程序代写代做代考 Java algorithm AI data structure Imperial College London – Department of Computing

Imperial College London – Department of Computing

MSc in Computing Science

580: Algorithms

Assessed Coursework 2

1. In this question you are asked to create solutions to the problem of determining whether
all the numbers in a given sequence are distinct. Each part of the question sets different
requirements on the solution. The general problem is as follows:

Given a sequence A = [A1, . . . , AN ] of integers, the procedure Distinct should return
True if A contains no duplicates and False otherwise.

(a) Write a version of the Distinct procedure that uses O(1) space and O(N2) time.
The space constraint refers to extra memory allocated by the Distinct procedure
itself. This excludes the memory used by the input sequence.

Discuss the space and time complexity of your solution. Briefly outline one alternative
solution that would satisfy the same requirements, and how you would expect its
performance to compare to your Distinct procedure.

(b) Can you trade off some space in order to obtain a faster solution to the problem?
Your next task is to write a version of the Distinct procedure that runs in O(N)
time, but is allowed to use O(N) space. You can assume the pre-existence of any of
the data structures covered in the course. You can also assume that the elements of
A are an average case input for any data structure used.

Discuss the space and time complexity of your solution. Include a discussion of the
way your procedure uses any data structure, the relevant operations of the data
structure, and the effect this has on the running time of Distinct.

2. Given a sequence A = [A1, . . . , AN ] of N integers, the procedure Longest should return
the length of the longest strictly increasing sequence within A. This sequence does not
have to be contiguous, but the ordering of A should be preserved, and each element must be
strictly less than the next. So, given A = [56,−12, 4, 34,−3, 5, 35], the longest increasing
sequence is either [−12, 4, 34, 35] or [−12,−3, 5, 35] or [−12, 4, 5, 35] (there might be more
than one longest sequence), and the length is 4.

Write a procedure for Longest that runs in O(N2) time.

To succeed in this task you will need to decompose the problem into subproblems. Start
by considering the following. If you know the length of the longest increasing sequence

1

within A that finishes with Ai, for all i < j, what is the length of the longest sequence that finishes with Aj? Submission Submit By: 1900, Monday 5th March 2018 Submit your typed answers to CATE in a file named cw2.pdf by the deadline above. Scanned copies of hand-written answers will not be accepted. Procedures can be written in either pseu- docode or Java. If you are using LATEX, then two suggested ways of typesetting procedures are to use a verbatim environment: egin{verbatim} Anything typed here will be output exactly as it is written in your source file end{verbatim} or an algorithmic environment which creates this sort of output: procedure Swap(A, i, j) if i ≤ j then temp = ai ai = aj aj = temp end if end procedure See https://en.wikibooks.org/wiki/LaTeX/Algorithms for details. 2 https://en.wikibooks.org/wiki/LaTeX/Algorithms# Typesetting_using_the_algorithmicx_package