Heaps and Heapsort 1 October 2020 OSU CSE 1 Heaps A heap is a - - PowerPoint PPT Presentation

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Heaps and Heapsort 1 October 2020 OSU CSE 1 Heaps A heap is a - - PowerPoint PPT Presentation

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Heaps and Heapsort 1 October 2020 OSU CSE 1 Heaps A heap is a binary tree of T that satisfies two properties: Global shape property: it is a complete binary tree Local ordering property: the label in each node is smaller than

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Heaps and Heapsort

1 October 2020 OSU CSE 1

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Heaps

  • A heap is a binary tree of T that

satisfies two properties:

– Global shape property: it is a complete binary tree – Local ordering property: the label in each node is “smaller than or equal to” the label in each of its child nodes

1 October 2020 OSU CSE 2

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SLIDE 3

Heaps

  • A heap is a binary tree of T that

satisfies two properties:

– Global shape property: it is a complete binary tree – Local ordering property: the label in each node is “smaller than or equal to” the label in each of its child nodes

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A complete binary tree is one in which all levels are “full” except possibly the bottom level, with any nodes on the bottom level as far left as possible.

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Heaps

  • A heap is a binary tree of T that

satisfies two properties:

– Global shape property: it is a complete binary tree – Local ordering property: the label in each node is “smaller than or equal to” the label in each of its child nodes

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Also in the picture is (as with BSTs, sorting, etc.) a total preorder that makes this notion precise.

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Simplification

  • For simplicity in the following illustrations,

we use only one kind of example:

– T = integer – The ordering is ≤

  • Because heaps are used in sorting, where

duplicate values may be involved, we allow that multiple nodes in a heap may have the same labels (i.e., we will not assume that the labels are unique)

1 October 2020 OSU CSE 5

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The Big Picture

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x

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The Big Picture

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y x

This tree’s root label y satisfies x ≤ y

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The Big Picture

1 October 2020 OSU CSE 8

y x

Observe: This tree is also a heap; and for each node in this tree with label z, we have: x ≤ y ≤ z.

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The Big Picture

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y x

This tree’s root label y satisfies x ≤ y

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The Big Picture

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y x

Observe: This tree is also a heap; and for each node in this tree with label z, we have: x ≤ y ≤ z.

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Examples of Heaps

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5 2 1 8 7 8 2 1 1 8

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Non-Examples of Heaps

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8 1 2 1 5 1 2 8 2 1 1 5

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Non-Examples of Heaps

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8 1 2 1 5 1 2 8 2 1 1 5

Shape property is violated here: not all nodes at the bottom level are as far left as possible.

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Non-Examples of Heaps

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8 1 2 1 5 1 2 8 2 1 1 5

Ordering property is violated here: value is out of order with that of its right child.

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Non-Examples of Heaps

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8 1 2 1 5 1 2 8 2 1 1 5

Shape property is violated: two levels are not “full”.

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Heapsort

  • A heap can be used to represent the

values in a SortingMachine, as follows:

– In changeToExtractionMode, arrange all the values into a heap – In removeFirst, remove the root, and adjust the slightly mutilated heap to make it a heap again

1 October 2020 OSU CSE 16

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SLIDE 17

Heapsort

  • A heap can be used to represent the

values in a SortingMachine, as follows:

– In changeToExtractionMode, arrange all the values into a heap – In removeFirst, remove the root, and adjust the slightly mutilated heap to make it a heap again

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Why should this work?

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How removeFirst Can Work

  • If the root is the only node in the heap,

then after removing it, what remains is already a heap; nothing left to do

  • If the root is not the only node, then

removing it leaves an “opening” that must be filled by moving some other value in the heap into the opening

1 October 2020 OSU CSE 18

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How removeFirst Can Work

  • If the root is the only node in the heap,

then after removing it, what remains is already a heap; nothing left to do

  • If the root is not the only node, then

removing it leaves an “opening” that must be filled by moving some other value in the heap into the opening

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The question is: which one?

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Example: A First Attempt

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4 3 2 1 5

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Example: A First Attempt

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4 3 2 5

If we remove the root, leaving this

  • pening ...
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Example: A First Attempt

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4 3 5

... we might move up the smaller child ...

2

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Example: A First Attempt

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4 3 5

... now creating another opening ...

2

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Example: A First Attempt

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4 5

... so, we might move up the smaller child.

2 3

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Example: A First Attempt

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4 5

Is the result necessarily a heap?

2 3

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Example: A Second Attempt

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4 3 2 1 5

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Example: A Second Attempt

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4 3 2 5

This time, let’s maintain the shape property ...

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Example: A Second Attempt

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4 2 5

... by promoting the last node on the bottom level.

3

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Example: A Second Attempt

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4 2 5

Now, we can “sift down” the root into its proper place ...

3

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Example: A Second Attempt

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4 3 5

... by swapping it with its smaller child ...

2

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Example: A Second Attempt

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4 3 5 2

... and then “sifting down” the root of that subtree.

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Example: A Second Attempt

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4 3 5 2

Is the result necessarily a heap?

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Pseudo-Contract

/** * Restores a complete binary tree to be a heap * if only the root might be out of place. * @updates t * @requires * [t is a complete binary tree] and * [both subtrees of the root of t are heaps] * @ensures * [t is a heap with the same values as #t] */ public static void siftDown (BinaryTree<T> t) {...}

1 October 2020 OSU CSE 33

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Building a Heap In the First Place

  • Suppose we have n values in a complete

binary tree, but they are arranged without regard to the heap ordering

  • How can we “heapify” it?

1 October 2020 OSU CSE 34

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Pseudo-Contract

/** * Makes a complete binary tree into a heap. * @updates t * @requires * [t is a complete binary tree] * @ensures * [t is a heap with the same values as #t] */ public static void heapify (BinaryTree<T> t) {...}

1 October 2020 OSU CSE 35

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Hint

  • To see how you might implement

heapify, compare the contracts of siftDown and heapify

  • The only difference: before we can call

siftDown to make a heap, both subtrees

  • f the root must already be heaps

– Once they are heaps, just a call to siftDown will finish the job

1 October 2020 OSU CSE 36

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Hint

  • To see how you might implement

heapify, compare the contracts of siftDown and heapify

  • The only difference: before we can call

siftDown to make a heap, both subtrees

  • f the root must already be heaps

– Once they are heaps, just a call to siftDown will finish the job

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How do we make the subtrees into heaps?

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Example

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1 4 5 3 2

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Example

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5 4 1 3 2

First, recursively “heapify” the left subtree.

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Example

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5 4 1 3 2

Then, recursively “heapify” the right subtree.

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Example

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5 4 1 3 2

Then “sift down” the root, because now

  • nly the root might

be out of place.

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Embedding a Heap in an array

  • While one could represent a heap using a

BinaryTree<T> (as suggested in the pseudo-contracts above), it is generally not done this way

  • Instead, a heap is usually represented

“compactly” using an array of T

1 October 2020 OSU CSE 42

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Interpreting an array as a Heap

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40 30 20 10 50

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1 October 2020 OSU CSE 44

40 30 20 10 50 4 3 1 2

Interpreting an array as a Heap

Because it’s a complete binary tree, the nodes can be numbered top-to- bottom, left-to-right.

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1 October 2020 OSU CSE 45

40 30 20 10 50 4 3 1 2 10 20 50 40 30 1 2 3 4

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1 October 2020 OSU CSE 46

40 30 20 10 50 4 3 1 2 10 20 50 40 30 1 2 3 4 At what index in the array is the left child of the node at index i?

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1 October 2020 OSU CSE 47

40 30 20 10 50 4 3 1 2 10 20 50 40 30 1 2 3 4 At what index in the array is the right child

  • f the node at index i?
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Resources

  • Wikipedia: Heapsort

– http://en.wikipedia.org/wiki/Heapsort

  • Wikipedia: Heap (data structure)

– http://en.wikipedia.org/wiki/Heap_(data_structure)

  • Big Java (4th ed), Sections 16.8, 16.9

– https://library.ohio-state.edu/record=b8540788~S7

1 October 2020 OSU CSE 48