CS106X – Programming Abstractions in C++

1. CS2 in C++ Peer Instruction Materials by Cynthia Bailey Lee is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org. CS106X – Programming Abstractions in C++ Cynthia Bailey Lee

2. Today’s Topics: Heaps! • Binary heaps • Insert, delete • Reasoning about outcomes in Binary heaps, and Big O analysis • Heapsort algorithm • (Schedule note: we’ll save stack and queue implementation for another day…)

3. Binary heap insert and delete

4. Binary heap insert + “bubble up”

5. Binary heap insert

6. Binary heap delete + “trickle-down”

7. Binary heap delete + “trickle-down”

8. Heap outcomes by insert order Create a MIN heap by inserting the elements, one by one, in the order given below for the second letter of your first 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}

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

10. 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}

11. How many distinct min-heaps are possible for the elements {3, 9, 18, 22, 34}? • 1-2 • 3-4 • 5-8 • 5! (5 factorial) • Other/none/more

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

13. Heapsort

14. Heapsort is super easy • Insert unsorted elements one at a time into a heapuntil all are added • 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!

15. Implementing heapsort Devil’s in the details

16. Build max-heap by inserting elements one at a time:

17. Build max-heap by inserting elements one at a time: What is the next configuration in this sequence? 12, 8, 2, 10 12, 10, 2, 8 12, 10, 8, 2 Other/none/more than one

18. Build max-heap by inserting elements one at a time:

19. Sort array by removing elements one at a time:

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

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

22. Complexity • How many times do we do insert() when building the heap? • How much does each cost? • How many times do we delete-max() when doing the final sort? • How much does each cost?