A2.dvi
School of Computer Science
University of Guelph
CIS*3490 The Analysis and Design of Algorithms
Winter 2023
Instructor: Fangju Wang
Assignment 2 (100%)
In this assignment, you practice algorithm design, expression, analysis, and implementation.
The analysis includes both theoretical analysis and empirical analysis.
For the following two questions, please design algorithms, and express them in the pseu-
docode that we are using (in the textbook and lecture slides), and then implement your algo-
ithms in the C programming language. For each question, you are required to submit both you
design in pseudocode and implementation in C, as well as algorithm analysis and comparison.
1. Counting Inversions (40%)
Let A[0..n− 1] be an a
ay of n distinct numbers. A pair of a
ay elements (A[i], A[j]) is called
an inversion if A[i] > A[j] for i < j.
1.1 Design a
ute force algorithm to count the number of inversions in an a
ay, analyze the
number of repetitions of its basic operation, and determine the efficiency class.
1.2 Design a recursive divide-and-conquer algorithm of Θ(n log n) to count the number of
inversions in an a
ay, set up a recu
ence to analyze the number of repetitions of its
asic operation of the best case, and determine the efficiency class. Use the Maste
Theorem to verify the efficiency class you determine.
1.3 Implement the two algorithms, and test them by using data A2 Q1.txt, which includes
50,000 integers. Your programs are required to display the numbers of inversions and
execution time. Compare the two algorithms in execution time and theoretical analysis.
A different file of the same size (same number of integers) will be used to grade you
programs. Your programs should prompt to enter a file name.
2. Finding Shortest Path Around (60%)
Let S be a set of points in a 2-dimensional plane. The convex hull of set S is the smallest convex
set containing S. (Please find more about the convex hull problem, especially the definition of
extreme point, on pages XXXXXXXXXXin the textbook.) It is assumed that not all the points in S
are on a straight line.
The points in the convex hull can be viewed as fence posts that support a fence su
ounding
all the points in S. Let s1 and s2 be two points in the hull. A path from s1 to s2 is a sequence
1
of points in the hull. By viewing points in the hull as fence posts, we may consider a path as
a sequence of posts holding a fence segment. The problem of finding shortest path around is to
find the shortest path from s1 to s2 that cannot cross inside the fenced area, but it goes along
the fence. From s1 to s2, there are two paths along the fence, going in the two directions. The
problem is to find the shorter one.
2.1 Design a
ute force algorithm to solve the shortest path around problem and analyze its
efficiency.
2.2 Design a recursive divide-and-conquer algorithm of Θ(n log n) to solve the shortest path
around problem, set up a recu
ence to analyze the number of repetitions of the basic
operation to compute the hull (for the best case), and determine the efficiency class. Use
the Master Theorem to verify the efficiency class that you determine.
2.3 Implement the two algorithms and test them using data A2 Q2.txt. The file contains 30,000
points (pairs of x-y coordinates). Your programs will be graded by using this data file
and two points s1 and s2 that are in the hull of the points in the file. Your programs
are required to display the number of points on the path, the length of the path, and x-y
coordinates of the points (in the order from s1 to s2 inclusive). Your programs are also
equired to display execution time for hull computing. Compare the two algorithms in
execution time and theoretical analysis.
Also, for grading your programs, we request that, when a program identify a hull point,
it displays the x-y coordinates.
You can hard-code the name of the data file in your programs. The file (data A2 Q2.txt)
will be used to grade your programs. Your program should prompt to enter s1 and s2.
Note: Write your own code for this assignment. NO code from any source is allowed.
Due time: 08:00am, Monday, Fe
uary 13, 2023. Submit your work as a tar file to Moodle.
2
A2_guide.dvi
School of Computer Science
University of Guelph
CIS*3490 The Analysis and Design of Algorithms
Winter 2023
Instructor: Fangju Wang
Assignment 2 Guide
1.1 Develop a
ute force algorithm based on the definition of inversion, which checks every
pair of (A[i], A[j]) for i < j.
1.2 Modify the mergesort algorithm to count the number of inversions.
2.1 Develop a
ute force algorithm based on the definition of convex hull. The algorithm is to
find the convex hull for a given set of points. Please see Figure 3.5 on page 111. Develop
your algorithm to find the nails holding the ru
er-band, that is, to find the extreme
points.
A
ute force algorithm checks every point in the given data set to see if it is an extreme
point (nail) of the convex polygon. An extreme point is an ending point of a straight
oundary line. Two points define a boundary line if all the other points are on one side
of the line.
Please read
• “Convex-Hull Problem” on page 109 for the definition and an example of convex
hull;
• the definition of extreme point (page 112);
• the equation of a straight line through two points (page 112);
• the method to check if all the points in a set are on one side of a line (page 112).
Then design an algorithm to find the shortest path.
• The equation for calculating the distance between points s1 = (x1, y1) and s2 =
(x2, y2) is
√
(x1 − x2)2 + (y1 − y2)2
2.2 Design a divide-and-conquer algorithm of Θ(n log n) based on the idea of quicksort. Please
ead
• “Convex-Hull Problem” on page 195 for the quick hull algorithm.
Then design an algorithm to find the shortest path.
1
• You can use the algorithm of 2.1 for finding the shortest path.
1.3, 2.3 Make sure your programs can be co
ectly compiled and executed on the Linux system
in SoCS.
Your work should be submitted as a tar file containing files like
eadme, design, P11.c, P12.c, P21.c, P22.c, makefile.
Please do not submit the data files.
Any compilation e
or or warning will result in a mark deduction. There will be some marks
allocated for documentation.
Each file should have a comment at the beginning containing your name, ID, date, and the
assignment name.
The readme file should contain the following:
• name, ID, and assignment numbe
• a
ief description of how to compile and run your programs.
The design file should include the algorithms you design for 1.1, 1.2, 2.1, and 2.2, efficiency
analysis, and comparisons. This can be a text, pdf or scanned file.
In your C files, each function should have a
ief comment describing its purpose. Also, any
section of code where it is not easily apparent what the code does should have a short comment.
You can use timespec get() to get program running time.
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