CS106X Programming Abstractions in C++ Dr. Cynthia Bailey Lee

Transcription

1 CS106X Programming Abstractions in C++ Dr. Cynthia Bailey Lee

2 2 Today s Topics: 1. Binary tree 2. Heap

3 Priority Queue Emergency Department waiting room operates as a priority queue: patients are sorted according to seriousness, NOT how long they have waited.

4 Priority queue implementations We want the best of both Fast add AND fast remove/peek We will investigate trees as a way to get the best of both worlds +

5 Binary trees

6 A binary tree In computer science, a binary tree is a tree data structure in which each node has at most two child nodes, usually distinguished as "left" and "right". (Thanks, Wikipedia!)

7 How many of these are valid binary trees? A. 0-3 B. 4 C. 5 D. 6 E. 7-8

8 A node object for binary trees Similar to a linked list node, it contains a pointer to data, and a pointer to the next elements Whereas a linked list node has just one next pointer, a binary node tree has two child pointers, left and right

9 A node object for binary trees Similar to a linked list node, it contains a pointer to data, and a pointer to the next elements Whereas a linked list node has just one next pointer, a binary node tree has two child pointers, left and right

10 Heaps

11 Heaps Heaps are one kind of binary tree They have a few special restrictions, in addition to the usual binary tree ADT: Must be complete Ordering of data must obey heap property Min-heap version: a parent s data is always its children s data Max-heap version: a parent s data is always its children s data

12 How many of these could be valid heaps? A. 0-1 B. 2 C. 3 D. 4 E. 5-8

13 How many of these are valid min-heaps? A. 0 B. 1 C. 2 D. 3

14 In how many places could the smallest number in this min-heap be located? A. 0-2 B. 3-4 C. 5-6 D. 7-8

15 In how many places could the number 35 be located in this min-heap? A. 0-2 B. 3-4 C. 5-6 D

16 In how many places could the number 35 be located in this min-heap? A. 0-2 B. 3-4 C. 5-6 D Min heaps also good for priority queues, if high priority in your system actually means low value (i.e. 1 means most important)

17 Heap in an array

18 Heap in an array We actually do NOT typically use a node object to implement heaps Because they must be complete, they fit nicely into an array, so we usually do that

19 Heap in an array For tree of height h, array length is 2 h -1 For a node in array index i: Parent is at array index: A. i 2 B. i / 2 C. (i 1)/2 D. 2i

20 Heap in an array For tree of height h, array length is 2 h -1 For a node in array index i: Left child is at array index: A. i +1 B. i + 2 C. 2i D. 2i + 1

21 Heap in an array For tree of height h, array length is 2 h -1 For a node in array index i: Parent is at array index: (i 1)/2 Left child is at array index: 2i + 1 Right child is at array index: 2i + 2

22 Heap insert and delete

23 Heap insert

24 Heap delete

25 TRUE OR FALSE There is only one configuration of a valid min-heap containing the elements {34, 22, 3, 9, 18} A. TRUE B. FALSE

26 Heap outcomes by insert order Create a MIN heap by inserting the elements, one by one, in the order given below for the first letter of your last name: A-F: {3, 9, 18, 22, 34} G-L: {3, 22, 18, 9, 34} M-R: {9, 22, 18, 3, 34} S-Z: {18, 22, 3, 9, 34}

27 Heap outcomes by insert order Create a MIN heap by inserting the elements, one by one, in the order given below for the first letter of your last name: A-F: {18, 9, 34, 3, 22} G-L: {3, 18, 22, 9, 34} M-R: {22, 9, 34, 3, 18} S-Z: {34, 3, 9, 18, 22}

28 How many distinct min-heaps are possible for the elements {3, 9, 18, 22, 34}? A. 1-2 B. 3-4 C. 5-8 D. 5! E. Other/none/more

29 Time cost What is the worst-case time cost for each heap operation: Add, Remove, Peek? A. O(n), O(1), O(1) B. O(logn), O(logn), O(1) C. O(n), O(logn), O(logn) D. Other/none/more

30 Heapsort

31 Heapsort is super easy 1. Insert unsorted elements one at a time into a heap until all are added 2. Remove them from the heap one at a time (we will always be removing the next biggest item, for max-heap; or next smallest item, for min-heap) THAT S IT!

32 Implementing heapsort Devil s in the details

33 We can do the entire heapsort in place in one array Unlike mergesort, we don t need a separate array for our workspace We can do it all in place in one array (the same array we were given as input)

34 Build heap by inserting elements one at a time:

35 Sort array by removing elements one at a time:

36 Build heap by inserting elements one at a time IN PLACE:

37 Sort array by removing elements one at a time IN PLACE: