Description
This assignment has 6 questions, for a total of 50 points. Unless otherwise specified, complete and reasoned arguments will be expected for all answers.
Question Points Score
Big oh and running times 10
Square vs. Multiply 5
Graph basics 8
Background: Probability 12
Tossing coins 7
Array Sums 8
Total: 50
1
Question 1: Big oh and running times ………………………………………………………. [10] (a) [4] Write down the following functions in big-oh notation:
1. 2.
(b) [6] Consider the following algorithm to compute the GCD of two positive integers a,b. Suppose a,b are numbers that are both at most n. Give a bound on the running time of Gcd(a,b). (You need to give a formal proof for your claim.)
Algorithm 1 Gcd(a,b)
if (a < b) return Gcd(b,a); if (b = 0) return a; return Gcd(b,a%b); (Recall: a%b is the remainder when a is divided by b)
1. a) O(n2), O(1/n2)
Question 2: Square vs. Multiply …………………………………………………………….. [5] Suppose I tell you that there is an algorithm that can square any n digit number in time O(nlogn), for all n ≥ 1. Then, prove that there is an algorithm that can find the product of any two n digit numbers in time O(nlogn). [Hint: think of using the squaring algorithm as a subroutine to find the product.]
Question 3: Graph basics …………………………………………………………………… [8] Let G be a simple undirected graph. Prove that there are at least two vertices that have the same degree.
Question 4: Background: Probability ……………………………………………………….. [12] (a) [3] Suppose we toss a fair coin k times. What is the probability that we see heads precisely once?
(b) [4] Suppose we have k different boxes, and suppose that every box is colored uniformly at random with one of k colors (independently of the other boxes). What is the probability that all the boxes get distinct colors?
(c) [5] Suppose we repeatedly throw a fair die (with 6 faces). What is the expected number of throws needed to see a ‘1’? How many throws are needed to ensure that a ‘1’ is seen with probability > 99/100?
Question 5: Tossing coins……………………………………………………………………[7] Suppose we have two coins, one of which is fair (i.e. prob[heads] = prob[tails] = 1/2), and another of which is slightly biased. More specifically, the second coin has prob[heads] = 0.51. Suppose we toss the coins N times, and let H1 and H2 be the number of heads observed (respectively).
(a) [3] Intuitively, how large must N be, so that we have H2 > H1 with “reasonable certainty”?
(b) [2] Suppose we pick N = 25. What is the expected value of H2− H1?
(c) [2] Can you use this to conclude that the probability of the event (H2−H1 ≥ 1) is small? [It’s OK if you cannot answer this part of the problem.]
Question 6: Array Sums ……………………………………………………………………. [8] Given an array A[1…n] of integers, find if there exist indices i,j,k such that A[i]+A[j]+A[k] = 0. Can you find an algorithm with running time o(n3)? [NOTE: this is the little-oh notation, i.e., the algorithm should run in time < cn3, for any constant c, as n →∞.] [Hint: aim for an algorithm with running time O(n2 logn).]
