#  problems 

1  CLRS Exercise 9.37 on page 223
Describe an O(n)time algorithm that, given a set S of n distinct numbers and a positive integer k ≤ n, determines the k numbers in S that are closest to the median of S. 
2  CLRS Exercise 4.51 on page 96
Use the master method to give tight asymptotic bounds for the following recurrences. a. T(n) = 2T(n/4) + 1 b. T(n) = 2T(n/4) + √n c. T(n) = 2T(n/4) + n d. T(n) = 2T(n/4) + n^{2} 
3  CLRS Problem 41 on page 107
Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for n ≤ 2. Make your bounds as tight as possible, and justify your answers.

4  DPV Exercise 2.4 on page 71 Suppose you are choosing between the following three algorithms:

5  The Strahler number of a binary tree is defined as
follows: An empty tree has Strahler number 0. If the binary tree T has subtrees T_{L} and T_{R}, the Strahler number S(T) of T is defined by max{ S(T_{L}),S(T_{R}) }, if S(T_{L}) ≠ S(T_{R}) S(T_{L}) + 1, otherwise 
#  hard problems 

6  CLRS Problem 43 on page 108
Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for sufficiently small n. Make your bounds as tight as possible, and justify your answers.
