ncue csie wireless communications and networking laboratory chapter 9 advanced search trees 1

NCUE CSIE Wireless Communications and Networking Laboratory

CHAPTER 9ADVANCED SEARCH TREESADVANCED SEARCH TREES

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Min-Max Heap

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A double-ended priority queuedouble-ended priority queue is a data structure that supports the following operation:

(1) Insert an element(2) Delete Max(3) Delete Min


Min-Max Heap

Definition: A min-max heapmin-max heap is a complete binary treecomplete binary tree such that if it is not empty, each element has a field called key. (1)Alternating levels of this tree are min levels and max levels

respectively. (2) The root is on a min level(3) Let x be any node in a min-max heap. If x is on a min (max) level

then the element in x has the minimum (maximum) key from among all elements in the subtree with root x.

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Min-Max Heap

minmin

maxmax

minmin

maxmax

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Procedure min-max heap (h, n, x)

n=n+1 p=n/2 if p==0 then h[1]=x else case level (p) min: if x < h[p] then swap (h[n], h[p]) verify-min (h, p, x) else verify-max (h, n, x)

max: if x > h[p] then swap (h[n], h[p]) verify-max (h, p, x) else verify-min (h, n, x)

Location

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Procedure verify-max (h, i, x)

Gp=(i/4) while (Gp≠0) do if x > h[Gp] then swap (h[i], h[Gp]) i=Gp Gp=(i/4) else Gp=0

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Procedure verify-min (h, i, x)

Gp=(i/4) while (Gp ≠0) do if x < h[Gp] then swap (h[i], h[Gp]) i=Gp Gp=(i/4) else Gp=0

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Time complexity

)(log nO = ½ Height

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Delete minimum element

Step1: Remove root

Step2: Let last node = x, x Insert a min-max heap and its root is empty

case 1: if root have no child, x placed in rootcase 2: if root have child but have no Gchild, Find child’s min k if (k < x) then swap (k, x) else x placed in root

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if (k < x) then swap (k, x) if (x > p) then swap (p, x) goto step 2else x placed in root

case 3: if root have Gchild, Find the min k of the Grandchild, and suppose k’s parent node is p

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Delete minimum element


Delete maximum element

Step1: Remove max (h[2] , h[3])

Step2: Let last node = x, x insert a min-max heap and its root is empty

case 1: if root have no child, x placed in rootcase 2: if root have child but have no Gchild, Find child’s max k if (k > x) then swap (k, x) else x placed in root

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if (k > x) then swap (k, x) if (x < p) then swap (p, x) goto step 2else x placed in root

case 3: if root have Gchild, Find the max k of the Grandchild, and suppose k’s parent node is p

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Delete maximum element


Time complexity

13

)(log nO


Double-ended Heap (Deap)

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A double-ended priority queue is a data structure that supports the following operation:

(1) Insert(2) Delete Max(3) Delete Min


DefinitionDefinition:

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A deap is a complete binary treecomplete binary tree that is either empty or satisfies the following 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, then let i be any node in the left

subtree. Let j be the corresponding node in the right subtree. If j does not exist, then let j be the node in the right subtree that corresponds to the parent of i. The key of node i is less than or equal to the key of node j.


min max

)2

()(

2 2)1log(

jjthennjif

ij i

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Insert

Step1: x placed behind last nodecase 1: if x is in min-heap, find its corresponding node (j) in max-heap if (x.key j.key) then min-heap-insert (D, n, x) else swap (x.key, j.key) max-heap-insert (D, j, x)

Step2:

case 2: if x is in max-heap, find its corresponding node (j) in min-heap if (x.key j.key) then≧ max-heap-insert (D, n, x) else swap (x.key, j.key) min-heap-insert (D, j, x)

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Time complexity

)(log nO

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Delete minimum

Step1: Remove Deap[2]

Step2: set last node =x

Step3: in min-heap , find the minimum sub-node, and then (repeatedly) replace the vacancy of the parent node until a vacancy occurred in the leaf node. (suppose this node is located in i)Step4: Deap_Insert (D, i, x)

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Delete maximum

Step1: Remove Deap[3]

Step2: set last node = x

Step3: in max-heap, Find the maximum sub node, and then (repeatedly) replace the vacancy of the parent point until a vacancy occurred in the leaf node. (suppose this node is located in i)Step4: Deap_Insert (D, i, x)

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Time complexity

)(log nO

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Summary

Min-Max Heap and DeapMin-Max Heap and Deap

InsertInsert O(logn)O(logn)

Delete MaxDelete Max O(logn)O(logn)

Delete MinDelete Min O(logn)O(logn)

Search MaxSearch Max O(1)O(1)

Search MinSearch Min O(1)O(1)

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Extended Binary Tree

If a binary tree has n nodes, then it will have n+1 null link. If we add a special node in each null link, then we call this node external node (or failure node), and the binary tree with external nodes is called extended binary tree.

: Internal node

: External node

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Internal path and External path

n

i

I1

)node internal theroot to thefrom length the(

1

1

)node external theroot to thefrom length the(n

i

E

I=0+1+1+2=4

E=2+2+2+3+3=12

0

1 1

32 2

2

23

E=I+2nE=I+2n

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Search tree

Internal and external nodes have no weighted value

External node have weighted value

Internal and external nodes have weighted value

Search cost

AVL Tree

B Tree

Balanced Binary Search Tree

Balanced m-way Search Tree

Huffman Tree

Optimal Binary Search Tree

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AVL Tree

1 RL hh

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DefinitionDefinition: An empty binary tree is height balanced. If T is a nonempty binarytree with and as its left and right subtrees, then T is heightbalanced iff (1) (3) and are height balanced

Balance factor

Recursive Definition

LT RT

RTLT


Searching cost = 2 Searching cost = 1.67

Internal and external nodes have no weighted value

should be as balanced as possible

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Balanced Binary Tree


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Balanced Binary Tree


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Non-Balanced Binary Tree


Rotation Types

LLLL LRLR RLRL RRRR

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Adjustment rules

Step1: Find the nearest un-balanced node, and then find the related three nodes

Step2: On related three nodes, pull up the medium node small node → on the left large node → on the right

Step3: Adjust subtrees

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Adjustment rules

new insert node


Time ComplexityTime Complexity:

OperationOperation Sequential listSequential list Linked listLinked list AVL TreeAVL Tree

Search for xSearch for x O(log n)O(log n) O(n)O(n) O(log n)O(log n)

Search for kth Search for kth itemitem

O(1)O(1) O(k)O(k) O(log n)O(log n)

Delete xDelete x O(n)O(n) O(1)O(1) O(log n)O(log n)

Delete kth itemDelete kth item O(n-k)O(n-k) O(k)O(k) O(log n)O(log n)

Insert xInsert x O(n)O(n) O(1)O(1) O(log n)O(log n)

Output in orderOutput in order O(n)O(n) O(n)O(n) O(n)O(n)

balance

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An AVL tree with height h, Requires at least node12 hF

Theorem

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M-way Search Tree

1k 2k 3k4k 1mk

Value < 1k< Value < 1k 2k

< Value1mk

oA

1A

1mA

(1) The key of each node is arranged from small to large , so ki < ki+1 , 1 i n≦ ≦

(2) Subtree Ai is also a m-way search tree35

4k


B Tree of order m

DefinitionDefinition: Balanced m-way search tree: commonly used in external search.

Can be empty. If not empty, it has to satisfy:(1)Root have at least two child nodes(2)Except the root and the failed node, the degree of a node is

between to m

(3) All failed nodes are on the same level

2

m

1__.12

deg2

mKeyofNom

mreem

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B tree of order 3

3deg2

32 ree

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B tree of order 4

4deg2

42 ree

38


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Step1: Find an appropriate place to insert x

Step2: Check overflowcase 1: if not overflow →stopcase 2: if overflow → split goto step 2 → (parent node) Split: Pull up the th key to the parent nodes, and the remaining nodes are divided into two nodes

2

m

Insert


Delete

Step1: Find the location node of x

Step2: case 1: if x is in leaf nodes (1) delete x (2) check underflow, if not → stop, else → rotation, if can’t do rotation → then do combination. goto (2) →check parent node)case 2: if x is in non-leaf nodes, relpace x with the minimum key in right subtree or the maximum key in left subtree goto case 1.

rotation:

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combination:

41

Combination


Weighted External Path Length

DefinitionDefinition: Give n+1 Different weighting values of the external node (j=1~(n+1)), then

jq

1

1

)j node externalan root to ofpath the(....n

jjqLPEW

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W.E.P.L. = 43 W.E.P.L. = 52

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Weighted External Path Length


Huffman Tree

Algorithm: 1. Construct a set of trees with root nodes that contain each of the

individual symbols and their weights2. Place the set of trees into a priority queue.3. while the priority queue has more than one item4. Remove the two trees with the smallest weights.5. Combine them into a new binary tree in which the weight of the

tree root is the sum of the weights of its children.6. Insert the newly created tree back into the priority queue.

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Procedure Huffman (w, n)

Begin for (i=1; i<n; i++) Begin new (t) t.left_child=least(w) t.right_child=least(w) t.weight=t.left_child.weight+t.right_child.weight Insert (w, t) EndEnd

Time complexity:Time complexity: )log( nnO45


Question:

Insert 5, then insert 80?

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Follow the data to build min-max heap26, 5, 33, 77, 2, 6, 19

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Question:


Delete minimum element twice, then delete maximum?

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Question:


Insert 4, 60?

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Question:


Follow the data to build Deap26, 5, 33, 77, 19, 2, 8

50

Question:


Delete minimum twice?

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Question:


Delete maximum?

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Question:


A Full Extened B.T. Height=k, I, E value？

))log((22)2( nnOkI k

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Question:


Mar, May, Nov, Aug, Apr, Jan, Dec, Jul, Feb, Jun, Oct, Sep,build an AVL Tree

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Question:


Give the following data to build an AVL tree? 26, 33, 77, 19, 2, 5, 4, 8, 6, 10

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Question:


An AVL Tree has 15 nodes, Maximum height? Minimum height?

Max ︰

4)115(log2 Min ︰

h 4 5 6

Fh+2-1 7 12 20

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Question:


Give the following sequence of data, how many rotations are needed to build an AVL tree and how many pointers are rearranged during rotation? Mon, Tue, Wed, Thu, Fri, Sat, Sun

57

Question:


DAS, JFK, ZRH, HKG, KHH, FRA, ARN, LBA, MEX,GLA, ORY, MAN, TPE, ORD, NAP(1) Draw the AVL tree(2) Insert NYC, what kind of rotation is required? Sketch this tree

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Question:


Insert 72

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Question:


M-way Search tree, Height= h, the maximum node is?maximum key is?

1

1... 110

m

mmmm

hh

Node ︰

)1()1

1(... 110

mm

mmmm

hhKey ︰

60

Question:


B tree of order 3

(1) Insert 38

(2) Insert 55

(3) Insert 37

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Question:


Give 26, 5, 33, 49, 17, 2, 8, 6, 10 build B tree of order 3

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Question:


Read the following data in the given order, and show the corresponding trees: 7, 8, 9, 2, 1, 5, 3, 6, 4(1) Binary Search tree(2) AVL tree(3) 2-3 tree

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Question:


Suppose in a B-tree must be at least one key and at most two keys in every node. Draw the B-tree after the following succession of events (i.e., event B is executed following event A, and event, C follows event B,etc. )

(A)Insert 10,50,40,80,90,100

(B)Delete 80

(C)Insert 20,Delete 50

(D)Delete 100

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Question:


Question is B tree of order 3

(A) (B)

65

Answer:


(C) Insert 20 Delete 50

(D)

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Answer:


B tree of order 3

(1) Delete 58

(2) Delete 65

(3) Delete 80

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Question:


B tree of order 3

(1) Delete 10

68

Question:


B tree of order 3

(1) Delete 10

69

Question:


B tree of order 3

(1) Delete 70

(2) Delete 50

70

Question:


B-tree of order 5(1) Insert 65(2) Delete 21 from the original B-tree (3) Delete 33 from the original B-tree

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Question:


95 q 136 q

The weighted value of the six external node is as follows ： minimum W.E.P.L is?

21 q 32 q 53 q 74 q

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Question:


Briefly describe Huffman encoding and illustrate if for the following data step by stepA B A D C D A C A B B C B B B A

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Question:


Construct Huffman binary codes for eight messages whose probabilities of appearance are 0.2, 0.19, 0.18, 0.14, 0.11, 0.08, 0.06, and 0.04

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Question:


Consider a new design cpu with six instructions which have the probability of occuranceI1=0.4, I2=0.3, I3=0.01, I4=0.08, I5=0.06, I6=0.06 (1) Construct the opcodes using Huffman’s method(2) Calculate the average number of bits per instruction

75

Question:


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Reference

Ellis Horowitz, Sartaj Sahni, and Susan Anderson-Freed〝 Fundamentals of Data Structures in C 〞 , W. H. Freeman & Co Ltd, 1992.

Ellis Horowitz, Sartaj Sahni, and Dinesh Mehta〝 Fundamentals of Data Structures in C++ 〞 Silicon Pr, 2006

Richard F.Gilberg, Behrouz A. Forouzan, 〝 Data Structures: A Pseudocode Approach with C 〞 , SBaker & Taylor Books, 2004

Fred Buckley, and Marty Lewinter 〝 A Friendly Introduction to Graph Theory 〞 Prentice Hall, 2002

〝資料結構 -使用 C語言〞蘇維雅譯，松崗， 2004 〝資料結構 -使用 C語言〞蔡明志編著，全華， 2004 〝資料結構 (含精選試題 )〞洪逸編著，鼎茂， 2005

ncue csie wireless communications and networking laboratory chapter 9 advanced search trees 1

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