Comp 251: Assignment 4

• You are provided some starter code that you should fill in as requested. Add your code only

where you are instructed to do so. You can add some helper methods, but this is at your own

risk; helper methods could cause your methods to crash when called from other programs if

not added responsibly. Do not modify the code in any other way and in particular, do not

change the methods or constructors that are already given to you, do not import extra code

and do not touch the method headers. The format that you see on the provided code is the

only format accepted for programming questions. Any failure to comply with these rules

will give you an automatic 0.

• The starter code includes a tester class. If your code fails those tests, it means that there is

a mistake somewhere. Even if your code passes those tests, it may still contain some errors.

We will grade your code with a more challenging set of examples. We therefore highly

encourage you to modify that tester class, expand it and share it with other students on the

myCourses discussion board. Do not include it in your submission.

• Your code should be properly commented and indented.

• Do not change or alter the name of one of the files you must submit. Files with the wrong

name will not be graded. Make sure you are not changing file names by duplicating them.

For example, main (2).java will not be graded. Make sure to double-check your zip file.

• Do not submit individual files. Include all your files into a .zip file and, when appropriate,

answer the complementary quiz online on MyCourses.

• You will automatically get 0 if the files you submitted on MyCourses do not compile.

• You should compile all your files directly from command line without using a package, by

using the command javac *.java

1. (50 points) Part 1 : Programming exercise

. We want to compare the naive and Karatsuba divide-and-conquer methods to multiply two

integers x and y. Download the java file multiply.java from the course web page. Here,

your task is to implement a recursive version of these algorithms in the methods naive(int

size, int x, int y) and karatsuba(int size, int x, int y), and to use

them to compare the efficiency of each algorithm (i.e. number of arithmetic operations). The

variable size is the size of the integers x and y, and is defined as the number of bits used to

encode them (Note: we assume that x and y have the same size).

Each method (i.e. naive and karatsuba) will return an integer array result that stores

the value of the multiplication in the first entry of the array (i.e. result[0]), and the cost of

this computation in the second entry (i.e. result[1]). We define the cost as the number of

brute force arithmetic operations of the (addition, subtraction, or multiplication) executed by

the algorithm multiplied by the size (in bits) of the integers involved in this operation (Note:

We ignore the multiplication by powers of 2 which can be executed using a bit shift. Of course,

this is a crude approximation).

In particular, for the base case (i.e. when the size of the integers is 1 bit), this cost will be 1

(brute force multiplication of two integers of size 1). In the induction case, the naive method

executes 3 arithmetic operations of integer of size m (i.e. cost is 3 ·m), in addition of the number of operations executed by each recursive call to the function. By contrast, the Karatsuba

algorithm requires 6 arithmetic operations of size m on the top of the cost of the recursion.

The output of your program will print a list of numbers such that, the first number of each row

is the size of the integers that have been multiplied, the second number is the cost of the naive

method, and the third number the cost of the Karatsuba method.

COMP 251 – HW4 Page 2 of 3 Fall 2018

You will test your methods on integers of size 1 to 15. Complete the provided CSV file,

karatsuba.csv with your results for both methods on integers of each size. Plot those

results, and write a short report describing and explaining what you observe. Include the plot

and the report in a file named 260xxxxxx.pdf, editing the name to match your student ID.

Part 2 : write your computations with all the steps and justified results in

a PDF file named 260xxxxxx.pdf. If this PDF is hand-written, make sure

you scan it. Pictures will not be accepted.

Enter your results on the MyCourses quiz

2. (25 points) We remind the master method for determining the asymptotical behaviour of a

recursive function.

Theorem 1 (Master method) Let a ≥ 1 and b ≥ 1 be two constants, and f(n) a function.

∀n ∈ N

+ we define T(n) as:

T(n) = aTn

b

?

+ f(n),where n

b

is interpreted as b

n

b

c or d

n

b

e.

Then, we can find asymptotical bounds for T(n) such that:

1. If f(n) = O(n

logb a−?

) with ? 0, then T(n) = Θ(n

logb a

).

2. If f(n) = Θ(n

logb a

· logp n), then T(n) = Θ(n

logb a

logp+1 n).

3. If f(n) = Ω(n

logb a+?

) with ? 0, and a · f

n

b

?

≤ cf(n), ∀n n0 with c < 1 and

n0 0. Then T(n) = Θ(f(n)).

When possible, apply the master theorem to find the asymptotic behaviour of T(n) from the

following recurrences. Show your work and justify your answer for each recurrence in the

PDF file. Answer in the MyCourses quiz.

(a) (5 points) T(n) = 25 · T(

n

5

) + n

(b) (5 points) T(n) = 2 · T(

n

3

) + n · log(n)

(c) (5 points) T(n) = T(

3n

4

) + 1

(d) (5 points) T(n) = 7 · T(

n

3

) + n

3

(e) (5 points) T(n) = T(n/2) + n(2 − cos n)

3. (25 points) Let TA and TB be two function returning the running time of algorithms A and B,

defined by the recusions TA(n) = 7TA(

n

2

) + n

2

and TB(n) = αTB(

n

4

) + n

2

. Find the largest

integer value of α for which algorithm B is asymptotically faster than A. Show your work and

justify your answer in the PDF file. Answer in the MyCourses quiz.

You will submit multiply.java, karatsuba.csv and student_id.pdf in a single

zip file in the MyCourses submission folder for Assignment 4. You will enter

your solutions to question 2 and 3 on the MyCourses assignment 4 quiz. Your

final answers must be entered in the quiz for you to receive a grade.