1Master of Computer Engineering Chiangmai University.

CPE261702 ALGORITHM ANALYSIS & DESIGNPRESENTATION TOPIC IN

MAP

1Master of Computer Engineering Chiangmai University

Talking Outline

Introduction Basic Abstract Data Type (ADT) How do implementation ?

Double Linked List Hash Table

Bucket Array Hash Function

Trie AVL Tree

Performing & Complexity Demo Programmer


Basic Array


12000 11000 15000 20000 18000Array salary

0 1 2 3 4index

We can build to platform ( php code )

$salary[i] = ?EX.

$salary[1] =$salary[3] =

But we can’t know. Who’s salary ?

1100020000

Associative Array


12000 11000 15000 20000 18000Array salary

Sara Bob Smith John Stevename

We can build to platform ( php code )

$salary[‘name’] = ?EX.

$salary[‘Smith’] =

We can replace index by name.

15000

Abstract Data Type

Since a map store a collection of object, it should be viewed as a collection of key-value pairs. As an ADT, a map M supports the following methods :

size () : Return the number of entries in M isEmpty () : Test whether M is empty. get (k) : If M contain an entry e with key equal to k, then return the value of e, else return null. put (k,v) : If M does not have an entry with key equal to k, then add entry (k,v) to M and return null; else, replace with v the existing value of the entry with key equal to k and return the old value. remove (k) : Remove from M the entry with key equal to k, and return its value; if M has no such entry, then return null.


Abstract Data Type (Cont.)

keys () : Return an iterable collection containing all the keys stored in M (so keys().iterator() returns an iterator of keys). values () : Return an iterable collection containing all the values associated with keys stored in M (so values().iterator() returns an iterator of values). entries () : Return an iterable collection containing all the key- value entries in M (so entries().iterator() returns an iterator of entries).


Example


Operation Output Map

isEmpty() true Ø

put(5,A) null {(5,A)}

put(7,B) null {(5,A),(7,B)}

put(2,C) null {(5,A),(7,B),(2,C)}

put(8,D) null {(5,A),(7,B),(2,C),(8,D)}

put(2,E) C {(5,A),(7,B),(2,E),(8,D)}

get(7) B {(5,A),(7,B),(2,E),(8,D)}

get(4) null {(5,A),(7,B),(2,E),(8,D)}

get(2) E {(5,A),(7,B),(2,E),(8,D)}

size() 4 {(5,A),(7,B),(2,E),(8,D)}

Example (Cont.)


Operation Output Map

remove(5) A {(7,B),(2,E),(8,D)}

remove(2) E {(7,B),(8,D)}

get(2) null {(7,B),(8,D)}

isEmply() false {(7,B),(8,D)}

Simple List-Based Map

A simple way of implementing a map is to store its n entries in list S,Implemented as an unsorted list ( base on doubly linked list ).


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nodes/positions

entries

Sara 11000

put ( Sara , 11000 )


put ( Bob , 12000 )


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nodes/positions

entries

Sara 11000 Bob 12000


put ( Smith , 15000 )


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nodes/positions

entries

Sara 11000 Bob 12000 Smith 15000


put ( Steve , 18000 )


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nodes/positions

entries

Sara 11000 Bob 12000 Smith 15000 Steve 18000


put ( Smith , 20000 )


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nodes/positions

entries

Sara 11000 Bob 12000 Smith Steve 180002000015000

Return value = 15000


get ( Steve )


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nodes/positions

entries




get ( John )


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nodes/positions

entries


Return value = null


remove ( Bob )


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nodes/positions

entries




remove ( Stephen )


trailerheader

nodes/positions

entries

Sara 11000 Smith 20000 Steve 18000

Return value = null

Simple List-Based Map Algorithm

Algorithm put ( k , v ) : Input : A key-value pair ( k , v ) Output : The old value associated with key k in M, or null if k is new for each position p in S.positions () do if p.element () .getkey () = k then t p.element () .getValue () B.set ( p, ( k , v )) return t S.addLast (( k , v )) n n + 1 return null


Simple List-Based Map Algorithm

Algorithm get ( k ) : Input : A key k Output : The value of key k in M, or null if there is no key k in M for each position p in S.positions () do if p.element () .getkey () = k then return p.element () .getvalue () return nullAlgorithm remove ( k ) : Input : A key k Output : The ( removed ) value of key k in M, or null if k is not in M for each position p in S.positions () do if p.element () .getkey () = k then t p.element () .getValue () S.remove ( p ) n n - 1 return t return null


Hash Table

Hash Table is data structure use to map identifying value, know as key (e.g., a person’s name ), to their associated value (e.g., their telephone number).

Two components in Hash Table :

Bucket Array

Hash Function


Hash Table Architecture


String Key

String

Hash Code

Compression Function

Hash Function

Hash Table

Addr. Key Data

Example Implement

A small phone book as a hash table.


Figure Reference : http://th.wikipedia.org/

Hash Table Algorithm

Map Methods with Separate Chaining used for Collisions.

Delegate operations to a list-based map at each cell.


Algorithm get(k):

Output: The value associated with the key k in the map, or null if there is no

entry with key equal to k in the map

return B[h(k)].get(k) {delegate the get to the list-based map at B[h(k)]}

Algorithm put(k,v):

Output: If there is an existing entry in our map with key equal to k, then we

return its value (replacing it with v); otherwise, we return null

t = B[h(k)].put(k,v) {delegate the put to the list-based map at B[h(k)]}

if t = null then {k is a new key}

n = n + 1

return t

Hash Table Algorithm


Algorithm remove(k):

Output: The (removed) value associated with key k in the map, or null if there

is no entry with key equal to k in the map

t = B[h(k)].remove(k) {delegate the remove to the list-based map at B[h(k)]}

if t ≠ null then {k was found}

n = n - 1

return t

The advantages of Hash Table

It has speed more than other table data

structures.

It has high performance when the number of entries

is large.

It has particularly efficient when the maximum

number of entries can be predicted in advance.


The disadvantages of Hash Table

Can’t be more difficult implement than self-balancing

Binary Search Tree ( BST )

Consuming more resources. If the data is less than the

surface area of reservation.

Not effective when the number of entries is very small.

Can’t find minimum or maximum data value.

Can’t sorting in Hash Table.


Trie

Ordered tree data structure.

Used to store an associative array.

The keys usually strings.

No node in the tree stores the key associated with

that node.

All the descendants of a node have a common prefix

of the string associated with that node.

The root is associated with the empty string.

Values are only associated with leaves and some inner

node that correspond to keys of interest.


Example

Tries


Figure Reference : http://en.wikipedia.org

Keys Values

to 7

tea 3

ted 4

ten 12

A 15

i 11

in 5

inn 9

Trie Algorithm

Pseudo-code for Insert:


Trie Algorithm


Trie searching method used same way as search tree structure.

In this case, we use preorder traversal.

Algorithm preOrder(v)visit(v)for each child w of v

preorder (w)

Tries Compare BST.

The main advantages of the tries over binary search trees(BSTs) :

Looking up keys is faster.

Tries can require less space when they contain a large number

of short string.

Tries facilitate longest-prefix matching.


Tries Compare Hash Table.

The main advantages of the tries over hash tables:

Tries can perform a “closest fit ” find almost as quickly as an

exact find.

Tries tend to be faster on average at insertion than hash table.

Tries can be implemented in a way which avoids the need for

additional ( dynamic ) memory.

Looking up keys can be much faster if a hash function can be

avoided.


Advantages of Tries

Looking up data in a trie is faster in the worst case, O(m) time,

compared to an imperfect hash table.

There are no collisions of different keys in a trie.

Bucket in a trie which are analogous to hash table bucket that

store key collisions are only necessary if a single key is

associated with more than one value.

There is no need to provide a hash function or to change hash

functions as more keys are added to a trie.

A trie can provide an alphabetical ordering of the entries by key.


AVL Tree

AVL Tree is a self-balancing binary search tree. In an AVL tree,

The height of the two child subtrees of any node differ by at

most one.

Search, insertion, and deletion all take O( log n ) time.

Insertion and deletion may require the tree to be re balanced by

one or more tree rotation.


AVL Tree Architecture


String KeyMathematic

FunctionInteger Key AVL Tree

Example Implement


44,A

17,B 78,D

48,F

88,H50,E32,C

63,G

AVL Tree

Insertion AVL Tree


Insert (15,Q) 44,A

17,B 78,D

48,F

88,H50,E32,C

63,G

15,Q

Insertion AVL Tree


Insert (50,Z) 44,A

17,B 78,D

48,F

88,H32,C

63,G

15,Q 50,E50,Z

Searching AVL Tree


Search (48) 44,A

17,B 78,D

48,F

88,H50,Z32,C

63,G

15,Q

Return Value = F

Searching AVL Tree


Search (60) 44,A

17,B 78,D

48,F

88,H50,Z32,C

63,G

15,Q

Return Value = null

Deletion AVL Tree


Delete (63) 44,A

17,B 78,D

48,F

88,H50,Z32,C

63,G

15,Q

Implement AVL Tree Algorithm


Insertion Algorithm

Algorithm Insert ( data ,v )

Input : Insert a data ( key-value pair ) in T at position v

Output : New node(data) or replace data at key.data = key(v)

if T.isExternal (v)

T.put(data,v)

return null

if key.data = key(v)

old value = T.getvalue()

T.put(data,v)

return (old value)



Else if key.data < key(v)

return Insert (k, T.left(v))

else if k.data > key(v)

return Insert (k, T.right(v))

if T is balancce

Return ()

else

Rebalancing Tree



Searching Algorithm

Algorithm Search ( k ,v )

Input : A key k and position v

Output : Return value of key in position v, else return null

if T.isExternal (v)

return null

if k < key(v)

return remove(k, T.left(v))

else if k = key(v)

return T.getValue(v)

else { k > key(v) }

return Search(k, T.right(v))



Deletion Algorithm

Algorithm delete ( k )

Input : searching by inorder traversal with key k

Output : return value (remove) at position v

if T.isExternal (v)

return null

if k < key(v)

delete (k , T.Left(v))

else if k > key (v)

delete (k , T.Right(v))

{

else

if T.Left(v) = null and T.Right(v) = null



value = T.getValue

T. Remove(k,v)

T.put (null,v)

Return (value)

else if T.Left(v) = null

temp = node(v)

node(v) = T.Right(v)

Remove temp

else if T.Right(v) = null

temp = node(v)

node(v) = T.Left(v)

Remove temp



else

tempnode = findmin(T.Right(v))

node.data = tempnode.data

delete(node.data,T.Right(v))

}

if T is balancce

Return ()

else

Rebalancing Tree

Performance & Complexity


Structure Insertion Searching DeletionDoubly link-List O(1) O(n) O(n)

Hash Table O(1) O(1) O(1)Trie O(dm) O(dm) O(dm)

AVL Tree O(log n) O(log n) O(log n)

* d = size of alphabet m = size of the string parameter of the operation

Thank you


1Master of Computer Engineering Chiangmai University.

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