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Presentation/PPT
27/08/2022
CS 5200 Homework -1
(Deadline: 09-17-2023 11:59pm)
Instructor: Huiyuan Yang
Q1. Complexity analysis [11%]
1) Consider the following algorithm (assume n >1)
int any-equal (int n, int A[ ][ ]) {
Index i, j, k, m;
for (i=1; i<=n; i++)
for (j=1; j<=n; j++)
for (k=1; k<=n; k++)
for (m=1; m<=n; m++)
if (A[i][j] == A[k][m] && !(i==k && j==m))
return 1;
return 0;
}
§ Show the best case; what is the best-case time complexity. [3%]
§ Show the worst case; what is the worst-case time complexity. [3%]
§ Try to improve the efficiency of the algorithm. [5%]
Q2. Growth of functions [15%]
1) Using the definition of O(.) and Ω(.), show that: [5%]
10???? + 8 ∈ ????(????!), ???????????? 10???? + 8 ∉ Ω(????!)
2) Using the definition of Θ(. ) To show that: [5%]
2????” + 2????! + Θ(????) ∈ Θ(????”)
3) Show by using limits that: [5%]
I. 4!#$% ∈ Θ(16#)
II. ????& − ????! + 2???? ∈ ????(????)
Q3. Master theorem [9%]
Find the asymptotic bound of the recurrence equations T(n) using master theorem when
applies:
I. ????(????) = 9????(????/3) + ????! + 4
II. ????(????) = 6????(????/2) + ????! − 2
III. ????(????) = 4????(????/2) + ????” + 7
Q4. Divide-and-Conquer [25%]
1) Solve the following recurrence equation using recursion tree. [7%]
????(????) = T(n/2) + T(n/4) + T(n/8) + n
2) We have a problem of size n, and two solutions to solve this problem: [8%]
a. Solution-I: runs ????! times to solve this problem directly (non-recursive algorithm).
b. Solution-II: is a recursive algorithm, which takes ???????????????????? times to split problem into
two sub-problems with equal size of ????/2, and ???????????? times to combine the results
together.
Show whether the solution-I or solution-II is more efficient.
3) Consider the following algorithm, which return a pair with minimum and maximum in
the list of A.[10%]
Min_Max (A, lt, rt) {
if (rt – lt ≤ 1)
Return (min(A[lt], A[rt]), max(A[lt], A[rt]));
(min1, max1) = Min_Max(A, lt, ⌊(lt + rt)/2⌋);
(min2, max2) = Min_Max(A, ⌊(lt + rt)/2⌋ + 1, rt);
Return (min(min1, min2), max(max1, max2));
}
§ Write the recurrence equation for the code above. [3%]
§ Solve the recurrence equation. [2%]
§ Show by example why the algorithm works. (hint: you can show each step when
call the Min_Max() function with a list of size 6, etc). [5%]
Q5. Programming [35%]
Implement the algorithm for finding the closest pair of points in two-dimension plane using the
Divide-and-Conquer strategy.
§ Compile without errors (by calling make) [5%]
§ Use the brute-force approach to solve the problem and print out the execution time. [5%]
§ Use the Divide-and-Conquer strategy to solve the problem and print out the execution
time. [25%]
Bonus: [Extra 20%]
Design an algorithm: Given an integer array of 2???? integers.
(1) Partition the list into two arrays of length ????, such that the difference between the
sums of the two sub-arrays is maximized; show the time complexity. [5%]
(2) Partition the list into two arrays of length ????, such that the difference between the
sums of the two sub-arrays is minimized; show the time complexity. [15%]
Submission requirements: [5%]
Put the theory part and code into a folder with the name: [YourStudentID-YourName-HW1].zip
§ Following all the submission requirements [5%]
§ Theory part:
o You solution should be clear, concise but also contain enough details.
o Only PDF format is allowed.
o Please make sure the scanned PDF file is high-resolution and easily readable if
you wrote your solution on paper.
§ Programming part:
o A sanity check to test the correctness of your algorithm.
o Your program will read an input file and write an output file.
o Your program will be called like this: prompt>./hw1 ./input.txt. ./output.txt
o Your code should be complied by calling the makefile.
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