Winter 2016 COMP-250: Introduction to Computer Science. Lecture 20, March 24, 2016

Transcription

1 Winter 201 COMP-250: Introduction to Computer Science Lecture 20, March 2, 201

2 Public Announcement

3 Public Announcement In order to have your exam printed by the Exam Office, it must be submitted by: Course : Title: Intro to Computer Science rch 2th, 201 COMP 250 Sect: 1 Mar Exam Date /28/2 Exam Date: Examiner: Associate Examiner: Name: Prof. C. Crépeau Examiner s signature: /28/201 Office #: Exam Time: 2:00:00 Cell #: PM Name: Tel #: Associate Examiner s signature: 1. Is this a CLOSED BOOK or OPEN BOOK exam? Cell #: INFORMATION AND AUTHORIZATION OF MATERIALS: 2. Is this a MULTIPLE-CHOICE Exam? Multiple versions must be submitted, regardless of the number of students, questions or whether the exam is only partially multiple-choice ( versions for enrolments of 200

4 Public Announcement

5 HEAPS I Heaps Properties Insertion and Deletion

6 Heaps Aheap is a binary tree T that stores a collection of keys (or key-element pairs) at its internal nodes and that satisfies two additional properties: - Order Property: key(parent) key(child) - Structural Property: all levels are full, except the last one, which is left-filled (complete binary tree)

7 Not Heaps bottom level is not left-filled

8 Not Heaps bottom key(parent)> level is key(child) not left-filled

9 Height of a Heap A heap T storing n keys has height h = log(n + 1), which is O(log n) n h = 2 h = 2 h h h 1 1 h

10 Height of a Heap A heap n 1 T + storing n keys... +2has h-1 = height 2 h - 1h = log(n + 1), which 0is O(log n) n h = 2 h = 2 h-1 0 h h h h h 1 1 Therefore 2 h-1 n 2 h - 1 h Taking logs, we get log (n + 1) h log n + 1 Which implies h = log(n+1)

11 Heap Insertion So here we go... The key to insert is

12 Heap Insertion Add the key in the next available position in the heap Now begin Upheap.

13 Upheap Swap parent-child keys out of order

14 Upheap Swap parent-child keys out of order

15 Upheap Continues

16 Upheap Continues

17 End of Upheap Upheap terminates when new key is greater than the key of its parent or the top of the heap is reached (total #swaps) (h 1), which is O(log n)

18 Removal From a Heap RemoveMin()

19 The removal of the top key leaves a hole We need to fix the heap First, replace the hole with the last key in the heap Then, begin Downheap

20 Downheap

21 Downheap Downheap compares the parent with the smallest child. If the child is smaller, it switches the two.

22 Downheap Continues

23 Downheap Continues

24 Downheap Continues

25 Downheap Continues

26 End of Downheap Downheap terminates when the key is greater than the keys of both its children or the bottom of the heap is reached. (total #swaps) (h 1), which is O(log n)

27 HEAPS II Implementation HeapSort Bottom-Up Heap Construction Locators

28 Implementation of a Heap public class HeapPriorityQueue implements PriorityQueue { } BinaryTree T; Position last; Comparator comparator;... heap last comp (,C) < = > (5,A) (,Z) (15,K) (9,F) (7,Q) (20,B) (1,X) (25,J) (1,E) (12,H) (11,S) (8,W)

29 Implementation of a Heap(cont.) Two ways to find the insertion position z in a heap: (2,C) (5,A) (,C) (15,K) (9,F) (7,Q) (,Z) z (1,X) (25,J) (1,E) (12,H) (11,S) (8,W) (20,B) (10,L)

30 Implementation of a Heap(cont.) Two ways to find the insertion position z in a heap: (5,A) (2,C) (,C) u (,C) (,Z) (15,K) (9,F) (7,Q) (,Z) (20,B) (1,X) (25,J) (1,E) (12,H) (11,S) (8,W) (20,B) (10,L) z w z

31 Vector Based Implementation Updates in the underlying tree occur only at the last element A heap can be represented by a vector, where the node at rank i has - left child at rank 2i and - right child at rank 2i

32 Vector Based Implementation Updates in the underlying 1 tree occur only at the last element A heap can be 2 represented by a vector, 3 where the node at rank i has - left child at rank 2i and - right child at rank 2i The leaves do no need 5to be explicitly stored 7 Insertion and removals into/from the heap correspond 8 to9 insertlast and 12removeLast 13 on the vector, respectively

33 Heap Sort All heap methods run in logarithmic time or better If we implement PriorityQueueSort using a heap for our priority queue, insertitem and removemin each take O(log k), k being the number of elements in the heap at a given time. We always have at most n elements in the heap, so the worst case time complexity of these methods is O(log n). Thus each phase takes O(n log n) time, so the algorithm runs in O(n log n) time also. This sort is known as heap-sort. The O(n log n) run time of heap-sort is much better than the O(n 2 ) run time of selection and insertion sort. In-Place Heap-Sort Do not use an external heap Embed the heap into the sequence, using the vector representation

34 Bottom-Up Heap Construction build (n + 1)/2 trivial one-element heaps

35 Bottom-Up Heap Construction build now build (n + 1)/2 three-element trivial one-element heaps top heaps of them

36 Bottom-Up Heap Construction downheap to preserve the order property

37 Bottom-Up Heap Construction downheap now form seven-element to preserve the heaps order property

38 Bottom-Up Heap Construction (cont.)

39 Bottom-Up Heap Construction (cont.)

40 Bottom-Up Heap Construction (cont.) The End

41 Analysis of Bottom-Up Heap Construction Proposition: Bottom-up heap construction with n keys takes O(n) time. - Insert (n + 1)/2 nodes - Insert (n + 1)/ nodes and downheap them - Insert (n + 1)/8 nodes and downheap them visual analysis: n inserts, n/2 upheaps with total O(n) running time

42 Chapter 8

43 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a2,a1,a3 a3,a1,a2 a1,a3,a2 a2,a3,a1 a3,a2,a1 a1,a3,a2 a3,a1,a2 a2,a1,a3 a2,a3,a1 a3,a2,a1 a1,a3,a2 a3,a1,a2 a2,a3,a1 a3,a2,a1 a2,a1,a3 a1,a3,a2 a3,a1,a2 a2,a3,a1 a3,a2,a1

44 N! N!/2 N!/2 log N! θ(n log N) N!/ N!/ N!/ N!/ 2,a3

45 Winter 201 COMP-250: Introduction to Computer Science Lecture 20, March 2, 201

46 Hardik Omar Faiz Lekan Faizy Chris David B. David B.R. DoYeon