Instructors: C. Y. Tang and J. S. Roger Jang All the material are integrated from the textbook...

Instructors: C. Y. Tang and J. S. Roger Jang

All the material are integrated from the textbook "Fundamentals of Data Structures in C" and some supplement from the slides of Prof. Hsin-Hsi Chen (NTU).

Chapter 9Heap Structures

Outline

MIN-MAX Heaps Deaps Leftist Trees Binomial Heaps Fibonacci Heaps

MIN-MAX Heaps

Definition

MIN-MAX Heaps

Complete binary tree. Set root on a min level. A node x in min level would have

smaller key value than all its descendents. (x is a min node.)

MIN-MAX Heaps

7

70 40

30 109 15

5045 2030 12

min

max

min

max

MIN-MAX Heaps

Insertion into a min-max heap


7

70 40

30 109 15

5045 2030 12

min

max

min

max

5


7

70 40

30 59 15

5045 2030 12

min

max

min

max

10


5

70 40

30 79 15

5045 2030 12

min

max

min

max

10


Check if it satisfies min heap.(Compare with its parent.)If no, move the key of current parent to current position.If yes, skip.


Initial

Heap={7,70,40,30,9,10,15,45,50,30,20,12}*n=13item=80parent=6

80>10→verify_max(heap,13,80)

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min

max

min

max

80


Input

verify_max(Heap,13,80)grandparent=380>40 →heap[13]=heap[3]

i=3grandparent=0

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min

max

min

max

80


Input

verify_max(Heap,3,80) → grandparent=null

break;heap[3]=80;

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70 80

30 109 15

5045 2030 12

min

max

min

max

40


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

g n) levels.

MIN-MAX Heaps

Deletion of min element


The smallest element is in the root.

We do the deletion as follows:1. Remove the root node and the node

x which is the end of the heap.2. Reinsert the key of x into the heap.


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min

max

min

max


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max

min

max

12


The reinsertion may have 2 cases:

1. No child. (Only one node in the heap)i i

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


2. The root has at least one childFind the min value. (Let this be node k.)

a. A item.key≦ heap[k].keyi i

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


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

i

Since k is in max level, it has no descendants.

k

k

i


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

i

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

p

k

k

p

i


Get the min value of the heap.

Move k to the root of this heap.

Swap i and p in case 2c.


Deletion of min element of a min-max heap with n elements need O(log n) time. In each iteration, i moves down two level

s. Since a min-max heap is a complete binary tree, heap has O(log n) levels.

Deaps

Definition

Deaps Complete binary tree. Either empty or satisfies the properties

1. The root contains no element.2. The left subtree is a min-heap.3. The right subtree is a max-heap.4. If the right subtree is not empty. Let i be any n

ode in left subtree, and j be the corresponding node in the right subtree. If no, choose the parent one. i_key≦ j_key.

Deaps The relation between i and j.

;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>12 j=6;

Deaps

5 45

10 258 40

1915 309 20

Deaps

Insertion into a deap

Insertion into a deap The insertion steps

1. max_heap(n): Check iff n is a position in the max-heap of the deap.

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

3. Compare key of i and j to satisfy deap.4. min_insert or max_insert.

1log22 nn 2/)2( 1log2 nn


5 45

10 258 40

1915 309 20 4

j


5 45

10 258 40

1915 309 20 4

ji


5 45

10 258 40

415 309 20 19


4 45

5 258 40

1015 309 20 19


Step 1. max_heap.

Step 2. min_partner.

Step 3. Compare key of i and j.

Step 4. min_insert or max_insert.

Insertion into a deap The time complexity is O(log n) as the

height of the deap is O(log n).

Deaps


Insertion into a deap The insertion steps

1. Save the last element as temp and remove this node from deap.

2. Find the node with smaller key from the children of removed minimum element and loop down until reaching leaves.

3. Insert temp into the left subtree of deap.


5 45

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1915 309 20temp


45

10 258 40

1915 309

j

temp:20

i


8 45

10 25 40

1915 309

j

i

temp:20


8 45

10 259 40

1915 30

i

temp:20


8 45

10 259 40

1915 3020


Step 1. Save the min element.

Step 2. Find the node with smaller key.

Step 3. (Exercise 2).


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

Leftist Trees

Definition

Leftist Trees Linked binary tree. Can do everything a heap can do and in th

e same asymptotic complexity. Can meld two leftist tree priority queues i

n O(log n) time. For any node x in an extended binary tree,

let shortest(x) be the length of a shortest path from x to an external node in the subtree rooted at x.

node externalan is if 0otherwise ))}(_()),(_(min{1)( x

ychildrightshortestxchildleftshortestxshortest

Leftist Trees

A G

B HC I

ED F J

Two binary trees

Extended binary trees

Leftist Trees

A G

B HC I

ED F J

The number inside each internal node x is shortest(x).

Leftist Trees

2 2

2 11 1

11 1 1

Leftist Trees

DefinitionA leftist tree is a binary tree such that if it is not empty, then for every internal node x: ))(_())(_( xchildrightshortestxchildleftshortest

2 2

2 11 1

11 1 1

Leftist Trees

By the definitionLet x be the root of a leftist tree that has n internal nodes.a) 12 )( xshortestn

x

shortest(x)

A binary tree whose shortest height is shortest(x) means for every path from root to leaf has at least shortest(x) nodes.

Such that it has at least 2shortest(x)-1 nodes.

Leftist Trees

b) The rightmost root to external node path is the shortest root to external node path.

x

shortest(x)

Base on the definition of leftist tree. ))(_())(_( xchildrightshortestxchildleftshortest

Leftist Trees

DefinitionA min-leftist tree (max leftist tree) is a leftist tree in which the key value in each node is no larger (smaller) than the key values in its children (if any).

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7 50

11 80

13

Leftist Trees

Combination of leftist trees


The combination step (min-leftist trees)1. Choose minimum root of the two trees, A a

nd B. 2. Leave the left subtree of smaller root (supp

ose A) unchanged and combine the right subtree of A with B. Back to step 1, until no remaining vertices.

3. Compare shortest(x) and swap to make it satisfy the definition of leftist trees.


Step 1. Choose smaller root and set as a.

Step 2. If a has no right_chlid, set b as a’s right_child. Else, recursively combine a’s right_child and b.

Step 3. Make the combined tree satisfied the leftist tree property.


2

7 50

11 80

13

5

9 8

12 10

20 1518


2

7

5011

8013

5

9 8

12 10

20 1518


2

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11

80

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5

9

812

1020

15

18


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2

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20 1518

2

1

1

1

2

2

1

1111

2

1

1


2

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5

98

1210

2015 18

2

1

1

1

2

2

1

11 11

2

1

1


Both insert and delete min operations can be implemented by using the combine operation. Insert: Treat the inserting node as a singl

e node binary tree. Combine with the original one.

Delete: Remove the node can get two separate subtrees. Combine the two trees.

Binomial Heaps

Definition

Binomial Trees

[Definition] Binomial trees A binomial tree Bk has 2k nodes with heig

ht be k.

Bk

Bk-1

Bk-1

B0 B1 B2 B3

Binomial Trees It has nodes at depth i. The ith child of root is the root of subtree

Bi-1. B4

B2B3 B1 B0

Depth 1 ： 4 nodes.



)(ki

Binomial Heaps

Collection of min (max) trees. The min trees should be Binomial

trees.

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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 lin

ked circular list of siblings. The position of pointer a is the min ele

ment.

Binomial Heaps

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Siblings

parent

child

a

pointera

Binomial Heaps

Combination of binomial heaps


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a

40

Pairwise combine


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a

20

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

After we delete the min element , we get several b-heaps that originally subtrees of the removed vertex. Then combine them together.

Time complexity

Binomial heaps Leftist 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)

Fibonacci Heaps

Definition

Fibonacci Heaps

Collection 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 node and decrease key.

Fibonacci Heaps

Deletion and Decrease key


Deletion

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Deletion

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Decrease key

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911


Decrease key

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Time complexity

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)

Instructors: C. Y. Tang and J. S. Roger Jang All the material are integrated from the textbook...

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Transcript of Instructors: C. Y. Tang and J. S. Roger Jang All the material are integrated from the textbook...