CMPSCI 711: Really Advanced Algorithms Micah Adler Summary: CMPSCI 711: Really Advanced Algorithms Micah Adler Problem Set 2 Out: March 4, 2003 Due: March 13, 2003 1. Let n and k be positive integers. Using the Lovasz Local Lemma and G(n; 1 2 ), derive a condition on n and k which ensures the following: there exists an n-vertex graph which contains neither a clique of size k nor an independent set of size k. 2. [MR95] Problem 7.12. Note that in part 1, well-de ned means that for every string x, this de nition speci es a unique value of M(x). 3. For any a in f1; 2; : : : ; p 1g, de ne the hash function h a (x) as (ax mod p) mod s, for some s < p. (a) Derive an upper bound on the probability that h a (1) = h a (s + 1) in terms of the parameter s. (b) Show that the probability of this pair of keys colliding is as bad as any distrinct pair of keys drawn from the key space f1; 2; : : : ; pg. 4. In our de nition of skip lists, we sampled the elements in M i with probability 1/2 to create the set M i+1 . Let's instead consider the more general case where we sample with probability p, for 0 < p < 1. (a) Let r be the resulting number of sets required. Describe a value R such that with high probability r  R. (b) Determine E[ P r Collections: Computer Technologies and Information Sciences