Assignment 5
Computer Science 231
Spring 2009
Due: Friday, March 13
Reading. CLRS Chapters 8, 9, pages 165 -- 195
CLR Chapters 9, 10, pages 172 -- 194
CLRS Exercise 8.1-1. What is the smallest possible depth of a leaf in a decision tree for
a comparison sort under the assumption that all elements are distinct?
CLRS Exercise 8.1-4. You are given a sequence of n elements to sort. The input
sequence consists of n/k subsequences, each containing k elements. The elements in a
given subsequence are all smaller than the elements in the succeeding subsequence and
larger than the elements in the preceding subsequence. Thus, all that is needed to sort the
whole sequence of length n is to sort the k elements in each of the n/k subsequences.
Show an Ω(n lg k) lower bound on the number of comparisons needed to solve this
variant of the sorting problem. (Hint: It is not rigorous to simply combine the lower
bounds for the individual subsequences.)
CLRS Exercise 8.2-4. Describe an algorithm that, given n integers in the range 0 to k,
preprocesses its input and then answers any query about how many of the n integers fall
into a range [a .. b] in O(1) time. Your algorithm should use Θ(n + k) preprocessing
time.
CLRS Exercise 8.3-5. Show how to sort n integers in the range 0 to n2 - 1 in O(n) time.
CLRS Exercise 9.1-1. Show that the second smallest of n elements can be found with
n + lg n - 2 comparisons in the worst cast. Hint: Build a "tournament tree" over the
elements in which each node has a winner (smaller element) and a loser (larger element).
Be careful that your algorithm accounts for all comparisons performed.
CLRS Exercise 9.3-7. 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.