DS(II) Ch09 PriorityQueues

8/10/2019 DS(II) Ch09 PriorityQueues

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C-C Tsai P.1

Chapter 9 Priority Queues

Single- and Double-EndedPriority Queues

Leftist Trees

Binomial Heaps

Fibonacci Heaps

Pairing Heaps Min-Max Heaps

Interval Heaps

C-C Tsai P.2

Single- and Double-EndedPriority Queues

A priority queue is a collection of elements suchthat each element has an associated priority.

For single-ended priority queues (SEPQ), theoperations supported by a min priority queue

are: SP1: Return an element with minimum priority.

minElement(); minKey();

SP2: Insert an element with an arbitrary priority.

insert();

SP3: Delete an element with minimum priority.

deleteMin();

Other ADTs: size(); isEmpty();


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C-C Tsai P.3

Illustration of Priority Queue

{}-deleteMin()

{(7,D),(9,C)}-deleteMin(){(9,C)}-deleteMin()

{(5,A),(7,D),(9,C)}AminElement()

{}trueisEmpty()

{}errordeleteMin()

{(5,A),(7,D),(9,C)}3size()

{(5,A),(7,D),(9,C)}-removeMin()

{(3,B),(5,A),(7,D),(9,C)}3minKey()

{(3,B),(5,A),(7,D),(9,C)}BminElement()

{(3,B),(5,A),(7,D),(9,C)}-insert(7,D)

{(3,B),(5,A),(9,C)}-insert(3,B)

{(5,A),(9,C)}-insert(9,C)

{(5,A)}-insert(5,A)

Priority QueueoutputOperation

C-C Tsai P.4

Heaps Applied for Priority Queue Max heap tree: a tree in which the key value in

each node is no smaller than the key values in itschildren. A max heapis a complete binary treethat is also a max tree.

Min heap tree: a tree in which the key value in

each node is no larger than the key values in itschildren. A min heapis a complete binary treethat is also a min tree.

[4]

14

12 7

810 6

[1]

[2] [3]

[5] [6]

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810 6

[1]

[2] [3]

[5] [6][4]

max heap tree min heap tree


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C-C Tsai P.5

Double-Ended Priority Queues

For double-ended priority queues (DEPQ), theoperations supported by a min priority queue are: SP1: Return an element with minimum priority.

SP2: Return an element with maximum priority.

SP3: Insert an element with an arbitrary priority.

SP4: Delete an element with minimum priority.

SP5: Delete an element with maximum priority.

Primary operations

Insert

Delete Max Delete Min

Note that a single-ended priority queue supports justone of the above remove operations.

C-C Tsai P.6

Leftist Trees

Leftist trees provide an efficientimplementation of meldable priority queues.

Linked binary tree.

Can do everything a heap can do and in the

same asymptotic complexity. Can meld two leftist tree priority queues in

O(log n) time.


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C-C Tsai P.7

Two extended binary trees

Extended Binary Trees

A G

B HC I

ED F J

Internal node

External node

C-C Tsai P.8

Height-Based Leftist Tree (HBLT)

For any node xin an extended binary tree. Let leftChild(x) andrightChild(x) denote the left and right children of the internal x,respectively. Define shortest(x) be the length of a shortest path from xto an external node in the subtree rooted at x.

If x is an external node, shortest(x) = 0;

otherwiseshortest(x) = 1+min {shortest(leftChild(x)), shortest(rightChild(x))}

2 2

2 11 1

11 1 1


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C-C Tsai P.9

The Definition of Leftist Trees

A leftist tree is a binary tree such that if it is notempty, then for every internal node x:

shortest(leftChild(x) >= shortest(rightChild(x))

2 2

2 11 1

11 1 1

Is not a leftist tree Is a leftist tree

C-C Tsai P.10

Lemma of Leftist Trees Let x be the root of a leftist tree that has ninternal nodes.

a) n >= 2shortest(x) 1

b) The rightmost root to external node path is the shortestroot to external node path. Its length is

shortest(x)


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C-C Tsai P.11

Min-Leftist Trees

Definition:A min-leftist tree (max-leftist tree) isa leftist tree in which the key value in eachnode is no larger (smaller) than the key valuesin its children (if any).

Two min-leftist trees:

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C-C Tsai P.12

Combination of leftist trees

The combination steps (min-leftist trees)

1. Choose minimum root of the two trees,Aand B.

2. Leave the left subtree of smaller root (supposeA)unchanged and combine the right subtree ofAwith

B. Back to step 1, until no remaining vertices.3. Compare shortest(x) and swap to make it satisfy

the definition of leftist trees.


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C-C Tsai P.13

Example1: Meld Two Leftist Trees

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5011

8013

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11 80

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combine

C-C Tsai P.14


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C-C Tsai P.15


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80

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C-C Tsai P.16

Example1: Meld Two Leftist Trees2

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80

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2

1

1

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1111

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98

1210

2015 18

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1111

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exchange1 exchange2


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C-C Tsai P.17


Step 1. Choose smallerroot and set as a.

Step 2. If a has noright_chlid, set b as asright_child. Else,recursively combine asright_child and b.

Step 3. Make thecombined tree satisfiedthe leftist tree property.

O(log n)

C-C Tsai P.18


Both insert and delete min operations can beimplemented by using the combine operation.

Insert: Treat the inserting node as a single nodebinary tree. Combine with the original one.

Delete: Remove the node can get two separatesubtrees. Combine the two trees.


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C-C Tsai P.19

Weight-Based Leftist Tree (WBLT) Define the weigth w(x) of node x to be the number of internal

nodes in the subtree with root x. The weight is 0 for an externalnode and is 1+ sum of the children weights of an internalnode.

A binary tree is a WBLT iff at every internal node the w value of theleft child is greater than or equal to the w value of the right child.

For any node xin an extended binary tree. The length, rightmost(x),of the rightmost path from xto an external node satisfies

rightmost(x)


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C-C Tsai P.21

Definition of Binomial Heaps

A min (max) binomial heap (B-heap) is acollection of min (max) trees.

The min trees should be Binomial trees.

Example: a B-heap consists of three min trees

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C-C Tsai P.22

Representation of Binomial Heaps The representation of B-heap:

Degree: number of children a node has. Child: point to any one of its children.

Left_link, Right_link: maintain doubly linked circularlist of siblings.

The position of pointer a is the min element.

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Siblings

parent

child

a

pointera


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C-C Tsai P.23

Combination of Binomial Heaps

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a

40

Combine a min tree of 40

C-C Tsai P.24


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a

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Combine a min tree of 40


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C-C Tsai P.25

Pairwise1 combine


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a

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C-C Tsai P.26

Pairwise1 combine


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a

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C-C Tsai P.27

Pairwise2 combine


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a

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C-C Tsai P.28

Pairwise2 combine


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C-C Tsai P.29

Pairwise2 combine


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a

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C-C Tsai P.30

Pairwise3 combine


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a

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C-C Tsai P.31

Pairwise3 combine


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a

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40

C-C Tsai P.32


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a

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Pairwise3 combine


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C-C Tsai P.33


The insertion is to combine a single vertex B-heap to original B-heap.

After we delete the min element, we get severalB-heaps that originally subtrees of the removedvertex. Then combine them together.

C-C Tsai P.34

Time Complexity of Binomial Heaps

Binomial heapsLeftist trees

Actual Amortized

Insert O(log n) O(1) O(1)

Delete min (or max) O(log n) O(n) O(log n)

Meld O(log n) O(1) O(1)


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C-C Tsai P.35

Fibonacci Heaps

A min (max) Fibonacci heap (F-heap) is acollection of min (max) trees.

The min trees need not be Binomial trees.

B-heaps are a special case of F-heaps.

So that what B-heaps can do can be done in F-heaps.

More than that, F-heap may delete an arbitrary nodeand decrease key.

C-C Tsai P.36

Deletion From an F-heap

The deletion of 12

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C-C Tsai P.37


After the deletion of 12

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C-C Tsai P.38


After the deletion of 12

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C-C Tsai P.39

Decrease Key From an F-heap

Decrease key of 15 to be 11

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15 30

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911

C-C Tsai P.40


After decrease key of 15=>11

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C-C Tsai P.41


After decrease key of 15=>11

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C-C Tsai P.42

Cascading Cut

When theNode is cut out of its sibling list in aremove or decrease key operation, follow pathfrom parent of theNode to the root.

Encountered nodes (other than root) with

ChildCut = true are cut from their sibling lists andinserted into top-level list.

Stop at first node with ChildCut = false.

For this node, set ChildCut = true.


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C-C Tsai P.43

Cascading Cut Example

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theNode

T

T

F

Decrease key by 2.

C-C Tsai P.44


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T

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F


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C-C Tsai P.45


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T

F

C-C Tsai P.46


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9 8F


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C-C Tsai P.47


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5 9

2 6

7 46

9

9 8T

Actual complexity of cascading cut is O(h) = O(n).

C-C Tsai P.48

Time Complexity of Fibonacci Heaps

Actual Amortized

Insert O(1) O(1)

Delete min (or max) O(n) O(log n)

Meld O(1) O(1)

Delete O(n) O(log n)

Decrease key (or

increase)

O(n) O(1)


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C-C Tsai P.49

Pairing Heaps

Fibonacci Pairing

Insert O(1) O(1)

Delete min (or max) O(n) O(n)

Meld O(1) O(1)

Delete O(n) O(n)

Decrease key (or

increase)

O(n) O(1)

C-C Tsai P.50

Pairing Heaps

Fibonacci Pairing

Insert O(1) O(log n)

Delete min (or max) O(log n) O(log n)

Meld O(1) O(log n)

Delete O(log n) O(log n)

Decrease key (or

increase)

O(1) O(log n)


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C-C Tsai P.51

Pairing Heaps

Experimental results suggest that pairing heapsare actually faster than Fibonacci heaps.

Simpler to implement.

Smaller runtime overheads.

Less space per node.

C-C Tsai P.52

Definition

A min (max) pairing heap is a min (max) treein which operations are done in a specifiedmanner.

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C-C Tsai P.53

Node Structure Child

Pointer to first node of children list.

Left and Right Sibling Used for doubly linked linked list (not circular) of

siblings.

Left pointer of first node is to parent.

x is first node in list iffx.left.child = x.

Data

Note: No Parent, Degree, or ChildCut fields.

C-C Tsai P.54

MeldMax Pairing Heap Compare-Link Operation

Compare roots.

Tree with smaller root becomes leftmost subtree.

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+ =

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7

Actual cost = O(1).


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C-C Tsai P.55

Insert

Create 1-element max tree with new itemand meld with existing max pairing heap.

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+insert(2) =

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C-C Tsai P.56

Insert

Create 1-element max tree with new itemand meld with existing max pairing heap.

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6 7

9

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+insert(14)=

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6 7

9

6

7

14

Actual cost = O(1).


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C-C Tsai P.57

Worst-Case Degree

Insert 9, 8, 7, , 1, in this order.

9

871

Worst-case degree = n 1.

C-C Tsai P.58

Worst-Case Height Insert 1, 2, 3, , n, in this order.

1

2

Worst-case height = n.3

4

5


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C-C Tsai P.59

IncreaseKey(theNode, theAmount)

Since nodes do not have parent fields, wecannot easily check whether the key in theNodebecomes larger than that in its parent.

So, detach theNode from sibling doubly-linkedlist and meld.

C-C Tsai P.60


If theNode is not the root, remove subtree

rooted at theNode from its sibling list.

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theNode

4=>18


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C-C Tsai P.61


Meld subtree with remaining tree.

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C-C Tsai P.62


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3

Actual cost = O(1).


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C-C Tsai P.63

Delete Max

If empty => fail.

Otherwise, remove tree root and meldsubtrees into a single max tree.

How to meld subtrees?

Good way => O(log n) amortized complexity forremove max.

Bad way => O(n) amortized complexity.

C-C Tsai P.64

Bad Way To Meld Subtrees

currentTree = first subtree.

for (each of the remaining trees)

currentTree = compareLink (currentTree,

nextTree);


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C-C Tsai P.65

Example

Delete max.

75268 4 1 3

Meld into one tree.

9

75268 4 1 3

8

6157 3 2 4

C-C Tsai P.66

Example

Actual cost of insert is 1.

Actual cost of delete max is degree of root.

n/2 inserts (9, 7, 5, 3, 1, 2, 4, 6, 8) followed by

n/2 delete maxs. Cost of inserts is n/2.

Cost of delete maxs is 1 + 2 + + n/21 = (n2).

If amortized cost of an insert is O(1), amortized costof a delete max must be (n).


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C-C Tsai P.67

Good Ways To Meld Subtrees

Two-pass scheme and Multipass scheme. Both havesame asymptotic complexity.

Two-pass scheme gives better observed performance.

Pass 1.

Examine subtrees from left to right.

Meld pairs of subtrees, reducing the number of subtrees tohalf the original number.

If # subtrees was odd, meld remaining original subtree withlast newly generated subtree.

Pass 2. Start with rightmost subtree of Pass 1. Call this the working

tree.

Meld remaining subtrees, one at a time, from right to left,into the working tree.

C-C Tsai P.68

Two-Pass SchemeExample

T2 T3 T4T1 T6 T7 T8T5

S1 S2 S3 S4

R1

R2

R3


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C-C Tsai P.69

Multipass Scheme

Place the subtrees into a FIFO queue.

Repeat until 1 tree remains.

Remove 2 subtrees from the queue.

Meld them.

Put the resulting tree onto the queue.

C-C Tsai P.70

Multipass SchemeExample

T2T1 T3 T4 T6 T7 T8T5

S2S1T6 T7 T8T5

S1T3 T4 T6 T7 T8T5

S3S2S1T7 T8


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C-C Tsai P.71

Multipass Scheme--Example

S3S2S1T7 T8

S4S3S2S1

S4S3 R1

R1 R2

C-C Tsai P.72

Multipass Scheme--Example

R1 R2

Q1

Actual cost = O(n).


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C-C Tsai P.73

Delete Nonroot Element

Remove theNode from its sibling list.

Meld children of theNode using either 2-pass ormultipass scheme.

Meld resulting tree with whats left of originaltree.

C-C Tsai P.74

Delete(theNode)

Remove theNode from its doubly-linked

sibling list.

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theNode


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C-C Tsai P.75

Delete(theNode)

Meld children of theNode.

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C-C Tsai P.76

Delete(theNode)

Meld with whats left of original tree.

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C-C Tsai P.77

Delete(theNode)

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Actual cost = O(n).

C-C Tsai P.78

MIN-MAX Heaps

Complete binary tree.

Set root on a min level.

A node xin min level would have smaller keyvalue than all its descendents. (xis a minnode.)

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min

max

min

max


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C-C Tsai P.79

Insertion into a min-max heap

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min

max

min

max5

Insert 5

C-C Tsai P.80


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min

max

min

max10

Insert 5


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C-C Tsai P.81


5

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min

max

min

max10

Insert 5

C-C Tsai P.82


Check if it satisfies min heap.(Compare with its parent.)

If no, move the key of currentparent to current position.If yes, skip.


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C-C Tsai P.83

Insertion into a min-max heap Initial Heap={7,70,40,30,9,10,15,45,50,30,20,12}

parent=7;

We want to insertion 80 into the position 13

*n=13, item=80;

80>10 verify_max(heap,13,80)

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min

max

min

max80

C-C Tsai P.84


Input:verify_max(Heap,13,80)

grandparent=3

80>40

heap[13]=heap[3]

i=3

grandparent=0

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max

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max80


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C-C Tsai P.85


Input

verify_max(Heap,3,80)

grandparent=null

break;

heap[3]=80;

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min

max

min

max40

C-C Tsai P.86


The time complexity of insertion into a min-max heap with n elements is O(log n).

A min-max heap with nelements has O(log n) levels.


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C-C Tsai P.87

Deletion of min element

The smallest element is in the root.

We do the deletion as follows:

Remove the root node and the node xwhich is theend of the heap.

Reinsert the key of x into the heap.

C-C Tsai P.88


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min

max

min

max

Delete 7


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C-C Tsai P.89


70 40

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max

min

max

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Delete 7

C-C Tsai P.90


i i

It means the item is the only one element, soit should be at root in heap.

The reinsertion may have 2 cases:

Case1: No child. (Only one node in the heap)


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C-C Tsai P.91


i i

Case2: The root has at least one child

Find the min value. (Let this be node k.)

a. A item.key heap[k].key

It means the item is the min element, and themin element should be at root in heap.

C-C Tsai P.92


b. item.key> heap[k].key and k is child of root.

i

Since kis in max level, it has no descendants.

k

k

i


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C-C Tsai P.93


c. item.key> heap[k].key and k is grandchild ofroot.

i

Example (p>i,): We should make sure the node inmax level contains the largest key. Then redoinsertion to the subtree with the root in red line.

p

k

k

p

i

C-C Tsai P.94


Get the min value of the heap.

Move kto the root of this heap.

Swap iandpin case 2c.


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C-C Tsai P.95

Deletion of min elementmin

max

min

max

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Delete 7

min

max

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max

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C-C Tsai P.96


Deletion of min element of a min-max heapwith n elements need O(log n) time.

In each iteration, imoves down two levels. Since amin-max heap is a complete binary tree, heap has

O(log n) levels.


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C-C Tsai P.97

Symmetric Min-Max Heaps

Symmetric Min-Max Heap (SMMH) is acomplete binary tree.

SMMH is either empty or satisfies theproperties:

The root contains no element.

The left subtree is a min-heap.

The right subtree is a max-heap.

If the right subtree is not empty. Let i be any nodein left subtree, and j be the corresponding node inthe right subtree. If no, choose the parent one.i_keyj_key.

C-C Tsai P.98

Example of a SMMH

5 45

10 258 40

1915 309 20

Min-heap Max-heap


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C-C Tsai P.99

SMMH

Let i be any node in left subtree and j be thecorresponding node in the right subtree. The relationbetween iandj.

;2/

)(if

;2 1log2

=

>

+=

j

nj

ij i

1

2 3

4 65 7

98 1110 12

Ex1:

i = 4;j = 4+2(2-1) = 6;

Ex2:

i = 9;j = 9+2(3-1) = 13;

j =13 > 12 j = 6;

C-C Tsai P.100

Insertion into a SMMH

The insertion steps1. max_heap(n): Check iffnis a position in the max-

heap of the SMMH.

2. min_partner(n) or max_partner(n) : Compute themin-heap/ max-heap node that corresponding to n.

( / )

1. Compare key of iandjto satisfy SMMH.

2. min_insert or max_insert.

1log22

n

n 2/)2(1log2 +

nn


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C-C Tsai P.101


5 45

10 258 40

1915 309 20 4

ji

Insert 4: i = 13-2(3-1) = 13-4 = 9 for min_heap

C-C Tsai P.102


5 45

10 258 40

415 309 20 19


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C-C Tsai P.103


4 45

5 258 40

1015 309 20 19

C-C Tsai P.104


Step 1. max_heap.

Step 2. min_partner.

Step 3. Compare key of i and j.

Step 4. min_insert or max_insert.

The time complexity is O(log n) as the height of the SMMH is O(log n).


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C-C Tsai P.105


The deletion steps

1. Save the last element as temp and remove thisnode from SMMH.

2. Find the node with smaller key from the children ofremoved minimum element and loop down untilreaching leaves.

3. Insert temp into the left subtree of SMMH.

C-C Tsai P.106


5 45

10 258 40

1915 309 20temp

Delete 5:


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C-C Tsai P.107


45

10 258 40

1915 309

j

temp:20

i

C-C Tsai P.108


8 45

10 25 40

1915 309

j

i

temp:20


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C-C Tsai P.109


8 45

10 259 40

1915 30

i

temp:20

C-C Tsai P.110


8 45

10 259 40

1915 3020


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C-C Tsai P.111


Step 1. Save the min element.

Step 2. Find the node withsmaller key.

Step 3. (Exercise 2).

The time complexity is O(log n) as the height of the SMMH is O(log n).

C-C Tsai P.112

Interval Heaps

Complete binary tree.

Each node (except possibly last one) has 2 elements. Lastnode has 1 or 2 elements.

Let a and b be the elements in a node P, a


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Example of an Interval Heap

28,55 35

25,60 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,70

15,80 30,60

10,90

Left end points define a min heap.

Right end points define a max heap.

28,55 35

25,60 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,70

15,80 30,60

10,90

Min and max elements are in the root.

Store as an array

Height is ~log2 n.

Example of an Interval Heap


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Insert An Element

28,55 35

25,60 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,70

15,80 30,60

10,90

Insert 27.3527,35

New element becomes a left end point.

Insert new element into min heap.

Another Insert

28,55 35

25,60 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,70

15,80 30,60

10,90

Insert 18.18,35




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28,55 25,35

25,60 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,70

15,80 30,60

10,90

Insert 18.

18,60



Another Insert

28,55 25,35

20,60 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,70

15,80 30,60

10,90

Insert 18.

,70



18,70

Another Insert


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Yet Another Insert

28,55 35

25,60 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,70

15,80 30,60

10,90

Insert 82.35,82New element becomes a right end point.

Insert new element into max heap.

After 82 Inserted

28,55 35,60

25,70 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,80

15,82 30,60

10,90


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28,55

25,70 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,80

15,82 30,60

10,90

One More Insert Example

Insert 8.

New element becomes both a left and a

right end point.


8

2528,55

20,70 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2015,80

10,82 30,60

8,90

After 8 Is Inserted


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C-C Tsai P.123

Remove Min Element

Remove Min element

If n = 0 => fail.

If n = 1 => heap becomes empty.

If n = 2 => only one node, take out left end point.

If n > 2 => not as simple.

Remove Min Element Example

28,55 35,60

25,70 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,80

15,82 30,60

10,90

Remove left end point from root.

Remove left end point from last node.

Reinsert into min heap, begin at root.

10,90

35,60

Delete last node if now empty.

35


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28,55 60

25,70 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,80

15,82 30,60

15,90

Swap with right end point if necessary.

,82

35


28,55 60

25,70 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6015,2020,80

15,82 30,60

15,90


,20

35



64/65

64

28,55 60

25,70 30,50 16,19 17,17 50,55 47,58 40,45 40,43

35,5045,6016,3520,80

15,82 30,60

15,90


,19

20


28,55 60

25,70 30,50 19,20 17,17 50,55 47,58 40,45 40,43

35,5045,6016,3520,80

15,82 30,60

15,90



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C-C Tsai P.129

Application Of Interval Heaps Complementary range search problem.

Collection of 1D points (numbers).

Insert a point. O(log n)

Remove a point given its location in the structure. O(log n)

Report all points not in the range [a,b], a

Documents

DS(II) Ch09 PriorityQueues