are problems that are known to require exponential run-times.  False – it is believed but has not been proved.

e)      If I discover a true polynomial algorithm that solves the Traveling Salesman Problem, I will have proved that P = NP.  True

3)      (12 points – 4 + 4 + 4) Consider RSA encryption

a)      Show (using pictures and explanation in detail) how an RSA Envelope (or digital envelope) works, and why it is used.

See Powerpoint slides

 

 

 

 

 

 

 

 

 

 

 

 

 

b)      Creating new RSA keys requires generating large random prime integers.  Explain the general approach discussed in lecture for generating large random prime numbers, and how its run-time can be determined (note: since this is only a general algorithm, a specific overall run-time is not required).

 

Consider the pseudcode below:

            Generate Large Random Integer X using a good random number generator

            While (!isPrime(X))

                        Generate Large Random Integer X

The overall run-time can be determined as follows:

            (Expected number of iterations of the loop) * (Time required to test primality)

Note that the Miller-Rabin Witness algorithm is an efficient way of testing primality, but it was not required in this answer.

 

 

 

 

 

 

 

 

 

 

c)      I'd like to send my friend a message such that 1) Only he can read it (no one else) and such that 2) He can verify that I was the sender and that it wasn't tampered with.  Explain how I could do this using RSA.  Assume that any RSA keys can be authenticated to their owners.

 

I need to encrypt and digitally sign my message.  Assume my keys are EM and DM and my friend's keys are EF and DF.  I can solve this problem in the following way:

            Encrypt the message with my friend's public key, EF

            Digitally sign the resulting message with my private key, DM

            Send the signed message to my friend

Upon receiving the message, my friend will

            Verify the signature using my public key, EM

            Decrypt the message using his private key, DF

 

 

4)      (10 points – 6 + 4)  Consider the simple, naïve algorithm (using a loop) we discussed in lecture for raising an N-bit integer to an integer power (i.e. XY for N-bit integers X and Y).

a)      Write a Java method to calculate this value, using the BigInteger class.  Your method should return a new BigInteger that is the current BigInteger raised to the argument BigInteger power.  Use the header below:

public BigInteger pow(BigInteger Y)  // Return this raised to the Y

{                                    // power

                  BigInteger ans = new BigInteger("1");

        BigInteger count = new BigInteger("1");

        while (count.compareTo(Y) <= 0)

        {

             ans = ans.multiply(this);

             count = count.add(BigInteger.ONE);

        }

        return ans;

   }

   Note that this method is assuming access to the BigInteger class.  In reality, we would have to write it with two arguments.

 

 

 

b)      Assuming the Gradeschool algorithm is used for multiplication, state and thoroughly justify the Theta run-time for exponentiation using the code you wrote in part a) above in terms of N.

The loop in the method iterates Y times.  At each iteration, a multiplication must be done.  Since Gradeschool requires Theta(N2) time, the total run-time will be YN2.  Since Y is an N-bit integer, it can have a value of up to ~2N in the worst case.  Thus, the total run-time is Theta(N22N).

 

 

5)       (10 points) An array representation of a min-heap data structure is shown below.  Draw the resulting arrays after the operations  Insert(28) followed by DeleteMin().  For full credit show the array as it would look after EACH operation.

1

2

3

4

5

6

7

8

9

10

 

15

30

20

35

60

55

25

45

50

80

 

 

1

2

3

4

5

6

7

8

9

10

11

15

28

20

35

30

55

25

45

50

80

60

 

 

 

 

 

6)      (16 points – 8 + 8)  Consider the graph below with the edge weights shown.  Assume the vertices are stored in alphabetical order, using an adjacency matrix.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


a)      Complete the table below, as it would look after a Depth-First Search Spanning Tree (starting from vertex A) were created for the graph.  val[] is the DFS visit order for the vertex, and dad[] is the parent vertex in the DFS tree.  Show your work above or in the space below the table for partial credit.

 

 

A

B

C

D

E

F

G

H

I

val

1

2

3

4

8

6

9

7

5

dad

-

A (1)

B (2)

C (3)

H (8)

I (9)

E (5)

F (6)

D (4)

 

 

 

 

 

 

 

b)      Complete the table below, as it would look after a Minimum Spanning Tree (starting from vertex A) were created for the graph, assuming Prim's Algorithm is used.  val[] is the edge weight associated with the vertex, and dad[] is the parent vertex in the MST.  Show your work above or in the space below the table for partial credit.

 

 

A

B

C

D

E

F

G

H

I

val

0

2

5

9

3

6

7

4

8

dad

0

A (1)

B (2)

C (3)

H (8)

C (3)

E (5)

F (6)

D (4)

 

 

 

7)      (8 points) Consider the weighted graph below. The numbers are the edge capacities. S is the source vertex and T is the sink vertex.

 

 

 

 

 

 

 

 

 

 

 

 


Using the BREADTH FIRST SEARCH implementation of the Ford-Fulkerson algorithm, show EACH AUGMENTING PATH generated (in the correct order that the paths are generated), the amount of flow for each path, and the Maximum Flow for the graph.  Assume that an adjacency matrix implementation is used, and that the vertices are listed in the adjacency matrix in the following order: S, A, B, C, D, E, T.   To ensure partial credit, be sure to SHOW YOUR WORK.

           

            SADT:             60

            SBDT:             40

            SCDAET:        60

            -----------------------

            TOTAL:          160

           

            Note that the last path has a backward edge in it.  See notes and text for more information.

 

8)      (8 points)  Consider the 2-OPT local search algorithm for the Traveling Salesman Problem. Consider the complete graph below and an initial tour of the graph ABCDEA of weight 25.

 

 

 

 

 

 

 

 

 

 

 

 


Show all of the neighbors of this tour, indicate the neighbor that is chosen and show the resulting tour.  Note that this is just ONE ITERATION of the 2-OPT algorithm. Show your work.

 

Neighbors possible:

 

OLD EDGES        NEW EDGES             DIFFERENCE

   AB, CD                AC, BD                         -- 3

   AB, DE                 AD, BE                          -- 3

   BC, DE                 BD, CE                          -- 2

   BC, EA                 BE, CA                          -- 4

   CD, EA                CE, DA                         -- 2

 

The best improvement of –4 results from the 4th neighbor shown above.  This results in the new tour: ABEDCA with length 21.

9)      (8 points – 4 + 4)  Consider the size and store arrays below, as discussed in lecture

 

index

1

2

3

4

5

6

size

3

10

5

6

4

14

 

index

1

2

3

4

5

6

7

8

9

10

11

12

store

-

-

1

5

3

4

5

3

4

2

4

5

 

 

a)      Fill the values into the store array as it would look after a dynamic programming solution to the Subset Sum problem were found for M = 12.  Show any work below.

 

 

 

 

 

 

 

b)      Assuming that there are N items in the size array, what is the run-time of this algorithm?  Why is this run-time only considered to be "pseudo" polynomial?  Explain.

 

Based on the code for the dynamic programming subset sum solution, the run-time is Theta(NM).  This is pseudo-polynomial because there is no limit on the value for the bound, M.  Thus, M could be exponential relative to N, and the overall run-time could therefore also be exponential.