CS 303 final

Last HW assignment problems

• assignment for Graphs
• assignment for Asymptotics

Additional problems

1. Recursion tree / recurrence relation. Suppose we have a recursive algorithm that solves a problem by splitting the size n problem into 4 different problems of size n/2, solves the smaller problems, and takes O(n) time putting the solutions together to be a solution to the size n problem. What is the running time of the algorithm? Use a recursion tree to demonstrate your solution.
2. NP. Show that the following problem is contained within NP. We are given a list of n students and the courses for each that they want to take in the spring, and we are also given a list of m teachers and the list of classes that they can teach. The goal is to determine whether there is a schedule so that all students are able to take all of the courses they want to, with the courses being taught by teachers who can teach them. Assume that courses are scheduled from 9am-4pm MWF and 9:30am-3:30pm on TR.
3. Complexity classes.

What is the smallest set it is in: P, NP, EXP, none of these.

• sorting
• factoring an integer
• matrix multiplication
• check if a given graph is connected
• given a program, and an input to the program of length n, does

 the program complete within time 2**n.

• given a program, and an input to the program of length n, does

 the program every halt.

• given a program, and an input to the program of length n, does

 the program complete within time n**2.

Binomial theorem -

Regular expression for something. DFA or NFA for something.

• binary strings with an even # of 1s
• binary strings that start and end with the same bit
• binary strings that somewhere contain 00 and 11
• binary strings that somewhere contain 00 or 11
• binary strings that are at most 4 bits long
• binary strings that are exactly 4 bits long

Countability - which of these are countable, which are not.

• integers
• rationals
• even integers
• multiples of 100
• real numbers
• rational #s between 0 and 1
• real #s between 0 and 1
• powerset of the integers
• powerset of the rationals
• powerset of the even integers

Undecidability - halting problem, others.

Combinations, permutations - quiz

Discrete probability