Homework Assignment 3
This assignment is due by Tuesday February 26 (in class). Assignments should be handed in before
the class begins.
Problem 1: Solve exercises 6.5-8, 6.5-9 (page 166), 8.2-2 (page 196), 8.4-2, 8.4-3 (page 204), and
9.3-2, and 9.3-8 (page 223) in the textbook.
Problem 2: Solve problem 8-6 (page 208) in the textbook.
Problem 3: In each of the following questions you are asked to design an O(n) algorithm that
takes as input: an array with ndistinct numbers and an integer k≤n(the array is not in sorted
order). Note that kis not a constant and it may depend on n(e.g. we may have k=√n) so an
algorithm with O(kn) run time does not satisfy the requirement. In each part you need to argue
both the correctness and run time of the algorithm.
(i) Give an O(n) algorithm that finds the ksmallest numbers in the array.
(ii) Give an O(n) algorithm that finds the elements of rank ⌈n
2⌉−k+1,⌈n
2⌉−k+2,...,⌈n
2⌉, . . . , ⌈n
2⌉+k
in the array (so we are looking for a “band” of 2kelements around the median).
(iii) Give an O(n) algorithm that finds the kelements that are closest in value to the median. For
example, if k= 5, n= 11 and the array is 1,2,3,4,5,6,8,14,15,16,17, then the median is 6 and
the 5 closest numbers are 2,3,4,5,8. Note that the array in the example is shown in sorted order
but the input array is not necessarily sorted.
Hints: You may assume a linear time algorithm for selection (as in chapter 9) but you can treat
it as a black box and do not need to know or explain its details in your solution for this question.
Parts (i), (ii) should be helpful for part (iii).
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