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MCIS 502

Mathematics in Computing

Exam 4

Due: Aug. 12, 2013, 11:55 PM <-- NOTE!!

Cutoff: Aug 12, 2013, 11:55 PM <-- NOTE!!!

The following questions are based on Chapters 10 and 11 of the

Rosen text.

Do your own work. Don't plagiarize!

Submit the completed exam to Blackboard, Assignments, Exam 4.

DO NOT INCLUDE PROOFS OR COMMENTS FOR YOUR WORK IN THIS SECTION.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PART A

Each question is worth 2 points. Choose True of False as the most appropriate answer.

1. The degree(v) of a pendant vertex may be either one or zero.

T or F

2. A tree is any connected, undirected graph with an odd number

of vertices.

T or F

3. A simple graph is an undirected graph with multiple edges but

no loops.

T or F

4. A multigraph is an directed graph with multiple edges and no

loops.

T or F

5. Consider the following directed relations on {1, 2, 3, 4} :

R = {(1,1), (2,2), (3,3), (4,4)}

T = {2,1), (3,1), (3,2)}

R is reflexive and T is transitive

T or F

6. Set A is divided into several disjoint partitions. The

UNION of these partitions is the original set.

T or F

7. A W16 has 17 vertices and 32 edges.

T or F

8. The root of any tree must be at either level 1 (one) or level

0 (zero).

T or F

9. A leaf is a vertex with no children.

T or F

10. A weighted graph has a value assigned to each edge.

T or F

11. The minimum spanning tree of a weighted graph is a graph

That is drawn with the length of each edge roughly

proportional to the value assigned to each edge.

T or F

12. Siblings must have the same parent but not necessarily the

same level.

T or F

13. Since Prim's and Kruskal's algorithms generate the minimum

spanning tree of a given weighted graph, each algorithm

would always provide identical MST solutions.

T or F

14. Bipartite graph, Kn,m, has (n+m) vertices and a maximum of

(n*m) edges.

T or F

PART B

Each question is worth 6 points unless noted otherwise. Provide your interim solutions to all problems.

Partial credit will be given for incorrect solutions if the process was shown and I can identify the error. Problems with incorrect answers and no intermediate work will be graded as zero.

Problems that require a particular approach - such as Prim and Kruskal - need a reasonable amount of intermediate work to identify and verify the solution process.

1. Form a binary search tree from the words of the following

sentence using alphabetical order and inserting words as they

appear in the sentence:

This test is easier than before because it is much shorter.

2. The expression below is in postfix expression form.

Determine its numerical value.

{ 3, -7, +, 6, 3, /, 2, *, / }

4. Define a postorder and preorder traversal of the following:

(3 points each)

[(-3 – 2y) / 5 ] + [(-y + 3) * 6) ]

a. postorder:

b. preorder:

5. Determine the Minimal Spanning Tree in Graph Y using

Kruskal’s Algorithm. All edges must be labeled from lower to

higher named vertices, e.g., from "c" to "d" but not from "d"

to "c".

6. Given the coding scheme:

a:001, b:0001, e:1, r:0000, s: 0100, t:011, x:01010

Find the words represented by: (1 point each)

a. 0001110000

b. 001010101

c. 01110100011

d. 0010000011

e. What is the best compression ratio (versus ASCII 8-bit

encoding) of the words in a through d above? (2 points).

7. Determine the Minimum Spanning Tree in Graph X. Use Prim's

Algorithm in which all edges must be labeled from lower to

higher named vertices, e.g., from "c" to "d" but not from "d"

to "c"

8. Construct a postorder, inorder and preorder transversal of

Tree T.(2 points each)

a. postorder:

b. inorder:

c: preorder:

10. Suppose that a full 37-ary tree has 4 internal vertices.

11. What is the shortest path in Graph S between "a" and "z".

Use Dijkstra's algorithm.

a. the shortest path is:

b. the shortest distance between "a" and "z" is:

12. A tree has 52 edges. How many vertices does it have?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

EXTRA CREDIT - OPTIONAL

DO ONE of the following:

A.

(3 points or no points)

Use a greedy algorithm to determine the shortest path in

Graph S. The algorithm starts at vertex “a” and ends at vertex “z” always selecting the shortest edge. The selection must be in ascending lexicographic order, i.e., m to n - not n to m. See discussion on pages 195, 232, and

798.

B.

(3 points or no points)

Develop the Basis Step of the algorithm to determine the

number of terms (cardinality) of the union of n mutually

For example, the cardinality of the union of three

mutually intersecting sets is

C(3,1) + C(3,2) + C(3,3) = 3+3+1 = 7.

C.

(1 point or no points)

Is the solution using Prim ’s Algorithm in Question B.5 the

same topology and length as the required Kruskal solution?

GRAPH INFORMATION

Graph G

Initially draw a hexagon with vertices a-b-d-f-e-c-a.

Connect vertices a to f; b to c; d to e.

b d a f

c e

Graph H

Initially draw a hexagon with vertices u-v-w-x-y-z-u.

Connect vertices u to x; v to y; w to z.

There is no connection in the center.

u

z v

y w

x

Graph S

Initially draw a hexagon with vertices a-b-d-z-e-c-a.

Connect vertices b to c; b to e; c to d; d to e.

Edge values are:

a-b = 3; a-c = 4;

b-c = 1; b-d = 5; b-e = 5

c-d = 2; c-e = 4;

d-e = 2; d-z = 5; e-z = 3.

b d

a z

c e

Tree T

Construct a Tree with

vertex a at level 0;

vertices b, c and d at level 1;

vertices e, f, i, j, and k at level 2;

vertices g, h, l and m at level 3.

Connect vertex a to b, a to c, and a to d.

Connect vertex b to e and f.

Connect vertex c (no further connection).

Connect vertex d to i, j and k.

Connect vertex e to g.

Connect vertex f (no further connection).

Connect vertex i to h.

Connect vertex j to l and m.

Connect vertex k (no further connection).

Connect vertex g, h, l and m (no further connection).

a

b c d

e f i j k

g h l m

Graph X

Initially draw a rectangle with vertices a-c-e-z-d-b-a.

Connect vertices a to d; c to d; d to e.

Edge values are:

a-b = 1; a-c = 4; a-d =3;

b-d = 3; c-d = 2; c-e = 2;

d-e = 2; d-z = 2; e-z = 2.

a c e

b d z

Graph Y

Draw a hexagon with vertices a-b-d-z-e-c-a.

Connect vertices b to c; b to z; d to e.

Edge values are:

a-b = 3; a-c = 3;

b-c = 2; b-d = 5; b-z = 4;

c-e = 5;

d-e = 1; d-z = 7; e-z = 3.

b d

a z

c e

Graph Z

Graph Z is a five-pointed figure.

Connect a to b, a to c and a to e.

Connect b to d.

Connect c to d.

Connect d to e.

b c

a d

e

Good Luck!

END.

502q4c13b