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1. A family, decides to repaint their home. There are four tasks to be performed in their house: paint the kitchen, den, bathroom and basement. The family members hold jobs, go to school etc, so each painting task has an associated opportunity cost (i.e. cost of painting a room instead of earning money from working or, going to school). The table below contains the tasks, family members and the associated opportunity costs.
We wish to assign one task to each family member.
a. (5 pts) How many possible solutions exist? Explain your answer.
b. (25 pts) Using the method presented in class provide an optimal solution (a set of task assignments) that assigns one task to each person, and minimizes the sum of the costs. You must represent your solution as a state space diagram with information such as the lower bound and each cost element that is added to generate the cost for the node.
c. (20 pts) How many partial solutions did you generate? Was it worth using this method? Explain your answer.
Hint: It is essential to demonstrate how the algorithm works. First, don't waste your time by generating the entire space tree. Second, please refrain from picking (or steering) your solution to the lowest task assignment sequence (if you can visually identify it). While this question is reduced in terms of possible parameters to make it manageable to solve on paper, the problem -and the associated technique- are regularly encountered in real logistics problems with hundreds of persons and hundreds of tasks. Therefore, it would be impossible to visually identify the lowest task assignment sequence.
2. (10 pts) Given an O(n2) algorithm, a data set with n = 16 takes 24 seconds. What is the largest size data set that can be executed by this algorithm in 60 seconds? Explain your answer.
3. Select ONE of the topics below.
a. (15 pts) Provide a one-page synopsis of the topic, explaining the problem(s) it solves, the method, asymptotic complexity, advantages/disadvantages or limitations.
b. (25 pts) Using the programming language of your choice4 implement the algorithm and provide sample input (if necessary).
● Miller-Rabin primality test algorithm5
● Finding all Hamiltonian cycles
● Horner's Rule algorithm (polynomial evaluation)6
● Langrange interpolation method
● Fast Fourier Transform
4 Available to compile/execute on PLUTO, WYRD.
5 In your synopsis you must explain how this technique improves on the Generalized Riemann hypothesis.
6 Your implementation should read a list of polynomial coefficients (the order should be in the increasing powers of x) along with a value of x to compute its value at, and return the value of the polynomial at that value.
