COMP4500/7500 Advanced Algorithms and Data Structures

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COMP4500/7500

Advanced Algorithms and Data Structures

School of Information Technology and Electrical Engineering

Assignment 1

This assignment is worth 20% (COMP4500) or 15% (COMP7500) of your final grade.

This assignment is to be attempted individually. It aims to test your understanding of graphs and graph algorithms. Please read this entire handout before attempting any of the questions.

Submission.   Answers to each of the questions in part A and Question 4(a), 4(b) and 4(c) from part B should be clearly labelled and included in a pdf file called a1.pdf.

You need to submit (i) your written answers to parts A and Question 4(a), 4(b) and 4(c) from part B in a1.pdf, as well as (ii) your source code file TradeFinder.java as well as any other source code files that you have written in the assignment1 package electronically using Blackboard according to the exact instructions on the Blackboard website: https://learn.uq.edu.au/

You can submit your assignment multiple times before the assignment deadline but only the last submission will be saved by the system and marked. Only submit the files listed above. You are responsible for ensuring that you have submitted the files that you intended to submit in the way that we have requested them. You will be marked on the files that you submitted and not on those that you intended to submit. Only files that are submitted according to the instructions on Blackboard will be marked - incorrect submissions will receive 0 marks.

Submitted work should be neat, legible and simple to understand – you may be penalised for work that is untidy or difficult to read and comprehend.

For the programming part, you will be penalised for submitting files that are not compatible with the assignment requirements. In particular, code that is submitted with compilation errors, or is not compatible with the supplied testing framework will receive 0 marks.

Late submission.   See section 5.3 of the course profile for details. If the assignment is submitted after the deadline, without an approved extension, a late penalty will apply. The late penalty shall be 10% of the maximum possible mark for the assessment item will be deducted per calendar day (or part thereof), up to a maximum of seven (7) days. After seven days, no marks will be awarded for the item. A day is considered to be a 24 hour block from the assessment item due time. Negative marks will not be awarded.

If there are medical or exceptional circumstances that will affect your ability to complete an assignment by the due date, then you can apply for an extension as per Section 5.3 of the electronic course profile (ECP). Extensions to assignments must be requested via my.UQ (https://my.uq.edu.au/). You can find instructions on how to submit your request online (https://my.uq.edu.au/information-and-services/manage-my-program/exams-and-assessment/applying-extension). Your extension application must be submitted on or before the assessment item’s due date and time.

School Policy on Student Misconduct.   You are required to read and understand the School Statement on Misconduct, available at the School’s website at: http://www.itee.uq.edu.au/itee-student-misconduct-including-plagiarism This is an individual assignment. If you are found guilty of misconduct (plagiarism or collusion) then penalties will be applied.

Part A (25 marks total)

Question 1: Constructing SNI and directed graph   [5 marks total]

(a) (1 mark) Creating your SNI. In this assignment you are required to use your student number to generate input.

Take your student number and prefix it by “98” and postfix it by “52”. This will give you a twelve digit initial input number. Call the digits of that number d[1], d[2], . . . , d[12] (so that d[1] = 9, d[2] = 8, . . . , d[12] = 2).

Apply the following algorithm to these twelve digits:

1   for i = 2 to 12

2       if d[i] == d[i − 1]

3           d[i] = (d[i] + 3) mod 10

After applying this algorithm, the resulting value of d forms your 12-digit SNI. Write down your initial number and your resulting SNI.

(b) (4 marks) Construct a graph S with nodes all the digits 0, 1, . . . , 9. If 2 digits are adjacent in your SNI then connect the left digit to the right digit by a directed edge. For example, if “15” appears in your SNI, then draw a directed edge from 1 to 5. Ignore any duplicate edges. Draw a diagram of the resulting graph. (You may wish to place the nodes so that the diagram is nice, e.g., no or few crossing edges.)

Question 2: Strongly connected components   [20 marks total]

Given a directed graph G = (V, E), a subset of vertices U (i.e., U ⊆ V ) is a strongly connected component of G if, for all u, v ∈ U such that v = u,

a) u and v are mutually reachable, and

b) there does not exist a set W ⊆ V such that U ⊂ W and all distinct elements of W are mutually reachable.

For any vertices v, u ∈ V , v and u are mutually reachable if there is both a path from u to v in G and a path from v to u in G.

The problem of finding the strongly connected components of a directed graph can be solved by utilising the depth-first-search algorithm. The following algorithm SCC(G) makes use of the basic depth-first-search algorithm given in lecture notes and the textbook, and the transpose of a graph; recall that the transpose of a graph G = (V, E) is the graph G^T = (V, E^T), where E^T = {(u, v) | (v, u) ∈ E} (see Revision Exercises 3: Question 6). (For those who are interested, the text provides a rigorous explanation of why this algorithm works.)

SCC(G)

1 call DFS(G) to compute finishing times u.f for each vertex u

2 compute G^T, the transpose of G

3 call DFS(G^T), but in the main loop of DFS, consider the vertices in order of decreasing u.f

4 output the vertices of each tree in the depth-first forest of step 3 as a separate strongly connected component

(a) (10 marks) Perform step 1 of the SCC algorithm using S as input. Do a depth first search of S (from Question 1b), showing colour and immediate parent of each node at each stage of the search as in Fig. 22.4 of the textbook. That means that you should draw the annotated graph for each stage of the search. (We want to make sure that you understand how the algorithm works.) Also show the start and finish times for each vertex.

For this question you should visit vertices in numerical order in all relevant loops:

for each vertex u ∈ G.V        and

for each vertex v ∈ G .Adj[u].

(b) (2 marks) Perform step 2 of the SCC algorithm and draw S^T.

(c) (8 marks) Perform steps 3, 4 of the SCC algorithm. In your solution you must list (and draw) the trees in the depth-first forest in the order in which they were constructed. (You do not need to show working.)

Part B (75 marks total): Trade manager

[Be sure to read through to the end before starting.]

You are in charge of regulating the trade of items of different possible types between a set of traders.

Each trader has one type of item that it produces, and a set of types of items that the trader is willing to trade. A trader is always willing to trade the type of item that they produce. More than one trader can produce the same type of item. The types of items that a trader is willing to trade never changes.

Just because a trader is willing to trade a particular type of item, doesn’t mean that they can trade that item. Initially, each trader can only trade the type of item that they produce. In order to be able to trade any other types of items, they need to form trade agreements with other traders.

For any two different traders, ta and tb, and two different types of item, gx and gy, we say that a trade agreement

(ta, tb, gx, gy)

can be formed between traders ta and tb in which ta agrees to trade items of type gx with tb in exchange for items of type gy from tb if and only if

● trader ta can trade items of type gx,

● trader tb can trade items of type gy,

● trader ta is willing to trade items of type gy, and

● trader tb is willing to trade items of type gx.

After agreement (ta, tb, gx, gy) is formed

● Trader ta can trade items of type gy, as well as all of the other types of items that it could trade before the agreement was formed, and

● Trader tb can trade items of type gx, as well as all of the other types of items that it could trade before the agreement was formed.

The agreement has no other side effects. (E.g. it does not change the types of items that any other trader can trade, nor does it change any of the types of items that any trader (including ta and tb) is willing to trade.)

Given a set of traders, T, and a (possibly empty) sequence of trade agreements between traders from T,

<(ta0, tb0, gx0, gy0),(ta1, tb1, gx1, gy1), . . . ,(tan, tbn, gxn, gyn)>

where, for i ∈ {0, 1, . . . , n}, tai ∈ T and Tbi ∈ T, tai = tbi and gxi = gyi, we say that the sequence of agreements can be formed if and only if for every i ∈ {0, 1, . . . , n}, the agreement (tai, tbi, gxi, gyi) can be formed, given that the only agreements that have already been formed so far are the ones that appear before it in the linear ordering, i.e. agreement (tai, tbi, gxi, gyi) can be formed, given that the only agreements that have already been formed so far are

(ta0, tb0, gx0, gy0),(ta1, tb1, gx1, gy1), . . . ,(ta(i−1), tb(i−1), gx(i−1), gy(i−i)) .

Recall here that before any agreements have been formed, each trader can only trade the type of item that they produce. This means, for example, that the first agreement in any non-empty sequence, (ta0, tb0, gx0, gy0), must be able to be formed before any other agreements have been formed, i.e. when each trader can only trade the type of item that they produce.

Example 1 As a running example, consider the following set, T, of traders:

t0 : (g0, [g0, g1, g2, g7, g8])

t1 : (g1, [g1, g5])

t2 : (g2, [g0, g2, g3, g4])

t3 : (g3, [g1, g2, g3, g4])

t4 : (g4, [g4, g6])

t5 : (g5, [g1, g4, g5])

t6 : (g6, [g0, g4, g6])

t7 : (g7, [g7, g8])

t8 : (g8, [g7, g8])

where each trader above is described first by their name, then the type of item that they produce, and then by the set of types of items that they are willing to trade. E.g. trader t0 can produce items of type g0 and is willing to trade items of either type g0, g1, g2, g7, or g8.

Initially, each trader can only trade the type of item that they produce. For example, initially t0 can only trade items of type g0. That is to say, t0 can trade g0 after the empty sequence of trade agreements, <>, have been formed.

After forming the following sequence of trade agreements,

<(t0, t2, g0, g2)>

we have that t0 can trade g2 as well as g0, and t2 can trade g0 as well as g2. Agreement (t0, t2, g0, g2) can be formed before any other agreements have been made because trader t0 can initially trade g0, the item it produces, and is willing to trade g2, and trader t2 can initially trade g2, the item that it produces, and is willing to trade g0.

If trader t0 would like to be able to trade items of type g1, then a longer sequence of trade agreements is required. In this example, there is only one producer of items of type g1, trader t1. No agreement can ever be formed directly between traders t0 and t1 because they have only one type of item in common that they are willing to trade (g1). There are two other traders, t3 and t5 who are also willing to trade items of type g1. Trader t0 will never be able to form an agreement directly with t5 because they have only one type of item in common that they are willing to trade (g1). Trader t3 is eventually able to form the trade agreement (t3, t0, g1, g2) with t0, after which t0 can trade g1, but this is only after a number of other trade agreements have been formed. (Note that initially, before any trade agreements have been formed, t3 cannot yet trade items of type g1, and t0 cannot yet trade items of type g2.). Trader t0 can trade g1 (as well as g0 and g2), for example, after the following sequence of trade agreements have been formed (between traders in T):

<(t0, t2, g0, g2),(t4, t6, g4, g6),(t2, t6, g0, g4),(t2, t3, g4, g3),(t1, t5, g1, g5),(t5, t3, g1, g4),(t3, t0, g1, g2)>

Note that the sequence of trade agreements above can be formed (in the order given), because:

● As per the earlier reasoning, agreement (t0, t2, g0, g2) can be formed before any other agreements have been made.

● Next, it is possible to form agreement (t4, t6, g4, g6), because trader t4 can trade g4, the item it produces, and is willing to trade g6, and trader t6 can trade g6, the item that it produces, and is willing trade g4.

● It is possible to next form agreement (t2, t6, g0, g4) because after forming agreement (t0, t2, g0, g2), t2 can trade g0, and after forming agreement (t4, t6, g4, g6), t6 can trade g4. Trader t2 is also willing to trade g4, and t6 is willing to trade g0.

● Next, it is possible to form agreement (t2, t3, g4, g3) because after forming agreement (t2, t6, g0, g4), t2 can trade g4, t3 can trade g3, the item that it produces; and t2 is willing to trade g3 and t3 is willing to trade g4.

● Next, it is possible to form agreement (t1, t5, g1, g5), because trader t1 can trade g1, the item it produces, and is willing to trade g5, and trader t5 can trade g5, the item that it produces, and is willing trade g1.

● Next, it is possible to form agreement (t5, t3, g1, g4) because after forming agreement (t1, t5, g1, g5), t5 can trade g1; after forming agreement (t2, t3, g4, g3), t3 can trade g4; and t5 is willing to trade g4 and t3 is willing to trade g1.

● Finally, it is possible to next form agreement (t3, t0, g1, g2), because after forming agreement (t5, t3, g1, g4), t3 can trade g1; after forming agreement (t0, t2, g0, g2), t0 can trade g2; and t3 is willing to trade g2 and t0 is willing to trade g1.

There is no sequence of trade agreements that can be formed such that trader t0 can trade items of either type g7 or g8. The only traders who are willing to trade items of these types are t0, t7 and t8. Even though t7 and t8 can form agreement (t7, t8, g7, g8), after which both t7 and t8 can both trade items of type g7 and g8, there is no way for t0 to form an agreement with either of them to trade g7 or g8 without first being able to trade one of these two items (which it cannot do, until one of these two agreements has been formed).

Your task is to design, implement and analyze an algorithm that takes as input:

● A set of traders, T,

● a trader t ∈ T, and

● the type of an item g that trader t is willing to trade and returns true if and only if there exists a (possibly empty) sequence of agreements between traders in the set T that can be formed, such that after those agreements are formed, trader t can trade items of type g; otherwise the algorithm should return false.

For example, given the set of traders T from Example 1,

1. given traders, T, trader t0, and item type g0, your algorithm should return true.

2. given traders, T, trader t0, and item type g1, your algorithm should return true.

3. given traders, T, trader t0, and item type g2, your algorithm should return true.

4. given traders, T, trader t0, and item type g7, your algorithm should return false.

5. given traders, T, trader t0, and item type g8, your algorithm should return false.

You algorithm must be designed and implemented as efficiently as possible.

Question 3: Design and implement an efficient solution   (50 marks)

Design and implement an algorithm that answers the question above. Your algorithm should be as efficient as possible. Marks will be deducted for inefficient algorithms (e.g. a brute-force approach is not appropriate). Clearly structure and comment your code. Use meaningful variable names.

● Your algorithm should be implemented in the static method TradeFinder.canTrade from the TradeFinder class in the assignment1 package that is available in the zip file that accompanies this handout. The zip file for the assignment also includes some other code that you will need to compile the class TradeFinder as well as some junit4 test classes to help you get started with testing your code.

● Do not modify any of the files in package assignment1 other than TradeFinder, since we will test your code using our original versions of these other files.

● You may not change the class name of the TradeFinder class or the package to which it belongs. You may not change the signature of the TraderFinder.canTrade method in any way or alter its specification. (That means that you cannot change the method name, parameter types, return types or exceptions thrown by the method.)

● Your implementation should be in Java 1.8. You are encouraged to use Java 8 SE API classes, but no third party libraries should be used. (It is not necessary, and makes marking hard.)

● Dont write any code that is operating-system specific (e.g. by hard-coding in newline characters etc.), since we will batch test your code on a Unix machine. Your source file should be written using ASCII characters only.

● You may write additional classes, but these must belong to the package assignment1 and you must submit them as part of your solution – see the submission instructions for details.

● The JUnit4 test classes as provided in the package assignment1.test are not intended to be an exhaustive test for your code. Part of your task will be to expand on these tests to ensure that your code behaves as required.

Your implementation will be evaluated for correctness and efficiency by executing our own set of junit test cases. Code that is submitted with compilation errors, or is not compatible with the supplied testing framework will receive 0 marks. A Java 8 compiler will be used to compile and test the code.

Question 4: Worst-case analysis   (25 marks)

This question involves performing an analysis of the worst-case time complexity and worst-case space com-plexity of your algorithm from Question 3.

(a) (5 marks) For the purpose of the worst-case time complexity analysis in Q4(b) and the worst-case space complexity analysis in Q4(c), provide clear and concise pseudocode that summarizes the algorithm you used in your implementation from Question 3. You may use the programming constructs used in Revision solutions, and assume the existence of common abstract data types like sets, maps, queues, lists, graphs, as well as basic routines like sorting etc.

Clearly structure and comment your pseudocode. Use meaningful variable names.

[It should be no more than two pages using minimum 11pt font. Longer descriptions will not be marked.]

(b) (12 marks) Let x be the number of traders in T, and let y be the number of different types of items in the set that is formed by merging all of the types of items that each trader in T is willing to trade into one set (i.e. y is the number of different types of items involved in the problem input).

Provide an asymptotic upper bound on the worst case time complexity of your algorithm in terms of parameters x and y. Make your bound as tight as possible and justify your solution using your pseudocode from Q4(a).

You must clearly state any assumptions that you make (e.g. on the choice of implementations of any data structures that you use, and their running time etc.).

To simplify your analysis, you should make the (incorrect but simplifying) assumption that HashSet (or HashM ap) operations that have expected-case time complexity O(1) actually have worst-case time complexity O(1). E.g. checking for set-membership in a HashSet has expected-case time complexity that is O(1).

[Make your analysis as clear and concise as possible – it should be no more than a page using minimum 11pt font. Longer descriptions will not be marked. Also note that to receive any marks for this question, you must justify your solution – it is not enough to only give an asymptotic upper bound without explanation.]

(c) (8 marks) As for Q4(b), let x be the number of traders in T, and let y be the number of different types of items in the set that is formed by merging all of the types of items that each trader in T is willing to trade into one set (i.e. y is the number of different types of items involved in the problem input).

Provide an asymptotic upper bound on the worst case space complexity of your algorithm in terms of parameters x and y. Make your bound as tight as possible and justify your solution using your pseudocode from Q4(a).

You must clearly state any assumptions that you make (e.g. on the choice of implementations of any data structures that you use, and their space usage etc.).

[Make your analysis as clear and concise as possible – it should be no more than a page using minimum 11pt font. Longer descriptions will not be marked. Also note that to receive any marks for this question, you must justify your solution – it is not enough to only give an asymptotic upper bound without explanation.]

Evaluation Criteria

Question 1

● Question 1 (a) (1 mark)

1 : correct answer to question given

0 : answer not given or contains one or more mistakes

● Question 1 (b) (4 marks)

4 : The answer to question 1(a) is 100% correct, and the correct graph is given.

0 : If the answer to question 1(a) is not 100% correct, or the answer contains at least one mistake.

Question 2

If the graph produced for Question 1 is given and 100% correct, then the following marking scheme applies. If the graph produced for Question 1 is mostly correct (but contains a minor error), then the student will receive 2/3 of the marks obtained using the following marking criteria. Else, if the graph produced for Question 1 contains more than a minor error, zero marks will be given for all aspects of this question.

● Question 2 (a) (10 marks)

10 : Correct answer given.

8 : All stages of the traversal shown on an annotated graph (like CLRS Fig. 22.4), including relevant features for each vertex (colour, immediate parent and start and finish times), but there are one or two minor mistakes.

6 : All stages of the traversal are shown on an annotated graph (like CLRS Fig. 22.4), however at most one of the relevant features for each vertex (e.g. start-time) may not be included and there may be up to three mistakes.

4 : Most stages of the traversal are shown on an annotated graph (like CLRS Fig. 22.4), however at most two of the relevant features for each vertex (e.g. start-time) may not be included and there may be up to four mistakes.

2 : Most stages of the traversal are shown on an annotated graph (like CLRS Fig. 22.4), however at most two of the relevant features for each vertex (e.g. start-time) may not be included and there may be up to five mistakes.

0 : Otherwise.

● Question 2 (b) (2 marks)

2 : correct answer to question given

0 : answer not given or contains one or more mistakes

● Question 2 (c) (8 marks)

This part of the question will be marked correct with respect to the finishing times for each vertex calculated in Q2(a). If those finishing times are not given in Q2(a) then no marks will be awarded for this section. Otherwise, the following marking criteria applies.

8 : All the trees in the depth-first forest are listed in the order in which they were constructed. All aspects of the depth-first forest are correct.

6 : All the trees in the depth-first forest are listed in the order in which they were constructed, but there was an error in the traversal that produced the forest.

4 : All of the trees in the depth-first forest are listed, however the order in which the trees were created may not be clear, or there may be one or two errors in the traversal that produced the forest.

2 : Either: (i) all of the strongly connected components of the graph are correctly listed, but the trees (from the depth-first forest) from which these connected components were derived are not clearly listed, or (ii) all of the trees in the depth-first forest are listed, however the order in which the trees were created may not be clear and there may be up to three errors in the traversal that produced the forest.

0 : Otherwise.

Question 3 (50 marks)

Your implementation will be evaluated for correctness and efficiency by executing our own set of junit test cases.

50 : All of our tests pass

45 : at least 90% of our tests pass

40 : at least 80% of our tests pass

35 : at least 70% of our tests pass

30 : at least 60% of our tests pass

25 : at least 50% of our tests pass

20 : at least 40% of our tests pass

15 : at least 30% of our tests pass

10 : at least 20% of our tests pass

 5 : at least 10% of our tests pass

 0 : less than 10% of our test pass or work with little or no academic merit

Note: Code that is submitted with compilation errors, or is not compatible with the supplied testing framework will receive 0 marks. A Java 8 compiler will be used to compile and test the code.