Due: 22nd of March 2019 at 11:59pm
Submit your assignment via Canvas. Submission must be in pdf format, and cannot be handwritten.
COMP 9007 – Assignment 1
As a first step go to the last page and read the section: Advice on how to do the homework.
The median of a list of 2n+ 1 distinct numbers is the element, m, of the list such that exactly nnumbers
are less than mand exactly nnumbers are greater than m. For example, in the list [3,1,7,2,6], the median
is 3.
a) Write an algorithm1that computes the median of a list of numbers. You may assume the list has an
odd number of distinct elements. Please note that correctness and understandability are more important
than efficiency. (4 marks)
b) Give an upper bound for the running time of your algorithm using big-O notation. (3 marks)
We will now develop a more efficient algorithm for computing the median.
The Partition(lst,x) function takes a list of numbers lst, and an element xof that list, and returns a reordering
of lst such that everything that occurs before xis less than xand everything that occurs after xis greater
than x. For example Partition([3,1,7,2,6],3) could return [2,1,3,6,7]. Note that the final output do es not
have to be completely ordered, though such an output would be a valid output of the Partition function.
c) Write an algorithm that implements the Partition(lst,x) function. Your algorithm should run in O(n) time
for a list of length n, and you should justify this. You may assume lst consists of distinct elements and x
is an element of the list. Hint: Start with a doubly-linked list with one element containing x.(9 marks)
d) Write an algorithm so that instead of returning a reordering of lst it returns the (0-indexed) place of xin
sort(lst) (e.g. returns 0 if xis the smallest element, or (n−1) if xis the largest element). Your algorithm
should not sort the list. Hint: you should be able to make a small modification to the algorithm in the
previous part. Call this function Position(lst,x). (4 marks)
e) If lst has 2n+ 1 elements and xis the median of lst, what will Position(lst,x) return? (2 marks)
The last question shows that finding the median is an instance of a more general problem: finding the element
of a list that is in position k. The Position and Partition functions give us some nice tools for dividing this
problem into smaller problems: Take an element xfrom the list. If Position(lst,x) =kthen we have found
the element. If Position(lst,x) is greater than kthen the required element lies in the sublist of Partition(lst,x)
before x. Otherwise, Position(lst,x) is less than kand the required element lies in the sublist of Partition(lst,x)
after x.
Suppose lst has nelements, Position(lst,x) =p, the output of Partition(lst,x) is lst1,x, lst2, and we are trying
to find the element ywith Position(lst,y)=k.
f) If p>kwe know that yis in lst1. What will Position(lst1,y) return? (2 marks)
g) If p<k we know that yis in lst2. What will Position(lst2,y) return? (2 marks)
1see the note on the last page about writing algorithms
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