Problem of the Day

To get into club, must get past bouncer Recorded correct responses of those

before you Guard said five & response was four Guard said three & response was five Guard said twelve & response was six Guard said six & response was three

Guard now says to you ten, what do you say?

Problem of the Day

To get into club, must get past bouncer Recorded correct responses of those before

you Guard said five & response was four Guard said three & response was five Guard said twelve & response was six Guard said six & response was three

Guard now says to you ten, what do you say?

Three!

LECTURE 39:COMPLETE BINARY TREES & HEAPS

CSC 212 – Data Structures

Priority Queue ADT

Prioritizes Entrys using their keys For Entrys with equal priorities, order not

specified Priority given to each value when added

to PQ Normally, the priority not changeable while

in PQ Access single Entry: one with the lowest

priority Returns Entry using min() or removeMin()

Location are imaginary – only smallest matters

Heaps

Binary-tree based PQ implementation Still structured using parent-child

relationship At most 2 children & 1 parent for each node

in tree Heaps must also satisfy 2 additional

properties Parent at least as important as its children Structure must form a complete binary tree

2

95

67

BinaryTree

Picturing Linked BinaryTree

B

CA

D

B

A C

D

BinaryTree

root

size4

Legal

CompleteBinaryTree

ADT which extends BinaryTree Add & remove methods defined plus those

inherited For this ADT, trees must maintain

specific shape Fill lowest level first, then can start new

level below it

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95

67

Illegal

2

95

7

6

CompleteBinaryTree

ADT which extends BinaryTree Add & remove methods defined plus those

inherited For this ADT, trees must maintain

specific shape Fill lowest level first, then can start new

level below it Lowest level must be filled in from left-to-

right

Legal

2

95

67

Illegal

2

95

67

What Is Purpose of a Heap?

Root has critical Entry that we always access Entry at root always has smallest priority in

Heap O(1) access time in min() without any real

effort CompleteBinaryTree makes insert()

easy Create leftmost child on lowest level When a level completes, start next one

Useful when:

Upheap

Insertion may violate heap-order property Upheap immediately after adding new Entry

Goes from new node to restore heap’s order Compare priority of node & its parent If out of order, swap node's Entrys Continue upheaping from parent node

Stop only when either case occurs: Found properly ordered node & parent Binary tree's root is reached

6

insert() in a Heap

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5

79

6

1

insert() in a Heap

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79

6

1

Start your upheaping!

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79

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1

Start your upheaping!

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79

1

6

Start your upheaping!

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79

1

6

Upheaping Must Continue

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79

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Upheaping Must Continue

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Upheaping Sounds Icky

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Check If We Should Continue

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79

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6

Stop At The Root

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79

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6

insert() Once Again

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79 3

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6

Upheaping Begins Anew

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79 3

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Maintain Heap Order Property

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79 3

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6

We Are Done With This Upheap!

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79 3

Removing From a Heap

removeMin() must kill Entry at heap’s root For a complete tree, must remove last node

added How to reconcile these two different

demands? Removed node's Entry moved to the

root Then remove node from the complete tree Heap's order preserved by going down

Removing From a Heap

removeMin() must kill Entry at heap’s root For a complete tree, must remove last node

added How to reconcile these two different

demands? Removed node's Entry moved to the

root Then remove node from the complete tree Heap's order preserved by going down

Censored

Downheap

Restores heap’s order during removeMin() Downheap work starts at root Swap with smallest child, if at least out-of-

order Downheaping continues with old smallest

child Stop at leaf or when node is legal

5

Before removeMin() is called

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97

5

Move Last Entry Up To Root

1

2

97

5

Move Last Entry Up To Root

1

2

97

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Move Last Entry Up To Root

9

2

7

5

Compare Parent W/Smaller Child

9

2

7

5

Compare Parent W/Smaller Child

9

2

7

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Compare Parent W/Smaller Child

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7

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Continue Downheaping W/Node

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Continue Downheaping W/Node

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Swap If Out Of Order

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9

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Check If We Should Continue

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9

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Stop When We Reach a Leaf

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Implementation Excuses

upheap & downheap travel height of tree O(log n) running time for each of these Serves as bound for adding & removing

from PQ What drawbacks does heap have?

PriorityQueue can be faster using Sequence

Only for specific mix of operations, however

PriorityQueue using heaps are fastest overall

What Is Purpose of a Heap?

Root has critical Entry that we always access Entry at root always has smallest priority in

Heap O(1) access time in min() without any real

effort CompleteBinaryTree makes insert()

easy Create leftmost child on lowest level When a level completes, start next one

Useful when: Will eventually add & remove everything Equally (large) numbers of adds & removes

Final Exam Schedule

Lab Mastery Exam is:Tues., Dec. 14th from 2:45PM – 3:45PM in OM 119

Final Exam is: Fri., Dec. 17th from 8AM – 10AM in OM 200