Exam paper (Supplementary) for first semeste

University of Johannesburg

University of Johannesburg Institutional Repository

FACULTY OF SCIENCEACADEMY OF COMPUTER SCIENCE AND SOFTWARE ENGINEERINGMODULE CSC3A10ADVANCED DATA STRUCTURES AND ALGORITHMSCAMPUS APKEXAM SSA EXAMINATION — JULY 2014DATE 2014-07 SESSION July 2014ASSESOR(S) DR ID ELLEFSENEXTERNAL MODERATOR DR GB O’REILLYDURATION 3 HOURS MARKS 150NUMBER OF PAGES: 12 PAGESINSTRUCTIONS:1. ALL QUESTIONS MUST BE ANSWERED2. ANSWER THE QUESTIONS IN NUMERIC ORDER3. CALCULATORS ARE NOT PERMITTED TO BE USED4. WRITE CLEARLY AND LEGIBLYREQUIREMENTS: NONE1CSC3A10 — July 2014QUESTION 1(a) [5]The choice of Abstract Data Type is depended on the input data that is being used. Discussthe generalised construction of Abstract Data Types and why the environment that theyare being used in will influence which ADT you choose.(b) [5]Compare and contrast the implementation of a Stack and a Queue. Provide three (3)points of comparison with regards to the structural differences between these structures,and discuss two (2) situations where a Stack or a Queue will be used.(c) [5]When you consider Worst-Case Runtime analysis, what are the benefits of using primitivecounting as a method of analysis?(d) [5]Consider the following function and using primiative counting express the runtime of thisfunction in Big-Oh notation. You should also state your assumptions and which operationsshould be considered to be primitive.public int a(int a[], int n) {int i = 0;int t = 0;for (i = 0; i < n; i++) {if (a%2 == 0) {int b = 0;while (b < n) {if (a[i] % 2 != 0) {t += a[i];}}} else {int c = a[i];c += a[i];t += c;}}return t;}[20]2CSC3A10 — July 2014QUESTION 2(a) [5]Provide a definition for the following components of a graph:1. End Vertices2. Incident Edges3. Adjacent Vertices4. Degree of a Vertex5. Self-loop(b) [5]Provide an algorithm that releases a Depth First Search of a graph.(c) [2]DFS makes use of the “Decorator Design Pattern”. Provide a brief explanation of thispattern.(d) [3]One application for a graph is to calculate the Minimum Spanning Tree. Elaborate on amethod for calculating the Minimum Spanning Tree for graph.(e) [5]Compare and contrast a digraph with a graph and elaborate on possible applications of adigraph over a graph.[20]3CSC3A10 — July 2014QUESTION 3Consider the following 2-4 tree provided below. Provide a graphic representation of the 2-4Tree resulting from the following operations. You must provide a graphic representation ofeach step in the process, including all intermediate operations. Relevant operations mustmake use of the Inorder predecessor. Removal operations should follow from the tree thatresulted from the insertion operations.1. Insert nodes that contain the following keys: (inserted one-by-one, in the given order)72, 36, 5, 682. Delete nodes that contain the following keys: (removed one-by-one, in the given order)72, 82, 24, 68The 2-4 tree is in the current state (leaf nodes are not shown, however they are assumed toexist):24 82 94The sequence of operations that result in the current state of the 2-4 tree can be found inAppendix A on Page 10.[15]4CSC3A10 — July 2014QUESTION 4Consider the following AVL tree provided below. Provide a graphic representation of theAVL Tree resulting from the following operations. You must provide a graphic representationof each step in the process, including all intermediate operations. Relevant operations mustmake use of the Inorder successor. Removal operations should follow from the tree thatresulted from the insertion operations.1. Insert nodes that contain the following keys: (inserted one-by-one, in the given order)93, 35, 692. Delete nodes that contain the following keys: (removed one-by-one, in the given order)23, 93, 69, 80, 35, 11, 1The AVL tree is in the current state:111 3223 80The sequence of operations that result in the current state of the AVL tree can be found inAppendix B on Page 10.[15]5CSC3A10 — July 2014QUESTION 5Consider the following Red-Black tree provided below. Provide a graphic representationof the Red-Black Tree resulting from the following operations. You must provide a graphicrepresentation of each step in the process, including all intermediate operations. Relevantoperations must make use of the Inorder successor. Removal operations should follow fromthe tree that resulted from the insertion operations.1. Insert nodes that contain the following keys: (inserted one-by-one, in the given order)34, 59, 23, 932. Delete nodes that contain the following keys: (removed one-by-one, in the given order)93, 45, 2, 62, 59, 23, 34The Red-Black tree is in the current state:262 6245The sequence of operations that result in the current state of the Red-Black tree can be foundin Appendix ?? on Page ??.[20]6CSC3A10 — July 2014QUESTION 6(a) [5]Explain why a Binary Tree that is not a Complete Binary Tree cannot have an efficiencyequivalent to O(logn).(b) [5]You have been approached to design a database of items that can be serialised to disk.The restriction is that you must make use of a tree and that you have to choose betweenan AVL or a 24-tree (or a direct modification of those trees). Explain which of these treesyou would choose being sure to justify your choices.(c) [3]Explain why Splay Trees have an amortised runtime of O(logn) even though their structureis very different to a Binary Search Tree.(d) [5]Complete the following table regarding a comparison between runtime and implementationof a Hash Table, Skip List, and (2,4) Tree:Search Insert Delete CommentHash Table (1) (2) ExpectedO(1)(7)Skip List (3) ExpectedO(logn)(4) (8)(2,4) Tree Worst-CaseO(logn)(5) (6) (9)(e) [2]Provide an algorithm to realise upheap in a heap.[20]7CSC3A10 — July 2014QUESTION 7(a) [4]A Priority Queue can be implemented using a heap or an array. Explain the two (2) possibleimplementations when using an array to implement a Priority Queue and the impact thatthese implementations have on the runtime of the Priority Queue structure.(b) [2]Discuss the two phases found in a Heap Sort function, and indicate which of these twophases will impact the runtime of a Heap Sort function.(c) [8]You have been approach by a company to sort an array of values using a heap. The use ofthe heap is required by the company as part of their corporate strategy for tree-based datastructures. However, they have the restriction that the sorting algorithm must not use anyextra space (other than the space initially allocated). Explain how you would achieve this,and also comment on the runtime of such a function.(d) [3]List three (3) methods of managing collisions in a hash table.(e) [3]Explain the role of “Comparators” for ADTs such as heaps.[20]8CSC3A10 — July 2014QUESTION 8Consider the following description of a graph.vertex adjacent verticies1 (2,3,4)2 (1,3,4)3 (1,2,4)4 (1,2,3,6)5 (6,7,8)6 (4,5,7)7 (5,6,8)8 (5,7)(a) [10]Provide a graphical representation of the above graph.(b) [5]What sequence of nodes would be output if a Breadth First Search was performed on theabove graph starting at node “1”?(c) [5]What sequence of nodes would be output if a Depth First Search was performed on theabove graph starting at node “1”?[20]9CSC3A10 — July 2014A 2-4 Tree OperationsInsert 82:82Insert 94:82 94Insert 24:24 82 94B AVL Tree OperationsInsert 11:11Insert 23:1123Insert 1:111 23Insert 32:111 2332Insert 80:10CSC3A10 — July 2014111 233280Rebalancing:111 3223 80C Red-Black Tree OperationsInsert 2:2Set Root Black:2Insert 26:226Insert 62:22662Double Red: 1. Trinode Restructure:11CSC3A10 — July 2014262 62Insert 45:262 6245Double Red: 2.1 Recolouring:262 6245— End of exam —12

Advanced Data Structures and Algorithms

FACULTY OF SCIENCEACADEMY OF COMPUTER SCIENCE AND SOFTWARE ENGINEERINGMODULE CSC3A10ADVANCED DATA STRUCTURES AND ALGORITHMSCAMPUS APKEXAM EXAMINATION — JUNE 2014DATE 2014-06-03 SESSION 12:30 – 15:30ASSESOR(S) DR ID ELLEFSENEXTERNAL MODERATOR DR GB O’REILLYDURATION 3 HOURS MARKS 150NUMBER OF PAGES: 11 PAGESINSTRUCTIONS:1. ALL QUESTIONS MUST BE ANSWERED2. ANSWER THE QUESTIONS IN NUMERIC ORDER3. CALCULATORS ARE NOT PERMITTED TO BE USED4. WRITE CLEARLY AND LEGIBLYREQUIREMENTS: NONE1CSC3A10 — June 2014QUESTION 1(a) [5]Provide a complete discussion on what is meant by the term Abstract Data Type andhow Abstract Data Types can be realised in Java.(b) [5]Compare and contrast the implementation of a Singly-linked and Doubly-linked list. Providetwo (2) points of comparison with regards to the structural differences between thesestructures, and discuss three (3) advantages and disadvantages of these structures.(c) [5]The performance of operations on Abstract Data Types is often expressed using Big-Ohnotation. Provide a discussion why this is the case and what advantages this approach hasover other alternatives.(d) [5]Consider the following function and using primiative counting express the runtime of thisfunction in Big-Oh notation. You should also state your assumptions and which operationsshould be considered to be primitive.public int f(int a[], int n) {int i = 0;while (i < n) {int b = n-i;float c = 0.0;int j = 0;while (j < b) {c += a[i] + a[j] / c;}for (int k = 0; k < n; k++) {c += a[j];}i++;}}[20]2CSC3A10 — June 2014QUESTION 2(a) [5]Discuss the following components of a graph and provide a detailed explination ofhow these components can be used to form a graph.1. Edges2. Vertices(b) [5]Provide an algorithm that releases a Breadth First Search of a graph.(c) [2]Compare and contrast Breadth First Search the Depth First Search in terms of applicationand runtime complexity.(d) [3]One application for a graph is to calculated the Shortest Path. Elaborate on a methodfor calculating the shortest path between two points in a graph.(e) [5]Graphs and Trees are common data structures used in Computer Science problems. Inmany cases the choice of a Tree or a Graph is dependent on the problem being solved.Provide a discussion of the differences between Trees and Graphs in terms of applicablealgorithms, runtime complexities, and uses.[20]3CSC3A10 — June 2014QUESTION 3Consider the following 2-4 tree provided below. Provide a graphic representation of the 2-4Tree resulting from the following operations. You must provide a graphic representation ofeach step in the process, including all intermediate operations. Relevant operations mustmake use of the Inorder predecessor. Removal operations should follow from the tree thatresulted from the insertion operations.1. Insert nodes that contain the following keys: (inserted one-by-one, in the given order)28, 98, 50, 222. Delete nodes that contain the following keys: (removed one-by-one, in the given order)93, 90, 49, 22The 2-4 tree is in the current state (leaf nodes are not shown, however they are assumed toexist):49 90 93The sequence of operations that result in the current state of the 2-4 tree can be found inAppendix A on Page 10.[15]4CSC3A10 — June 2014QUESTION 4Consider the following AVL tree provided below. Provide a graphic representation of theAVL Tree resulting from the following operations. You must provide a graphic representationof each step in the process, including all intermediate operations. Relevant operations mustmake use of the Inorder successor. Removal operations should follow from the tree thatresulted from the insertion operations.1. Insert nodes that contain the following keys: (inserted one-by-one, in the given order)84, 87, 50, 42. Delete nodes that contain the following keys: (removed one-by-one, in the given order)84, 50, 79, 48, 23, 25The AVL tree is in the current state:48252379The sequence of operations that result in the current state of the AVL tree can be found inAppendix B on Page 10.[15]5CSC3A10 — June 2014QUESTION 5Consider the following Red-Black tree provided below. Provide a graphic representationof the Red-Black Tree resulting from the following operations. You must provide a graphicrepresentation of each step in the process, including all intermediate operations. Relevantoperations must make use of the Inorder successor. Removal operations should follow fromthe tree that resulted from the insertion operations.1. Insert nodes that contain the following keys: (inserted one-by-one, in the given order)18, 6, 77, 36, 59, 472. Delete nodes that contain the following keys: (removed one-by-one, in the given order)83, 50, 36, 6The Red-Black tree is in the current state:8350The sequence of operations that result in the current state of the Red-Black tree can be foundin Appendix C on Page 11.[20]6CSC3A10 — June 2014QUESTION 6(a) [5]Provide a diagram that demonstrates the relationship between the structure of a Red-BlackTree and a 2-4 Tree.(b) [5]Binary Search Trees have an ordering property that determines how the nodes in the treeare structured. Heaps also have an ordering property that determines how the nodes inthe tree are structured. Explain the difference between the Binary Search Tree OrderingProperty and the Heap Ordering Property and the result that these properties have on theresulting trees.(c) [10]Provide a proof to show that the height of an AVL tree storing n keys is bound by O(logn).[20]7CSC3A10 — June 2014QUESTION 7(a) [2]Explain the phases of a “Divide and Conquer” algorithm.(b) [3]Provide brief pseudocode that explains how a Merge Sort operates.(c) [5]Using your knowledge of algorithmic analysis explain why QuickSort is described as hav-ing a worst-case runtime of O(n2), but an expected runtime of O(n logn). Under whatconditions will QuickSort exhibit a O(n2) runtime?(d) [2]Explain what is meant by “Stable Sorting Algorithm”.(e) [2]Under what conditions would a “Stable Sorting Algorithm” be important? Justify youranswer.(f) [3]A Lexicographic Sort is a type of Stable Sorting Algorithm. Provide and example of aLexicographic Sort and explain how it would work.(g) [3]You have been approached by a company to write a function that can be used to returnthe number of bit that are “set” in a binary number. For instance, the binary number00110012 has 3 bits set, therefore the function should return 3. You should write a functionthat operates as quickly as possible, and memory usage is not a concern. Explain how youwould approach such a problem, what data structures you would use and what the expectedruntime of your approach would be.[20]8CSC3A10 — June 2014QUESTION 8(a) [10]Bob loves programming languages and wants to plan his university course schedule for thenext few years (assuming that the is allowed to choose). He is interested in the followingnine programming language courses: ASM, C, C++, Java, C#, Erlang, Scala, SML andSmalltalk. The prerequisites for the courses are:• ASM - none• C - ASM• C++ - ASM, C• Java - C, C++• C# - Java, ASM, C++• Erlang - SML, Smalltalk• Scala - Java, SML, Smalltalk• SML - ASM• Smalltalk - noneDraw a graph that demonstrates the dependencies between the above courses, and thenoutline the order of courses Bob should take to cover all the languages he wants to learn.(b) [10]A Bitset is a common structure that allows set operations to be defined in terms of anumber of bits that are set or not set in an array of integer values. In most cases, theimplementation of a bitset makes use of integer arrays (meaning that you can have atmost 32-bits per integer in the array - assuming that you are using 32-bit integer values).Provide an implementation for a class that realises the ISet interface described below.Your implementation should be as efficient as possible and you only need to provide theimplementation for the functions below. You can assume that the constructors have beenimplemented for you and that the bitset array has been initialised where every bit is unset.1. set(...) — will set the indicated bit in the bitset.2. unset(...) — will unset the indicated bit in the bitset.3. union(...) — will create a new bitset where bits set in the current bitset arecombined with the bits set in the passed bitset (union operation).4. difference(...) — will create a new bitset where the bits are set if they are set inthe current bitset but not in the passed bit set (difference operation).pub l i c i n t e r f a c e I S e t {p r i v a t e i n t [ ] b i t s e t ;pub l i c vo id s e t ( i n t b i tnumber ) ;pub l i c vo id unse t ( i n t b i tnumber ) ;pub l i c I S e t un ion ( I S e t o t h e rS e t ) ;pub l i c I S e t d i f f e r e n c e ( I S e t o t h e rS e t ) ;}[20]9CSC3A10 — June 2014A 2-4 Tree OperationsInsert 49:49Insert 93:49 93Insert 90:49 90 93B AVL Tree OperationsInsert 25:25Insert 48:2548Insert 79:254879Rebalancing:4825 79Insert 23:10CSC3A10 — June 201448252379C Red-Black Tree OperationsInsert 83:83Set Root Black:83Insert 50:8350— End of exam —11

Advanced Data Structures and Algorithms

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University of Johannesburg Institutional Repository

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