代写COMP9020 Assignment 3 – Guide 2024 Term 1代做留学生SQL 程序

COMP9020

Assignment 3 – Guide

2024 Term 1

Due: Thursday, 4th April, 18:00 (AEDT)

Submission is through inspera. Your assignment will be automatically submitted at the above due date. If you manually submit before this time, you can reopen your submission and continue until the deadline.  If you need to make a submission after the deadline,  please use this link to request an extension: https://www.cse.unsw.edu.au/ cs9020/extension_request.html. Unless you are granted Special Consid- eration, a lateness penalty of 5% of raw mark per 24 hours or part thereof for a maximum of 5 days will apply. You can request an extension up to 5 days after the deadline.

Answers are expected to be provided either:

•  In the text box provided using plaintext, including unicode characters and/or the built-in formula editor (diagrams can be drawn using the built-in drawing tool); or

•  as a pdf (e.g.  using LATEX) – each question should be submitted on its own pdf, with at most one pdf per question.

Handwritten solutions will be accepted if unavoidable, but we don’t recommend this approach as the assessments are designed to familiarise you with typesetting mathematics in preparation for the final exam and for future courses.

Discussion of assignment material with others is permitted, but the work submitted must be your own in line with the University’s plagiarism policy.

Objectives and Outcomes

The aim of this assignment is to build your understand of recurrence, induction and the asymptotic analysis of algorithms.  Most questions are presented at a highly abstract level so that the consequences are very general, and can be applied in a variety of situations:  not just later on in the course, but also beyond. The specific motivation for each problem can be summarised as follows:

Problem 1: This question looks at the complexity of three algorithms for various matrix operations – two iterative algorithms for matrix addition and multiplication and one recursive algorithm for matrix multiplication. This recursive algorithm takes the first steps towards the Strassen Algorithm for matrix multiplication.  In that algorithm, the recursive submatrices (SW + TY, etc) are cleverly computed via a different set of intermediate matrix calculations (e.g.  one step is to calculate (S + V)(W + Z)) rather than the more direct approach we take here.  The Strassen Algorithm is also closely related to the Karatsuba algorithmfor integer multiplication – an elementary form. of this is taught in primary school as the Box method.

Problem 2: In this question you are asked to work on a series of identities of gradually increasing com- plexity. The aim is to practice applying basic induction on algebraic expressions to prove properties of the natural numbers.

Problem 3: This question introduces a commonly used recurrent structure used in computer science, binary trees. The question will guide you through how you define a recursively defined mathemat- ical structure along with functions on that structure, and how to use structural induction to reason about such structures and functions.

Problem 4: In this question you will be tested on your ability to solve a recurrence relation asymptoti- cally. The goal is to test your ability to reason about asymptotic growths beyond the master theorem, and use the idea of asymptotics to solve relations that might be difficult to solve exactly.

After completing this assignment, you will:

•  Be able to make rigorous arguments about the foundational structures used in discrete mathematics [All problems, particularly 1 and 3]

•  Understand the fundamental Computer Science concepts of recursion and induction [All problems]

•  Analyze the correctness and efficiency of algorithms [Problem 1 and 4]

Specific advice on how to do this assignment

Problem 1

• For Question 1, you are asked for asymptotic upper bounds.  You should aim to give the best upper bound you can derive as this indicates your level of ability and understanding.   Sub- optimal upper bounds may still be eligible for full marks if optimisations fall outside the expected knowledge of the course; but sub-optimal bounds where there is a reasonable expectation that students can find improvements will only be eligible for partial marks.

• As an example, recall the following algorithm outline for insertionSort from lectures:

sort(L) :

if L.isEmpty() :

return   L else :

L2 := sort(L.next)     insert   L.data into   L2 return L2

In particular, the line:

insert   L.data into   L2

If you were asked to provide an asymptotic upper bound for the running time of this line, then an HD-level answer eligible for full marks might be:

To insert L.data into L2 we could iterate through L2, comparing each element of the list with L.data, until we find the correct place to insert it. In the worst case this will require checking every element of L2, which means at most n − 1 = O(n) comparisons. Therefore an upper bound on the running time for this line would be O(n).

Note that since L2 is sorted, it is possible to use binary search to find the correct place to insert L.data, and this gives a solution that will only require O(log n) comparisons.  So the upper bound of O(n) is sub-optimal; however as we don’t assume knowledge of binary search, the well-reasoned solution given above would be eligible for full marks.

• As always, it is recommended that you justify your answer. For parts a) and b) a sensible way to show how you obtain the running time would be to use the process used in lectures – calculating the running time of each individual line and combining them alltogether.  Textual descriptions for non-trivial running times are a good idea. It is also a good idea to annotate the algorithm to assist with referencing. Again, taking the example above, here is an HD-level answer calculating the recurrence relation for the running time of that algorithm:

sort(L) :

if L.isEmpty() :

return   L else :

L2 := sort(L.next)     insert   L.data into   L2 return L2

Let the running time of the above algorithm on a list of length n be T(n). Then:

– Lines 1, 2 and 6 consist of a constant number of elementary operations, so have

running time O(1) (as indicated)

– Line 4 involves a recursive call to sort with a list of length n − 1. So it has running time T(n − 1).

– Line 5 has running time O(n) as discussed above.

– The “if” part is only executed when L is the empty list (i.e.  when n = 0); for all other n only the “else” part is executed.

Putting this all together gives the following recurrence relation for the running time: T(0) = O(1)             T(n) = O(1) + O(n) + T(n − 1) = T(n − 1) + O(n).

In inspera, the algorithms in the question are presented in tables.  Copying and pasting the algorithms from the question to your solution will give you the algorithm in a table form. making annotation easier

• For part c) you are only provided with a description of the algorithm idea. To provide a justifica- tion for your recurrence it is sufficient to identify what parts of the algorithm are contributing to the running time and what their algorithmic cost is. When the algorithm description is informal, you may have to workout a way to fill in some missing steps (e.g. the insert example above) but if this is necessary, a simple/obvious approach is acceptable.

Remark

The recursive algorithm takes the first steps towards the Strassen Algorithm for matrix multi- plication. In that algorithm, the recursive submatrices (SW + TY, etc) are cleverly computed via a different set of intermediate matrix calculations (e.g. one step is to calculate (S + V)(W + Z)) rather than the more direct approach we take here.

Problem 2

• The problem encourages you to follow the typical layout for an inductive proof.  You are asked to prove P(n), Q(n) and R(n) for all n ∈ N.  To do so follow find an appropriate base case, for which proving the identity should be fairly simple, and then show that P(n)  =⇒ P(n + 1).

• You will probably want to do the questions in order, as the answers found in the previous questions allow you to simplify the sums you will be working within the subsequent ones. This is however not strictly necessary.

• There are multiple ways you can rearrange the sums you will obtain. In particular, for question b), you may use the answer obtained in question a) but it is also possible that you can take another route.  It might be of use (in particular for question c) to use the following identity (derived in the lectures starting from 0): ∑i n =1 i = n(n 2 +1) .

• The goal of the question is to familiarise yourself with basic induction.  Although you do not strictly need to use induction, and will get full marks for a non-inductive proof, you are encour- aged to think about how you would use induction to work on such a problem, and how the proofs might differ whether you use induction or not. Often induction makes it simpler to find a formal proof, however you might find that using a) to prove b) without using induction is somewhat simpler.

• If you approach a question inductively, but find that you did not make use of your inductive hypothesis in your inductive step, then you are not actually using induction, so you might want to reframe. your proof as a non-inductive proof.

Problem 3

• The abstract definition of a binary tree is similar to the abstract definition of a linked list – except we have two “tails”, and no data item in the head.  Visually, it may be useful to think of the recursive case as:

• An implementation of the functions count, leaves and internal in the programming language of your choice, rather than as an abstract mathematical function, is acceptable and will not incur any penalties. However, such an answer for (b) and (c) will make it almost impossible to provide a suitable answer for part (d) (see comments below).

• Here are two examples of recursively defined functions for computing the length of a linked list, and concatenating two linked lists (compare to the definition given in lectures of similar functions for words):

• For a distinction level answer or higher, a short (one or two sentence) justification for correctness is expected. Here is an example of an HD-level answer giving a recursive definition for comput- ing the height of a binary tree: the length of the longest path from the root to a leaf (and defined to be −1 for the empty tree):

• Instead of providing justification as above, it may be worth checking your answer against the example tree given in the assignment description.  Indeed, such a check would demonstrate a level of understanding expected of a credit-level student, so partial marks are available for a check like the following:

• Even though the recursive definition of binary trees consist of one base case and one recursive case, some of the functions in parts (a)–(c) involve other cases:  in particular  additional base cases/special cases of the recursive case (it is not important which category it falls under).

• Part (d) is about proving the specified relationship between the functions you have defined, it is not about showing a particular property of binary trees (i.e.  that the number of leaves is one more than the number of fully-internal nodes).  In particular, your proof should be relying on the definitions of leaves and internal provided in parts (b) and (c) [and the recursive definition of binary trees on which these are based].  An induction proof based on “the number of nodes in the tree” is unlikely to be proving the correct result.  As we are proving a result for an infinite set of structured objects, proof by structural induction is a recommended approach.  Here is an example of an HD-level structural induction proof involving the functions length and concat defined above:

• Note that when using structural induction, the induction hypothesis is that the proposition holds for all smaller structures required to construct the complex structure. For example, when trying to prove P(T) when T = (Tleft, Tright) the standard inductive step would be to show:

If P(Tleft) and P(Tright) both hold, then P(T) holds.

Problem 4

• In this problem you are asked to find the asymptotic solution of a recurrence relation. The master theorem works when you have a single recurrent term on the right-hand side of the equation, however in this relation you have several recurrent terms with different coefficients.

• A common way of solving such problems, is to guess the solution and substitute your guess in the equation and see if it holds.

• Since you are only asked to find asymptotic behaviour, your guess doesn’t need to worry about details such as the exact values of coefficients.

• Here is an example of how you might analyse the asymptotic growth of a recurrence relation without using the master theorem:

• This example gives you some indications as to what your proof should look like. In the exercise however you might have to be more careful as some steps might not be as straightforward.