CS106X – Programming Abstractions in C++Cynthia Bailey Lee

              CS2 in C++ Peer Instruction

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2

Today’s Topics: Heaps!1. Binary heaps

Insert, delete Reasoning about outcomes in Binary

heaps, and Big O analysis2. Heapsort algorithm3. (Schedule note: we’ll save stack and

queue implementation for another day…)

Binary heap insert and delete

Binary heap insert + “bubble up”

Binary heap insert

Binary heap delete + “trickle-down”

Binary heap delete + “trickle-down”

Heap outcomes by insert orderCreate 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}

TRUE OR FALSE There is only one configuration of a valid

min-heap containing the elements {34, 22, 3, 9, 18}

A. TRUEB. FALSE

Heap outcomes by insert orderCreate 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}

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

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(logn), O(1), O(logn)D. O(n), O(logn), O(logn)E. Other/none/more

Heapsort

Heapsort is super easy1. Insert unsorted elements one at a time

into a heap until all are added2. 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!

Implementing heapsortDevil’s in the details

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

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

What is the next configuration in this sequence?

A.12, 8, 2, 10B.12, 10, 2, 8C.12, 10, 8, 2D.Other/none/more

than one

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

Sort array by removing elements one at a time:

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

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

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?