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## PowerPoint Slideshow about ' Heaps' - macon

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What is a heap?

- Like a binary search tree, but less structure within each level.
- Guarantees:
- Parent better than child
- That’s it!
- What does better mean?
- What do we use heaps for?

What can we do with a heap

- Keep the things we are most interested in close to the top (and fast to access)
- For instance: suppose we have some data.
- We want to prioritize it.
- We want to keep the most important thing at the top, at all times.
- Min heap: priority 1 is more important than 100
- Max heap: other way around

Abstract data type (ADT)

- We are going to use max-heaps to implement the priority queue ADT
- (for us, priority 100 is more important than priority 1)
- A priority queue Q offers (at least) 2 operations:
- Extract-max(Q): returns the highest priority element
- Insert(Q, e): inserts e into Q
- (and maintain the heap-order property)
- Can do same stuff with BST… why use heaps??
- BST extract-max is O(depth); heap is O(log n)!

Max-heap order property

- Look at any node u, and its parent p.
- p.priority >= u.priority
- After we Insert or Extract-max, we make sure that we restore the max-heap order property.

Maintaining heap-ordering

- FYI: We can prove that the heap-order property is always satisfied, by induction (on the sequence of Inserts and Extract-max’es that happen).
- Initially, there are no nodes, so it is true.
- Consider an Insert or Extract-max.
- Suppose it is true before this operation.
- Prove it is true after. (2 cases; Insert or Extract-max)
- By the magic (wonder, beauty, etc.) of induction, that’s all you have to show.

Movie time

- Using a heap to get the largest 31 elements from a list (played from 1:39 on)
- Notice that, after inserting, we have to “percolate” the larger elements down
- That’s the rough idea of maintaining the heap-order property
- We do it a little differently. (Insert at the bottom, and then fixup the heap-ordering.)
- (but we don’t care about that right now)

Step 1: represent the heap as an array

- Consider element at index i
- Its children are at 2i and 2i+1

Building a heap: a helper function

I:3

Precondition: trees rooted at L and R are heaps

Postcondition: tree rooted at I is a heap

MaxHeapify(A,I):

L = LEFT(I)

R = RIGHT(I)

If L <= heap_size(A) and A[L] > A[I]

then max = L

else max = I

If R <= heap_size(A) and A[R] > A[max]

then max = R

If max is L or R then

swap(A[I],A[max])

MaxHeapify(A,max)

Case 1: max = L

Need to fix…

L:7

R:5

I:7

Case 2: max = I

Heap OK!

L:3

R:5

I:5

Case 3: max = R

Need to fix…

L:3

R:7

The main function

BUILD-MAX-HEAP(A):

for i = heap_size(A)/2 down to 1

MaxHeapify(A,i)

- What does this look like?
- MaxHeapify animation

- Node at depth d has height <= h-d
- Cost to “heapify” one node at depth d is <= c(h-d)
- Don’t care about constant c
- Cost to heapify all nodes at depth d is <= 2^d(h-d)

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