1
Discrete A 0728
Final Exam Review (Version A)
Name: ___________________________________
Chapter 1
For an election with 4 candidates A,B, C, and D, we have the following
preference schedule:
1.
Who is the winner by Plurality?
Rank the players using EXTENDED Plurality.
2.
Who is the winner by Borda Count?
Rank the players using EXTENDED Borda Count.
3.
Who is the winner by Plurality –With-Elimination?
Rank the players using EXTENDED Plurality With Elimination.
4.
Who is the winner by Pairwise comparisons?
Rank the players using EXTENDED Pairwise comparisons.
5.
Is there a Condorcet Candidate? Who?
Number of voters
1st choice
2nd choice
3rd choice
4th choice
20
A
B
C
D
14
C
B
D
A
8
D
B
A
C
7
C
D
B
A
Chapter 2
6.
Consider the weighted voting system [12:11, 5, 5].
Are any players dictators? Do any players have veto power? Are any players dummies?
7.
Consider the weighted voting system [q:10, 8, 6] in which a strict majority of the votes is needed
to pass a motion. What is the value of the quota?
8.
Consider the weighted voting system [30: 24, 12, 8, 4, 2]. What is the minimum percentage of
votes needed to pass the motion?
9.
Evaluate:
200!
198!
Consider the weighted voting system: [35: 32, 15, 10, 3] with four players P 1, P2, P3, and P4.
10.
What is the weight of the coalition: [P2, P3, P4]?
Banzhaf Coalitions: 4 Players
Is this a winning coalition?
{P1}
{P1,P2}
{P1, P2, P3}
11.
Does any player have veto power in the weighted voting system?
{P2}
{P1,P3}
{P1, P2, P4}
What does this imply about any winning coalitions?
{P3}
{P1,P4}
{P1, P3, P4}
{P4}
{P2,P3}
{P2, P3, P4}
12.
List the winning coalitions. How many are there?
{P2,P4}
{P1, P2, P3, P4}
{P3,P4}
13.
Which players in the coalition [P1, P3] are critical players?
14.
Which players in the coalitions [P1, P3, P4] are critical players?
15.
Find the Banzhaf Power Distribution for the weighted voting system.
2
D
A
B
C
2
Sequential
Coalitions:
3 Players
Consider the weighted voting system: [10: 7, 5, 4] with three players P 1, P2, and P3.
16.
Which player in the sequential coalition <P1, P2, P3> is pivotal?
17.
Which player in the sequential coalition <P3, P2, P1> is pivotal?
18.
In how many sequential coalitions is P2 the pivotal player?
19.
Find the Shapley-Shubik Power Distribution for the weighted voting system.
[P1,P2,P3]
[P1,P3,P2]
[P2,P1,P3]
[P2,P3,P1]
[P3,P1,P2]
[P3,P2,P1]
Chapter 3
Ned buys a chocolate-strawberry-vanilla cake for
$14.00. He cuts the cake into six 60-degree
wedges as shown. Ned values strawberry twice as
much as vanilla. Ned values chocolate twice as
much as strawberry.
20.
4
chocolate
strawberry
How much is each flavor of the cake worth
to Ned?
3
5
30°
vanilla
2
6
30°
1
21.
How much is each 60-degree slice of cake
worth to Ned?
Four players (A, B, C, and D) agree to divide a cake fairly using the lone-divider
method. The table shows how each player values each of the four slices that have
been cut by the divider.
22.
Assuming all players play honestly, who was the divider?
23.
What should each players bid be?
24.
Describe a possible fair division of the cake.
A
B
C
D
S1
20%
25%
15%
24%
S2
32%
25%
15%
24%
S3
28%
25%
30%
24%
S4
20%
25%
40%
28%
Five players agree to divide a cake using the last diminisher method. The players play in the following order: A, B, C,
D, E. Suppose there are no diminishers in round 1. In Round 2, C and D are the only diminishers.
25.
Which player gets his fair share at the end of round 1?
26.
Who cuts the cake at the beginning of round 2?
27.
Who gets his fair share at the end of round 2?
28.
Who cuts the cake at the beginning of round 3?
29.
How many rounds does it take to divide the cake among five people?
3
Four heirs (A,B,C, and D) must fairly divide an estate consisting of two items—a house and a cabin—using the method
of sealed bids. The players’ bids (in dollars) are:
A
B
C
D
House 195,000 212,000 201,000 182,000
30.
What is the original fair share of each player?
Cabin 45,000 36,000 35,000 42,000
31.
What does the initial allocation look like?
32.
Is there a surplus? If so, how much?
33.
Describe the final allocation of goods and money.
Four players agree to divide the 12 items below using the method of markers. The players’ bids are as indicated:
1
2
A
34.
3
C,D
4
5
B
6
B
7
A,C
8
D
9
B
10
C
11
A
12
D
Describe a fair division of the items:
A:
B:
C:
D:
Leftovers:
Chapter 4
Consider the following: The following question refers to a country with five states. There are 240 seats in the
legislature, and the population of the states are given in the table:
35. What is the standard divisor?
36. What is the standard quota for state A?
37. What is the standard quota for state D?
38. Use Hamilton’s Method to apportion the 240 seats. What is the final apportionment to each state?
39. Using a modified divisor of D = 40.1 , use Webster’s Method to apportion the 240 seats. What is the final
apportionment to each state?
4
Chapter 5
Consider the following figure:
40.
How many paths are there from A to C?
41.
How many paths are there from A to F?
Graph A
B
F
C
A
42.
Vertex C is adjacent to which vertices?
43.
What is the degree of each vertex?
E
D
A
An undercover police officer is assigned the job of once a night walking
each of the 48 blocks of a certain section of town described by the street
grid shown below. The walk starts and ends at A. The officer wants to
minimize the total number of blocks he has to walk each night.
44.
How many vertices of odd degree are there in the graph
representing this problem?
45.
An optimal eulerization of this problem can be
obtained by adding how many edges?
46.
Suppose it takes the officer 5 minutes to walk a block.
In an optimal trip, the officer will cover the entire
neighborhood in how long?
In a certain city, there is a river running through the middle of the city.
There are 4 islands and 11 bridges as shown in the figure.
47.
North Bank
Draw a graph that appropriately models this situation.
How many vertices does it have? How many edges?
48.
What is the degree of the vertex that represents the North Bank?
49.
Are you able to start on the South Bank, cross each bridge
exactly once and end back on the South Bank? In other words,
does this graph have an Euler Circuit?
50.
PARK
B
D
C
A
South Bank
Are you able to start on the South Bank, cross each bridge exactly
once, and end somewhere else besides the South Bank? In other words, does this graph have an Euler Path? If it
does, where would you end up?
5
Chapter 6
Formulas/Vocab:
KN = the complete graph with N vertices
# Hamilton Circuits (including mirrors) in KN
Draw K5 :
 N 1!
#Edges in KN
N ( N  1)
2
51.
How many edges are there in K11 ?
52.
How many Hamilton Circuits are there in K7?
53.
How many Hamilton Circuits are there in K5, not including mirror images?
A truck must drop off furniture at 4 different homes (A,B,C,D) as shown in the
graph, starting and ending at A. The numbers on the edges represent distances
(in miles) between locations. The truck driver wants to minimize the total
length of the trip.
54.
The nearest-neighbor algorithm applied to the graph
yields what solution? What is the weight of the circuit?
55.
The cheapest-link algorithm applied to the graph yields what solution?
What is the weight of the circuit?
A
4
5
9
D
1
56.
The repetitive nearest neighbor algorithm yields what solution?
What is the weight of the circuit?
57.
What is the optimal solution to the problem?
58.
What is the relative error of the nearest neighbor solution?
8
C
B
2
Chapter 7
59.
How many different spanning trees does the following graph have?
A
18
60.
Find the minimum spanning tree and its weight:
8
7
E
14
B
5
19
17
D
15
2
4
C
6
61.
Find the minimum spanning tree and its weight.
Boston Buffalo Chicago Columbus Louisville
Boston
******
446
963
735
941
Buffalo
446
******
522
326
532
Chicago
963
522
******
308
292
Columbus
735
326
308
******
209
Louisville
941
532
292
209
******
62.
The shortest network between three points is either a Minimum Spanning Tree or a Steiner Tree.
How can you tell?
63.
What is the length of the shortest network connecting points A , B, and C?
A
A
140˚
37.9ft
C
25.6 in
35˚
35˚
62.1 ft
B
20˚
C
B
48.1 in
Chapter 8
Assume you have a digraph with five vertices (A, B, C, D, and E) and nine arcs. A is incident from C, D, and E; B is
incident to D and E; B is incident from C; D is incident to C; D is incident from E.
64.
What is the indegree and outdegree of vertex A?
65.
What is the indegree and outdegree of vertex D?
Suppose you have the following
digraph (The numbers in parenthesis
represent hours).
66.
How many tasks are in the
project?
67.
How many direct precedence
relationships are in the
project?
68.
Find the length of the critical
path from each vertex.
A(6)
D(3)
G(2)
Start(0)
B(4)
C(8)
E(6)
End(0)
F(4)
7
69.
Find the critical path and its length for the entire project.
70.
What priority list would you use to schedule using the decreasing time algorithm?
71.
What priority list would you use to schedule using the critical path algorithm?
72.
Using the decreasing time algorithm, you would start by scheduling task _____ to one processor and task _____
to another.
73.
Using the critical path algorithm, you would start by scheduling task _____ to one processor and task _____ to
another.
74.
Scheduling the project on two processors, using the decreasing time algorithm, gives you a
project completion time of how long?
Mini Lessons
75.
What is the fewest number of colors you need to color any planer map?
(example: the county map of Indiana)
A
76.
True or False: To color the following graph using vertex coloring techniques,
vertex A and B cannot be the same color.
77.
Which two vertices CAN be the same color?
77.
What are the first 10 Fibbonacci numbers?
78.
What is the golden ratio? What is a golden rectangle?
79.
Encode using a Caesar cipher with Key 5: “You need to study to get an A.”
80.
What is 78 in Binary? What is 11011 in base 10?
81.
You have an unlimited amount of Christmas stockings. Each will hold 7 pounds of goodies before they fall off
the wall. You have goodies that weigh 3, 4, 1, 5, 2, 2, 5, 6, 3, 3, and 1 pounds. Pack the stockings using each of
the three bin-pack techniques discussed in class: Next Fit, First Fit, and Worst Fit.
C
D
B