1. Chapter 12
Abstract Data Type

2. OBJECTIVES After reading this chapter, the reader should be able to:
Understand the concept of an abstract data type (ADT).
Understand the concept of a linear list as well as its operations and applications.
Understand the concept of a stack as well as its operations and applications.
Understand the concept of a queue as well as its operations and applications.
Understand the concept of a tree as well as its operations and applications.
Understand the concept of a graph as well as its operations and applications.

3. 12.1
BACKGROUND

4. Abstract Data Type (ADT)
An Abstract Data Type (ADT) is a data structure and a collection of functions which operate on the data structure.
Stack – The operations new(), push(v, S), top(S), and pop(S) may be defined with axiomatic semantics as following.
new() returns a stack
pop(push(v, S)) = S
top(push(v, S)) = v
where S is a stack and v is a value.

5. Note:
The concept of abstraction means: 1. You know what a data type can do How it is done is hidden.

6. Queue problem
Queue simulation
enqueuing
dequeuing
queue

7. Note:
Abstract Data Type
Declaration of data
Declaration of operations
Encapsulation of data and operations

8. Stack problem
Stack – The operations new(), push(S, v), top(S), and popoff(S) may be defined with axiomatic semantics as following.
new() returns a stack
popoff(push(S, v)) = S
top(push(S, v)) = v
where S is a stack and v is a value.
The pop operation is a combination of top, to return the top value, and popoff, to remove the top value.

9. Model for ADT
Figure 12-1

10. 12.2
LINEAR
LISTS

11. Linear list
Figure 12-2
Linear List is a list in which each element has a unique successor.

12. Categories of linear lists
Figure 12-3
Categories of linear lists
Data can be inserted and deleted anywhere.
Data can only be inserted and deleted at the ends of the structure.

13. Insertion in a general linear list with ordered data
Figure 12-4

14. Deletion from a linear list
Figure 12-5
Deletion from a linear list

15. Retrieval from a linear list
Figure 12-6
Retrieval from a linear list

16. Traversal of a linear list
Figure 12-7
Traversal is an operation in which all elements in the list are processed sequentially, one by one.

17. Implementation
Two common methods:
Array
Linked list

18. 12.3
STACKS

19. Three representations of a stack
Figure 12-8
Stacks
A restricted linear list in which all additions and deletions are made at one end, called the top.
A LIFO (Last In, First Out) data structure.

20. Push operation in a stack
Figure 12-9

21. Pop operation in a stack
Figure 12-10

22. Example 1
Show the result of the following operations on a stack S. push (S , 10) push (S , 12)
push (S , 8) if not empty (S), then pop (S) push (S , 2)
Solution

23. Implementation Applications Two common methods: Reversing data Parsing
Array
Linked list (more common)
Applications
Reversing data
1,2,3,4 → 4,3,2,1
Parsing
Unmatched parentheses in an algebraic expression
Postponing data
Postfix: ab+c* → (a+b)*c, abc+* → a*(b+c)
Backtracking steps
Computer gaming

24. 12.4
QUEUES

25. Queue representation Queues
Figure 12-12
Queues
A restricted linear list in which data can only be inserted at one end (rear) and deleted from the other end (front).
A FIFO (First In, First Out) data structure.

26. Enqueue operation
Figure 12-13

27. Dequeue operation
Figure 12-14

28. Example 2
Show the result of the following operations on a queue Q. enqueue (Q , 23) if not empty (Q), dequeue (Q) enqueue (Q , 20) enqueue (Q , 19) if not empty (Q), dequeue (Q)
Solution

29. Implementation
Two common methods:
Array
Linked list (more common)
Applications found in virtually every OS and network and in countless other areas.

30. 12.5
TREES

31. Trees A Tree consists of
a finite set of elements called Nodes
a finite set of directed lines, called Branches, that connect the nodes
Degree of a node – number of branches associated with the node.
Indegree branch
Outdegree branch
Root –
The first node of a tree
Indegree : 0
Other nodes –
Indegree : 1
All nodes –
outdegree : 0, 1 or more.

32. Representation of a tree
Figure 12-16
Representation of a tree

33. Trees Leaf – Internal nodes – a node that is not a root or a leaf.
Outdegree : 0
Internal nodes – a node that is not a root or a leaf.
Parent –
A node with successors
outdegree : > 0
Child –
A node with a predecessor
indegree : 1
Siblings – two or more nodes with the same parent.
Ancestor – any node in the path from the root to the node.
Descendant – any node in the path from the node to a leaf.

34. Trees
Path – a sequence of nodes in which each node is adjacent to the next one.
Level of a node – its distance from the root.
Root : 0
Height (Depth) of the tree –
the level of the leaf in the longest path from the root plus 1.
Subtree – any connected structure below the root.

35. Figure 12-17
Tree terminology
Height : 3

36. Figure 12-18
Subtrees

37. 12.6
BINARY
TREES

38. Binary Trees
Binary tree – A tree in which no node can have more than two subtrees.
Null tree - A tree with no node.
Height of a Binary Tree – (with N nodes)
Hmax = N
Hmin = [log2N] +1
# of nodes of a binary tree – (Given height H)
Nmin = H
Nmax = 2H - 1

39. Binary tree
Figure 12-19

40. Examples of binary trees
Figure 12-20

41. Balanced Trees
The distance of a node from the root determines how efficiently it can be located.
Balance Factor of a Binary Tree – is the difference in height between its left and right subtrees. B = HL - HR
A binary tree is balanced if the height of its subtrees differs by no more than 1 (B = -1, 0, or +1) and its subtrees are also balanced.

42. Binary Tree Traversal
Binary Tree Traversal each node of the tree be processed once and only once in a predetermined sequence.
Depth-First Traversal –
Preorder Traversal (NLR) –
Inorder Traversal (LNR) –
Postorder Traversal (LRN) –
Need a stack
Breadth-First Traversal –
process all of the children of a node before proceeding with the next level.
Need a queue

43. Depth-first traversal of a binary tree
Figure 12-21
NLR LNR LRN

44. Preorder traversal of a binary tree
Figure 12-22
A, B, C, D, E, F

45. Inorder traversal of a binary tree
Figure 12-23
C, B, D, A, E, F

46. Postorder traversal of a binary tree
Figure 12-24
C, D, B, F, E, A

47. Breadth-first traversal of a binary tree
Figure 12-25
Breadth-first traversal of a binary tree

48. Implementation Applications
A binary tree is normally implemented as a linked list.
Applications
Expression tree – a binary tree with the following properties:
Each leaf is an operand
The root and internal nodes are operators.
Subtrees are subexpressions, with the root being an operator.
Three different expression formats:
Infix expression
Postfix expression – produced by Postorder traversal
Prefix expression – produced by Preorder traversal

49. Expression tree Infix expression: Prefix expression: + * a + b c d
Figure 12-26
Infix expression:
Prefix expression: + * a + b c d
Postfix expression: a b c + * d +

50. 12.7
GRAPHS

51. Graphs A Graph consists of
A set of vertices (sing. vertex) and
A set of lines (connecting pairs of vertices)
Directed graph (digraph) – each line has a direction to its successor - arc
Undirected graph – each line has no direction – edge
Adjacent vertices (neighbors) – two vertices directly connected by a line
Path – a sequence of vertices in which each vertex is adjacent to the next one.
Cycle – a path consisting of at least three vertices that starts and ends with the same vertex.

52. Graphs
Two vertices are said to be connected if there is a path between them.
A graph is said to be connected (suppressing direction) if there is a path from any vertex to any other vertex.
A directed graph is strongly connected if there is a path from each vertex to every other vertex.
A directed graph is weekly connected if at least two vertices are not connected.

53. Directed and undirected graphs
Figure 12-27

54. Add vertex
Figure 12-28

55. Delete vertex
Figure 12-29

56. Add edge
Figure 12-30

57. Delete edge
Figure 12-31

58. Find vertex
Figure 12-32

59. Traverse Graph
You must somehow ensure that you process the data in each vertex only once.
Visited flag
Two standard graph traversals
Depth-first traversal – you process a vertex’s descendants before you move to an adjacent vertex.
Breadth-first traversal – you process all adjacent vertices of a vertex before going to the next level.

60. Depth-first traversal of a graph
Figure 12-33
在不同的資料結構下，即使起始點相同，產生的順序亦有所不同。

61. Breadth-first traversal of a graph
Figure 12-34
在不同的資料結構下，即使起始點相同，產生的順序亦有所不同。

62. Implementation Applications Two common methods: Network
Array (adjacency matrix)
Linked list (adjacency list)
Applications
Network
Weighted graph - a graph with weighted lines.
Minimum Spanning Tree
A spanning tree is a tree that contains all of the vertices in the graph.
A minimum spanning tree is a spanning tree such that the sum of its weights is the minimum.

63. Graph implementations
Figure 12-35: Part I
Weight (e.g. distance between networks in an internetwork)

64. Graph implementations
Figure 12-35: Part 2

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