Advance Data Structure Review of Chapter 4

1
Advance Data StructureReview of Chapter 4

• ???

2
Overview of Chapter 4

• Type of List
• 4.1 Singly Linked Lists
• 4.2 Representing Lists in C
• 4.3 Reusable Linked List Class
• 4.4 Circular Lists
• 4.9 Doubly Linked Lists
• 4.10 Generalized Lists
• 4.12 Heterogeneous Lists
• Example
• 4.5 Linked Stacks and Queues
• 4.6 Polynomials
• 4.7 Equivalence Classes
• 4.8 Sparse Matrices

3
Array-Based Ordered List

• ??
• ????????? ? O(1)
• ??
• ??????,?????????
• ??????????? ? O(n)

?????
4
Array-Based Ordered List Insert
Insert (2, 44)
5
Array-Based Ordered List Delete
Delete(2)
6
Linked List
data
link
1
2
3
4
5
6
7
8
9
10
11
7
Linked List ????

• ???????????????,????????
• ????????,???????
• ???????,????????
• ?????????? O(1)
• ?????????????,?????????????????,???????
• ????? O(n)

8
Singly Linked Lists
Circular Lists
9
Doubly Linked Lists
Head Node
data
LeftLink
RightLink
-
BAT
CAT
EAT
A
0
0
F
T
T
B
F
C
0
F
Generalized Lists
P
10
Heterogeneous Lists
11
Singly Linked Lists Insert
first
BAT
CAT
EAT
HAT
0
FAT
(3)
(2)
(1)
12
Singly Linked Lists Delete
(2)
(3)
first
BAT
CAT
EAT
HAT
0
FAT
(1)
13
Implementing Node of Linked Lists

• The Node of Linked list include
• Data field ????
• Link field ????? node
• ????,??? List Node ? class
• Example
• class ThreeLetterNode
• private
• char data3
• ThreeLetterNode link

14
Implementing Linked Lists

• ?? List Node ???,?? Lists,????????????
• Design Attempt 1
• Use a global variable which is a pointer of
  ListNode
• Example ThreeLetterNode first
• Unable to access to private data members data
  and link.
• Design Attempt 2
• Make public member functions in class ListNode.
• Defeat the purpose of data encapsulation.
• Design Attempt 3
• Composition Class (HAS-A)
• Friend Classes.
• Nested Classes.
• Inheritance with private

15
Design Attempt 3 Composition Classes

• Composition by Friend Classed
• //forward delcarion
• class ThreeLetterList
• class ThreeLetterNode
• friend class ThreeLetterList
• private
• char data3
• ThreeLetterNode link
• class ThreeLetterList
• public
• //List Manipulation operations
• .
• private
• ThreeLetterNode first

• Composition by Nested Classed
• class ThreeLetterList
• public
• //List Manipulation operations
• .
• .
• private
• //nested class
• class ThreeLetterNode
• public
• char data3
• ThreeLetterNode link
• ThreeLetterNode first

16
Design Attempt 3 Composition Classes
List Manipulation operations interface
???
ThreeLetterNode
ThreeLetterList
???
ThreeLetterNode
17
Design Attempt 3 Composition Classes

• Linked Lists ???????????,???????????????,????(Enca
  psulation)????
• Linked Lists ?????(member functions)?????? Linked
  Lists Node ????
• ???????? Liked List ????????????????
• Linked Lists Node ???????????,?? Linked Lists
  ????????!?????,?????

18
Design Attempt 3 Composition Classes ????

• Linked Lists Node ????????????,?????
• ? C ? template ??!
• Linked Lists ??????????????????? Linked List Node
• ?????? iterators class,?????????????,Linked List
  ??????? iterators????????

19
Design Linked Lists with Template

• template ltclass Typegt class List //forward
  declaration
• template ltclass Typegt
• class ListNode
• friend class ListltTypegt
• private
• Type data
• ListNode link
• template ltclass Typegt
• class List
• public
• List() first 0 //constructor initializing
  first to 0
• // List manipulation operations
• private
• ListNodeltTypegt first

20
Container and Iterators

• Containers (??)
• A container class is a class that represents a
  data structure that stores a number of data
  objects. (see book p121)
• Iterators (???)
• An iterator is an object that is used to traverse
  all the elements of a container class.

21
Linked Lists Iterators

• template ltclass Typegt class ListIterator
• public
• ListIterator(const ListltTypegt l)list(l),current
  (l.first)
• bool NotNull()
• bool NextNotNull()
• Type First()
• Type Next()
• private
• const ListltTypegt list // refers to an
  existing list
• ListNodeltTypegt current // points to a node
  in list

Member function see book p180. Program 4.9
22
Linked Lists Iterators Member Function

• template ltclass Typegt
• //check that the current element in list is
  non-null
• Boolean ListIterator ltTypegtNotNull()
• if (current) return TRUE
• else return FALSE
• template ltclass Typegt
• //check that the next element in list is non-null
  
• Boolean ListIterator ltTypegtNextNotNull()
• if (current current-gtlink) return TRUE
• else return FALSE

23
Linked Lists Iterators Member Function

• template ltclass Typegt
• //return a pointer to the first element of list
• Type ListIterator ltTypegtFirst()
• if (list.first) return list.first-gtdata
• else return 0
• template ltclass Typegt
• //return a pointer to the next element of list
• Type ListIterator ltTypegtNext()
• if (current)
• current current-gtlink
• if (current) return current-gtdata
• else return 0

24
Design Linked Lists with Template
List Node

• template ltclass Typegt class List //forward
  declaration
• template ltclass Typegt
• class ListNode
• friend class ListltTypegt
• friend class ListIterator ltTypegt
• private
• Type data
• ListNode link

Lists
template ltclass Typegt class List friend class
ListIterator ltTypegt public List() first
0 //constructor initializing first to 0
//List manipulation operations private
ListNodeltTypegt first
25
Reusing the Linked Lists Class

• ????????,??????????????? Linked Lists Class
  ??????
• ?????,??????????? Linked Lists class,??
• ???????
• stacks ? queues ???
• ??????????????
• ?? equivalence class ?

26
Circular Lists
Check Last Node current-gtlink first
Use the last node instead of the first node
27
Circular Lists with dummy head node
Empty List
28
Deconstruct a Linked Lists

• ?? Linked Lists ? Node ??? new ???,??????????
  delete ??,??? memory leak ????
• ????
• ????
• ????????????,?????? delete first.
• Free Pool
• ?????,?????????????? available-space list or av
  list (???? static ListNodeltTypegt av)
• ???????,??? Free Pool,? Free Pool???,?? Free Pool
  ??????????,??,? Free Pool ??,? new
  ???????????????
• Free Pool ? Liked Lists?????????????

29
Deconstruct a Linked Lists

• ??????
• template ltclass Typegt
• ListltTypegtList()
• //Free all nodes in the chain
• ListNodeltTypegt next
• for ( first first next)
• next first-gtlink
• delete first

Time Complexity O(n)
30
Deconstruct a Linked Lists

• Free Pool

(1)
(3)
(4)
first
0
(2)
av
0
31
Doubly Linked Lists

• To efficiently delete a node, we need to know its
  preceding node. Therefore, doubly linked list is
  useful.
• A node in a doubly linked list has at least three
  fields left link field (llink), a data field
  (item), and a right link field (rlink).

32
Doubly Linked Lists

• class DblList
• class DblListNode
• friend class DblList
• private
• int data
• DblListNode llink, rlink
• class DblList
• public
• //List manipulation operations
• private
• DblListNode first //points to head node

33
Doubly Linked Lists
first
data
LeftLink
RightLink
-
BAT
CAT
EAT
p-gtllink-gtrlink p p-gtrlink-gtllink
-
first
Empty doubly linked circular list with head node
34
Doubly Linked Lists Delete
first
-
x
CAT

• void DblListDelete(DblListNode x)
• if (xfirst)
• cerrltltDeletion of head node not
  permittedltltendl
• else
• x-gtllink-gtrlink x-gtrlink
• x-gtrlink-gtllink x-gtllink

35
Doubly Linked Lists Insert
first
-
x
p
void DblListInsert(DblListNode p,DblListNode
x) //insert node p to the right of node x
p-gtllink x p-gtrlink x-gtrlink
x-gtrlink-gtllink p x-gtrlink p
36
Generalized List

• Generalized List
• A generalized list, A, is a finite sequence of
  ngt 0 elements, a0,,an-1, where ai is either an
  atom or a list.
• The elements ai, 0?i?n-1, that are not atoms are
  said to be sublists of A.
• Examples
• D()
• A (a, (b, c))
• B (A, A, ())
• C (a, C)
• Consequences
• Lists may be empty (Example D)
• lists may be shared by other lists (Example B)
• lists may be recursive (Example C)

37
Generalized List
D() A (a, (b, c)) B (A, A, ()) C (a, C)

• D0

A
a
0
F
T
b
F
c
0
F
B
T
T
0
0
T
C
a
0
F
T
38
Important Generalized List Functions

• List Copy
• See textbook pp225-227
• Program 4.36
• List Equality
• See textbook pp228
• Program 4.37
• List Depth
• See textbook pp229
• Program 4.38

39
Lists Copy

• ??????????
• ???(shallow copy)???????
• ???(deep copy)??????

L
???
20
45
51
76
L
L
20
45
51
76
???
L
20
45
51
76
40
List Copy with Recursive Algorithms

• Indirect Recursive
• A recursive algorithm consists of two components
• The recursive function (the workhorse)
• Declared as a private function
• A second function that invokes the recursive
  function at the top level (the driver)
• declared as a public function.

41
List Copy

• void GenListCopy(const GenList l)
• first Copy(l.first)
• GenListNode GenListCopy(GenListNode p)
• //Copy the nonrecursive list with no shared
  sublists pointed at by p
• GenListNode q 0
• if (p)
• q new GenListNode
• q-gttag p-gttag
• if (!p-gttag)
• q-gtdata p-gtdata
• else
• q-gtdlink Copy(p-gtdlink)
• q-gtlink Copy(p-gtlink)
• return q

Driver
Workhorse
O(M)
42
Example List Copy A (Generalized Lists)
A
r
b
t
t
0
t
u
v
s
f
a
f
b
0
t
f
e
0
w
x
f
c
f
d
0
A((a,b),((c,d),e))
43
Example List Copy A (Generalized Lists)
GenListCopy(A)
Level of recursion Value of p Continuing level p Continuing level p
1 b 2 r 3 u
2 s 3 u 4 v
3 t 4 w 5 0
4 0 5 x 4 v
3 t 6 0 3 u
2 s 5 x 2 r
1 b 4 w 3 0
2 r
1 b
44
List Depth

• The Depth of list is defined as follows.
• An empty list has depth 0.

45
Heterogeneous list

• A heterogeneous list is one that contains nodes
  of different types.
• If merging nodes by using union, then each node
  is allocated for the largest node type. This
  would waste space.
• Use of public inheritance and virtual functions
  can resolve this issue.
• Let S(x) denote the space occupied by an object
  of type x, then
• S(CombinedNode)S(Data)S(CombineNode )
• S(int) max(S(char), S(int), S(float))
  S(CombineNode )

46
Heterogeneous list with union

• struct Data
• // id 0, 1, or 2 if the node contains a char,
  an int, or a float
• int id
• union
• int i
• char c
• float f
• class CombinedNode
• //Use union to merge different node types into
  one class definition
• friend class List
• friend class ListIterator
• private
• Data data
• CombinedNode link

Space allocation is based on the largest data
type, which is float.
47
Heterogeneous list with union

• class list
• friend class ListIterator
• public
• // List manipulation operations follow
• .
• private
• CombinedNode first
• //the return type of class ListIterator is Data

48
Heterogeneous list with Public Inheritance

• class Node
• friend class List friend class ListIterator
• protected
• Node link
• virtual Data GetData() 0
• templateltclass Typegt
• class DerivedNode public Node
• friend class List friend class ListIterator
• public
• DerivedNode(Type item) data(item) link 0
• private
• Type data
• Data GetData()
• Data DerivedNodeltchargtGetData()

49
Heterogeneous list with Public Inheritance
class List friend class ListIterator public N
ode first class ListIterator
public ListIterator(const List l) list(l),
current(l.first) Data First() // minor
change in homogeneous list implementation Data
Next() // minor change in homogeneous list
implementation Boolean NotNull() //
implemented as for homogeneous lists Boolean
NextNotNull() // implemented as for homogeneous
lists private const List list Node
current Data temp
50
Heterogeneous list with Public Inheritance

• Key point
• using the dynamic typing through public
  inheritance, a pointer to a Node may be used to
  point to nodes of type
• DerivedNodeltchargt
• DerivedNodeltintgt
• DerivedNodeltfloatgt.
• This eliminates the problem that necessitated the
  artificial union of the different node types

51
Example

• Stacks and Queues
• Polynomial
• Equivalence Classes
• Sparse Matrix

52
Example Stacks and Queues
data
link
top
first
rear
data
link
Linked Queue
Linked Stack
53
Example Stacks and Queues

• class stack //forward declaration
• class StackNode
• friend class stack
• private
• int data
• StackNode link
• //???,??????????
• StackNode(int d0, StackNode l0)data(d),
  link(l)
• class Stack
• public
• Stack() top0 //constructor
• void Add(const int)
• int Delete(int )
• private
• StackNode top

????? textbook pp188-189
54
Example Polynomial
Polynomial representation of 3x142x81
Polynomial representation of 8x14-3x1010x6
Polynomial representation of 3x142x81 with
Circular linked lists
55
Example Polynomial
Zero polynomial with dummy head to avoid special
case
3x142x81 with dummy head to avoid special case
56
Example Polynomial

• struct Term
• // all members of Terms are public by default
• int coef // coefficient
• int exp // exponent
• void Init(int c, int e) coef c exp e
• class Polynomial
• friend Polynomial operator(const
  Polynomial,const Polynomial)
• private
• ListltTermgt poly

????,class Polynomial ?????????? class List !
57
Example PolynomialRepresentation of 3x2y with
Generalized Lists
trio
vble
exp
link
trio
vble
exp
link
var
y
0
ptr
1
0
var
x
0
no
3
2
0
P
About class definition of polynomial with
Generalized Lists Please see book pp222-223
58
Example Polynomial with Generalized Lists

• trio var the node is a head node.
• vble indicates the name of the variable. Or it is
  an integer point to the variable in a variable
  table.
• exp is set to 0.
• trio ptr
• coefficient itself is a list and is pointed by
  the field dlink.
• exp is the exponent of the variable on which the
  list is based on.
• trio no
• coefficient is an integer and is stored in coef.
• exp is the exponent of the variable on which the
  list is based on.

59
Example PolynomialRepresentation of P(x,y,z)
with Generalized Lists
v
z
0
p
2
p
1
0
v
y
0
v
y
0
p
3
p
2
0
p
4
p
1
0
v
x
0
n
2
0
0
v
x
0
n
3
8
0
v
x
0
n
1
10
n
2
8
0
v
x
0
n
1
4
n
6
3
0
P(x,y,z) x10y3z2 2x8y3z2 3x8y2z2 x4y4z
6x3y4z 2yz ((x102x8)y3 3x8y2)z2
((x46x3)y42y)z
60
Example Equivalence Classes

• Equivalence Relation
• Definition
• A relation over a set S, is said to be an
  equivalence relation over S if and only if it is
• Reflexive(???)
• xx
• Symmetric(???)
• xy ? yx
• Transitive (???)
• xy and yz ? xz
• over S.
• Example
• is an equivalence relation over Z (how prove?
  )

61
Example Equivalence Classes

• Give a set S0,1,2,3,4,5,6,7,8,9,10,11 and
  define the relation as follows
• 04, 31, 610, 89, 74, 68, 35, 211, 110
• If is a equivalence relation over S, then we
  get partitioning of S into three equivalence
  classes
• 0,2,4,7,11 1,3,5 6,8,9,10
• How do we determine the equivalence classes over
  S?

62
Determine the Equivalence Classes

• Two phases to determine equivalence classes.
• Phase one
• the equivalence pairs (i, j) are read in and
  stored.
• ????????????
• Phase two
• Begin at 0 and find all pairs of the form (0, j).
  
• Continue until the entire equivalence class
  containing 0 has been found, marked, and printed.
• Next find another object not yet output, and
  repeat the above process

63
Example Equivalence Classes
0
1
2
3
4
5
6
7
8
9
10
11
data
11
3
11
5
7
3
8
4
6
8
6
0
link
0
0
0
0
0
0
data
4
1
0
10
9
2
link
0
0
0
0
0
0
64
Example Sparse Matrix

• Circular linked list representation of a sparse
  matrix has two types of nodes
• head node head, down, right, and next
• entry node head, down, row, col, right, value
• Head node i is the head node for both row i and
  column i.
• Each head node is belonged to three lists a row
  list, a column list, and a head node list.
• For an nxm sparse matrix with r nonzero terms,
  the number of nodes needed is maxn, m r 1.

65
Example Sparse Matrix
down
head
right
down
head
row
col
right
f
i
j
next
value
Setup for aij
Head node
Typical node
H1
H0
H2
H3
Matrix head
H0
11
H1
12
2
1
H2
-4
H3
-15
??head data member of a node is not shown
66
Example Sparse Matrix
A 4x4 sparse matrix