Quiz Questions

[Note: this webpage last modified Tuesday, 26-Apr-2011 15:53:06 EDT]

This page collects sample quiz questions, or questions from past quizzes.

1. Choose 4 out of 5, due by email by May 7. Take home quiz, so please no sharing (except with Jeff).
   1. Explain why there can be no one way functions if P=NP.
   2. True or False, there exists a computable pseudorandom generators that are secure against any computable function. Why?
   3. Suppose G is a pseudorandom generator that is secure against an algorithm A. Suppose G takes (log(n))² bits and stretches that to n bits, and that A takes n bits as input. Give an algorithm that uses G to determine the majority output of A over all its possible inputs (that will be either 0 or 1). What is the running time of your algorithm?
   4. Explain the definition of a pseudorandom generator and how it can be used in an encryption scheme. Hint, think of the one time pad.
   5. Suppose we knew that there is an EXP-complete function f so that the following holds. For any poly-time randomized algorithm A, A fails to invert f with probability at least 1-1/n^c for all constants c. Does this mean that f is a one way function? Why?

2. (2 Points)

   (a) What is the probability of tossing 10 coins and getting 0 heads?

   (b) What is the probability of tossing 10 coins and getting exactly 1 heads?

3. (4 Points)

   (a) Suppose you flip n coins. What is the expected number of times that you will see two coins in a row that are either both heads or both tails?

   (b) Use Markov's inequality to give a bound on the probability that more than 3*(n-1)/4 times you will see either two heads in a row or two tails in a row.

4. (4 Points)

   (a) Let A be a randomized algorithm that solves a BPP problem and that runs in time n². Give a randomized algorithm A' that solves the same problem and has a probability of error less than 1/2ⁿ.

   (b) Prove your algorithm has at most this probability of error.

   (c) What is the running time of A'?

• What is the probability of tossing 10 coins and getting exactly 1 heads?

  What is the expected number of coins that come up heads when tossing 10 coins?

  What is the expected number of 0's in a sequence of 10 random bits?

  You look at n random bits. What is the expected number of 1's? Use Markov's inequality to give a bound on the probability that the number of 1's is greater than (n/2 + n/10). Use the Chernoff bound to give a bound on the same probability.

  Which of the following are (known to be) contained in each other? L, P, BPP, NP, PSPACE, EXP.

  Let A be a randomized algorithm that solves a BPP problem. Give a deterministic algorithm to solve the problem. What is the running time of your algorithm?

  Let A be a randomized algorithm that solves a ZPP problem. Give a randomized algorithm A' that solves the same problem and has a probability of error less than 1/2ⁿ, and prove your algorithm has at most this probability of error.

  Give a randomized algorithm to test whether a given polynomial is identically zero. You do not need to prove the algorithm is correct, just give the steps of the algorithm, and explain the running time.

  Suppose you flip n coins. What is the expected number of times that you will see two coins in a row that are either both heads or both tails? The highest this could possibly be is n-1. What is the expected number of times you will see k heads or k tails in a row? The highest this could possibly be is n-(k-1).

  Consider the following algorithm for determining if a Boolean formula is satisfiable.

        Input: formula phi
        Repeat k times:
          Choose an assignment a to phi at random.
          If phi(a) is true, return true
        If phi(a) was false for all randomly chosen assignments, return false
      

  True or false - this algorithm shows that SAT is in BPP. Explain.

• Consider this algorithm.

  Isolated-vertex
  Input: adjacency matrix A for a graph G
  Output: true if the graph contains an isolated vertex (vertex with no neighbors/edges).
  Algorithm:
    for i=1 up to n {
      hasNeighbor=false
      for j=1 up to n {
        if (A[i,j]) {
          hasNeighbor=true
      }
      if (hasNeighbor == false) { return true}
    }
    return false
  

  Question: What is the running time and memory space usage for this algorithm? For the memory space, only count the amount of "working memory space" - do not count the amount of memory that A takes.

• Question: Which complexity classes does the Isolated-vertex problem belong to due to this algorithm? Circle all that it is in.
  L, NL, P, NP, PH, PSPACE, EXP.

• Let L be a language that can be solved in PSPACE, and let M be a Turing machine that decides L while using at most n^c memory space, for some constant c. Explain how we can decide L in EXP. (This shows that PSPACE is contained in EXP).

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