Homework 1. This problem set is due Monday January 13 at 11:59 pm. It must be submitted electronically to Canvas.

Reading: CLRS chapters 1–3; chapter 4, sections 4.1, 4.3–4.5. Reading for lecture 2: chapters 5, 7, and 11.

Problem assignment:

• "DO" exercises are strongly recommended to check your understanding of the concepts. Do them but do not submit them.
• Problems labeled "HW" are standard homework problems that you are required to submit.

"Do" Exercises (do but do not submit):

1. "DO" Exercise 2.3-5, page 39.
   • Write pseudocode for binary search as a recursive procedure.
   • Write pseudocode for binary search as a iterative procedure.
   • Write a loop invariant for your iterative procedure and prove it correct.

2. "DO" Exercise 2.3-7, page 39. Write pseudocode for your algorithm.

3. "DO" Problem 2-1, pages 39–40.

4. "DO" Problem 2-4, pages 41–42.

5. "DO" Exercise 3.1-1, page 52.

6. "DO" Exercise 4.5-1, page 96.

7. "DO" Problem 30-1, parts a and c, pages 920–921.

Homework Assignment: DUE Monday January 13 at 11:59 pm

• Collaboration policy: There is no penalty for acknowledged collaboration. To acknowledge collaboration, give the names of students with whom you worked at the beginning of your homework submission and mark any solution that relies on your collaborators' ideas. Note: you must write up each problem solution by yourself without assistance. DO NOT COPY or rephrase someone else's solution. The same requirement applies to books and other written sources: you should acknowledge all sources that contributed to your solution of a homework problem. DO NOT COPY or rephrase solutions from written sources.
• Internet policy: Looking for solutions to homework problems on the internet, even when acknowledged, is STRONGLY DISCOURAGED. If you find a solution to a homework problem on the internet, do not not copy it. Close the website, write up your solution by yourself, and cite the url of website in your writeup.
• Copied solutions obtained from a written source, from the internet, or from another person will receive ZERO credit and will be flagged to the attention of the instructor.
• Write out your work. If you just write your answer without showing your work, you will not receive credit.
• When describing an algorithm in pseudocode, explain the meaning of your variables in English. Comment the lines of your pseudocode. Your pseudocode should be short, clear, and complete to receive full credit.
• Use the master theorem to solve recurrences if it applies.

Homework Problems:

1. HW We are given an array A[1.. n] of n real numbers, n ≥ 3, with the property that A[1] ≥ A[2] and A[n − 1] ≤ A[n]. We say that an element A[x] is a local minimum if A[x − 1] ≥ A[x] and A[x] ≤ A[x + 1].
   Example: There are 6 local minima in the following array:

   
   
   We obviously can find a local minimum in O(n) time by scanning through the array. Give a divide and conquer algorithm that finds a local minimum in O(lg n) time. Your algorithm should return the index of a local minimum. You may use the following test data as an example to help you understand this problem and write correct pseudocode: readme   input   output.
   • Describe your algorithm in pseudocode. Comment your code. No credit for Ω(n) time algorithms. (5 points)
   • Give a worked example or diagram to show more precisely how your algorithm works. (2 points)
   • Argue that your algorithm is correct. Simply restating the steps of the algorithm will receive no credit. (1 point)
   • Analyze the asymptotic running time of your algorithm (this includes writing and solving the associated recurrence). (1 point)
   Now suppose we generalize the problem to two dimensions. The input is an m × n matrix M of real numbers. An element M[x, y] is a local minimum if no neighbor has a strictly lower value (equivalently, if all neighbors have equal or higher values). Give an efficient algorithm to find a local minimum on this input. Your algorithm should return an ordered pair of indices of a local minimum.
   • Describe your algorithm in pseudocode. Comment your code. Maximum points for the most efficient algorithms. No credit for Ω(mn) time algorithms. (6 points)
   • Give a worked example or diagram to show more precisely how your algorithm works. (2 points)
   • Argue that your algorithm is correct. Simply restating the steps of your algorithm will receive no credit. (1 point)
   • What is the asymptotic running time of your algorithm in terms of m and n? Analyze the running time of your algorithm; justify your claim for its running time. (2 points)

2. HW (a) Given an array A[1.. n] of n real numbers, give an algorithm to compute the number of "distinct pairs", which are ordered pairs of indices (i, j) such that A[i] ≠ A[j], in O(n lg n) time.
   Example:
                         A = [10,   10,   19]
                         distinct pairs: (1, 3), (2, 3), (3, 1), (3, 2)
                         count: 4
   
   • Describe your algorithm in pseudocode. Comment your code. (5 points)
   • Give a worked example or diagram to show more precisely how your algorithm works. (2 points)
   • Argue that your algorithm is correct. Simply restating the steps of the algorithm will receive no credit. (1 point)
   • Analyze the asymptotic running time of your algorithm. Simply stating the running time will receive no credit. (1 point)
   (b) Given an array A[1.. n] of n real numbers, give an algorithm to compute the number of "distinct triples", which are indices (i, j, k) such that A[i], A[j], A[k] are distinct, in O(n²) time.
   • Describe your algorithm in pseudocode. Comment your code. (6 points)
   • Give a worked example or diagram to show more precisely how your algorithm works. (2 points)
   • Argue that your algorithm is correct. Simply restating the steps of the algorithm will receive no credit. (1 point)
   • Analyze the asymptotic running time of your algorithm. Simply stating the running time will receive no credit. (1 point)

3. HW Consider two square matrices X and Y of size n = 2^k, where k is a positive integer. Let Z denote their product, i.e.,
   XY = Z, where X, Y, and Z are all of size n, i.e., n × n matrices, with real-number entries. (See here for a refresher on matrices and matrix multiplication. Of particular interest is the section on "Block matrices and submatrices" on page 9.) Observe that Z can be computed using a divide-and-conquer approach. The first step is to divide the two input matrices into quadrants, as follows:
   For example, if then Note that these quadrants are square matrices of size n/2. Furthermore, they can be used to compute the quadrants of Z as follows:
   
   
   • Write pseudocode for a divide-and-conquer algorithm that takes X and Y as input and returns their product, Z. You may use the following subroutines, all of which run in O(n²) time:
     1. Divide-Matrix which takes a square matrix A and returns its quadrants A₁, A₂, A₃, A₄.
     2. Add-Matrices which takes two matrices A and B and returns their sum A + B.
     3. Combine-Matrices which takes quadrants A₁, A₂, A₃, and A₄ returns a square matrix A.
     (5 points)
   
   A second, more clever algorithm involves defining 7 new square matrices of size n/2 as follows:
   • Using the new n/2 × n/2 matrices Q₁, …, Q₇ defined above, show that the following equations for the quadrants of Z match those given in the first part of the problem. (3 points)
   • Write pseudocode for a divide-and-conquer algorithm that takes X and Y as input and computes Z using the seven matrices Q₁, …, Q₇ given above. You may use the subroutines Divide-Matrix, Add-Matrices, and Combine-Matrices. (5 points)
   • Analyze and compare the asymptotic running times of the two divide-and-conquer algorithms above. Count the number of additions and multiplications of n/2 × n/2 square matrices required for each algorithm. Write out the recurrences and solve them using the master theorem. Specify the base cases. Which algorithm is asymptotically faster? (3 points)