1. University of Michigan
EECS 281 Data Structures and Algorithms
Homework 4
Spring Semester, 2014
Assigned: 06/14/14, 12:00pm
Due: 06/20/14, 11:55pm
Instructions: Homework is on 4 pages and is worth 20 total points. You must complete all
problems. Enter all responses into the CTools Test Center. For Test Center questions, be sure
to press the finish button to officially submit your answers. Please be sure to double check your
submissions via a confirmation from CTools before deciding that the homework is submitted. A
key for the homework set will be posted after the due date.
Clarifications:
● Assume all C++ code to be C++11standard compliant (compiled with std=c++11).
Homework Goals:
● Demonstrate understanding of Dynamic Programming problems.
Honor Code: This work is to be completed individually. Collaboration is not allowed.
Potential violations will be reported to the honor council. By submitting, you confirm
that you have neither given nor received unauthorized help on this assignment.
2. Problem 1 (5 points)
The TSA has introduced a new rule at the Detroit Airport: you may only bring a maximum of K
items in your carryon. Each item is assigned a value that represents how badly you want to
bring that item on the plane. Given that you want to bring N items (where N is greater than K)
and that you want to maximize the total value of the items you bring, answer the following
questions.
1.1 Assuming the best possible algorithm, which algorithm family is suitable to find the most
optimal solution for your problem?
1.2 In terms of K, N, or both, what is the worstcase runtime complexity of this algorithm?
For the rest of the questions, assume the TSA decided to revise their rule. The new rule allows
you to carry as many items as you want, as long as the total weight of all the items you carry on
is less than some weight W. Like in the previous questions, you are to figure out which different
items to take and how many of each.
1.3 Assuming the best possible algorithm, which algorithm family is suitable to find the most
optimal solution?
1.4 Assume that the weight of each unit item is always a multiple of 1 pound (1 lbs, 2 lbs, etc,
and not, say, 1.5 lbs), what is the worstcase runtime complexity of this algorithm in terms of W,
N, or both?
4. 2.3
int Algorithm3(int m, int size[], int val[])
bool possSet[0..n] //(0: absent, 1: present)
int maxVal = 0
for int i = 0 to 2^n 1
possSet[] = genNextPower(n)
int setSize = findSize(possSet[])
int setValue = findValue(possSet[])
if setSize < m and setValue > maxVal
bestSet[] = possSet[]
maxVal = setValue
return maxVal
A. Bruteforce
B. Greedy
C. Dynamic Programming
D. Divide and Conquer
E. Branch and Bound
F. Backtracking
2.4
int Algorithm4(int n)
int a[0..n]
a[0] = a[1] = 1
for int i = 2 to n
a[i] = a[i 2] + a[i 1].
return a[n]
A. Bruteforce
B. Greedy
C. Dynamic Programming
D. Divide and Conquer
E. Branch and Bound
F. Backtracking
5. Problem 3: On Piracy and its Consequences (5 points)
Captain Drew “The Beard” DeOrio and his faithful crew of pirates are sailing the high seas,
when they happen upon a merchant ship full of treasure 100 gold coins! After scuttling the
merchant ship and marooning the sailors on a deserted island, the pirates gather to decide how
to, in the eloquent words of James, “divvy the booty, yarr!”.
Although pirates are not generally known for following rules, when it comes to treasure there is
a very specific protocol that must be followed. Pirates take treasure seriously.
1. The pirate in charge proposes a distribution of coins.
2. All the pirates vote on whether or not to accept this distribution.
3. If more than half the pirates vote against the distribution, the pirate in charge is thrown
overboard, the next pirate in line is made captain and the process is begun anew.
The hierarchy of command on Captain DeOrio’s vessel is as follows:
1. Captain Drew “The Beard” DeOrio
2. First Mate Dave “FooBar” Paoletti
3. Helmsman Matt “Diffy” Diffenderfer
4. Galley Cook James “StewPot” Juett
5. Lookout Kevin “Arrrrr!” Kiningham
6. Kevin’s parrot, “Squawky”
7. Deck Urchin Steve “Swabbie” Merritt
With this process in mind, how many coins will Captain DeOrio keep for himself?
Clarifications:
● These are extremely logical pirates.
● Pirates do not want to be thrown overboard.
● Each pirate wants to maximize the amount of gold they receive.
● Each pirate would prefer to throw another pirate overboard, if all other results would
otherwise be equal.
● Squawky is a logical pirate parrot.
● Yes, there are hungry sharks in the water.
● As pirates do not trust each other, no alliances or deals will be made.
● The pirate in charge gets to vote too.
Hints:
● Start from the base case and work your way backwards. With only two pirates, how
many coins can the captain keep? With three pirates?
● It may be helpful to make a table.
6. Question 4: The Lost Boys (5 points)
After pillaging the merchant ship and arguing over who gets what, the pirates decide to throw
Captain DeOrio overboard anyway. Unfortunately, as they realize after the fact, Captain DeOrio
was the only one who knew the secret pirateshipsailing algorithm that they need to get home!
These are now some very lost pirates. But just how lost are they? Calculate the probability that,
if they choose a random course, these poor pirates will end up back at their secret
piratelairhideout (marked with an X on this map). Keep in mind that this is a sailing ship, and
can only sail with the wind (you may only move right or down).
What is the probability for:
1. A map of size 3 x 3? (1 pt)
2. A map of size 12 x 12? (2 pts)
3. A map of size 26 x 26? (2 pts)
Clarifications:
● It’s okay to sail past/through whales.
● Give answers as a percentage (0%100%) rounded to the nearest whole number.
● The pirate ship always begins in the upper left corner of the map.
● Pirates will not sail off of the map. They’re not that dumb.
● The pirate lair is always located in the lower right corner of the map.
● As an example, a 2 x 2 map would have 4 possible paths (RIGHT, RIGHTDOWN,
DOWN, and DOWNRIGHT), and 2 would lead home: 50%
Hints:
● Remember, the probability of the pirates finding their way home is equivalent to the
number of routes that lead home divided by the number of possible routes.
