CS 600 Test I

October 4, 2000

1. Find the big-oh relationships for the following functions. Give your calculations. [30]

n^1/3, 3ⁿ, (ln n)², 5^n/2, lnln n.

2. Let L be an array of size n, let L[i] denote the i-th key of L, let x be the key being searched for in L, and let p(i) be the probability that x = L[i].
Assuming the following (1) - (3), answer the questions (a) and (b) below for the binary search (see next page):

(1) x is always found in L.
(2) The probability that x is found with 5 or less number of comparisons is 40%, and all the other cases together occur 60% of the time. Also assume that each number of comparisons is equally probable within each group.
(3) The number of keys n is given by n = 2^k - 1 for some integer k.

Answer the following questions:

(a) Find the probability p(i) in terms of n. [15]
(b) Compute the average time (number of comaprisions) of the binary search algorithm in terms of n. [35]

You may use the following formulas if you need them:

$\Sigma_{i=1}^{n} i2^{i} = (n - 1) 2^{n + 1} + 2$, $\Sigma_{i=1}^{n} i^{2} = n( n + 1 )(2n + 1)/6$,
$\Sigma_{i=1}^{n} i^{3} = ( n( n + 1 )/2 )^{2}$, $\lg (n!) = \Theta(n\lg n)$.

3. Find a certificate for each of the following problems: [20]

(a) Set Cover
(b) String Matching


Set Cover: Given a set X and a collection C of subsets of X, is there a subcollection of C such that every element of X is in some member of the subcollection ?
String Matching: Given two strings S and T, is S a substring of T ?