Question

SECTION B (4 QUESTIONS, 90 MARKS) Question 4. [32 marks in total] For this question, assume the definition of Java class Heap given in the Appendix. The heaps referred to in this question are all maxHeaps. a. (5 marks) Insert into an empty (binary) heap the values 35, 4, 72, 36, 49, 50. Draw all intermediate steps b. (3 marks) Carry out one deletion operation on the final heap in (a.) above. c. (2 marks) Give the worst-case time complexity of insertion and deletion operations for a heap with n elements, and justify this complexity (in two sentences or so). d. (4 marks) Imagine a variation to the binary heap of lectures called terHeap, where instead of each node having two children, it has three children. The tree is still required to be complete, and the heap property that the parent is at least as large as its children must be maintained. An example of a terHeap is as follows 70 60 40 50 10 45 10 40 30 25 There is similarly an array representation index 0 60 6 25 10 45 5 7 1 8 value 70 40 50 10 40 30 Call the children of each node leftChild, midChild, and rightChild. What are the equations that relate the index of a node to the indices of its children in this array repre sentation? e. (7 marks) Using your answer to (d.) above or otherwise, modify the code for heapRebuild (in the Appendix) to handle terHeaps rather than regular binary heaps. You should write out the full code in the answer book. (You may leave out the comments from the original code.) f. (5 marks) Imagine a further generalisation of the heap, so that rather than the standard Heap where each node has two children, or the terHeap of (d.) where each node has three children, you have an naryHeap where each node has n children (subject to being a complete tree). the children are represented In the origin and rightChild. For an naryHeap's heapRebuild, describe a possible representation for heapRebu: process by riables child Page 2

Answer 1

(a)

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(b)

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Raplocing 6ith Iast node hoole 50 36 49 49 is greator than 36. So 36 36 35 (35 49 interchanging get 4 we Nots, deleting replacing Yoot hode 49 and the with 4,which is the lat node 36 greator than 4. IS 36 35 B6 So intorchanging shlld 35 done be 35 w hich is lest node and veplace toith Now delate 36 35 Now dolating Toot node is xeplacing toith and 35 last node. j.e 4 which
(c)

(d)

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(e)

fallows Cd) Gi ven axYay representation is ar index 6 7 4 5 89 3 2 value 10 40 30 45 70 60 40 50 25 10 xepresend Aopth (hoig ht) of e We Can tree th level nth node So fox index of xepresnted of noder be t ax Can 2n+1, 2n+ 2 _d+ and 2 +2n+ So indicer lenel 2dt1 2n 2 are On and +2n+ 2 children is mentioned it hexe Since single node . preent fox are 2n left chied Fox 2 nt1 Midd le child 2nt 2 Rigkt child

Should have uploaded the appendix programme for modification as mentioned.