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Data Structure
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Che-Rung Lee
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CS135601 Introduction to Information Engineering
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Data abstraction
• Main memory is organized as a sequence
of addressable cells, but the data we want
to model is usually not.
• Use “model” and “simulation”
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Pointers
• What is a pointer?
– A special data that records memory address
• Example in C
int a = 3;
int *p = NULL;
p = &a;
variable
a
p
address
0x03
0x04
value
3
5
0
0x03
*p = 5;
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Outline
•
•
•
•
•
Customized data type
Array and list
Stack and queue
Trees
Hash table
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Customized Data Type
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How to model a warrior?
•
•
•
•
•
•
•
Class
Skills
Equipments
Life point
Magic point
Money
…
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Diablo III
But computers only have primitive data
types: integer, real, character, and Boolean.
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User-defined data types
• Conglomerate of primitive data types
collected under a single name
• Example in C: struct
User-defined data type
typedef struct {
char class[10]; // Barbarian, Witch, Wizard or Monk
int lifePoint; // min is 0, max is 100
int level;
// min is 1, max is 72
…
} Warrior;
An instance of
type Warrior
Warrior player1;
player1.lifePoint = 100;
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Abstract data type
• A full model of abstract data type should
include the operations of the model
– Like +-*/, input, output for primitive data types
• Example in C++: class
class Warrior {
char class[10]; // Barbarian, Witch, Wizard or Monk
…
void fight(….); // function that defines the action “fight”
};
– This is called an object, which we will talk
more in the programming language lesson.
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Heterogeneous array
• The storage that contains different types of
data is called a heterogeneous array
– struct and class are heterogeneous arrays
– The items are called components.
– The storage that contains the same type of
data is called a homogeneous array
• Example
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struct {
char Name[25];
int
Age;
int
SkillRating;} Employee;
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Storage of heterogeneous array
• Static method:
– components are stored
one after the other in a
contiguous block
Meredith W Linsmeyer
• Dynamic method:
– components are stored
in separate locations
identified by pointers
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23 6.2
pointers
Meredith W Linsmeyer
23
6.2
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Array and List
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When to use arrays?
• Stock prices, student names, temperature
readings
– One dimensional array
• Matrix, images, the grades of class, train
schedule
– Two dimensional array
• Computed Tomography(斷層掃描)
– Three dimensional array
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Storing arrays
• Use a variable to denote the address of
the first element
– Ex: int Readings[24];
Relative address
called “index”
0
1
2
3
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In C, the index
starts from 0
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Two dimensional array
• Two dimensional array is stored in a one
dimensional memory cells.
• Two ways to order the data
column
row
Row major order
a11
a12
a13
a11 a12 a13 a21 a22 a23 a31 a32 a33 a41 a42 a43
a21
a22
a23
Column major order
a31
a32
a33
a41
a42
a43
a11 a21 a31 a41 a12 a22 a32 a42 a13 a23 a33 a43
– What is the memory location of A[2][3] in the
row (column)
major order?
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High dimensional array
• Consider the dimensional array A[m][n][k]
– What is the size of the array?
– What is the memory location of A[1][2][3] in
the row major order?
This changes first
• The row major order
– What is the memory location of A[1][2][3] in
the column major order?
• The row major order
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This changes first
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When to use list?
• List is a collection of data which are
arranged sequentially.
– One dimensional array is a list of elements
– Two dimensional array can be viewed as a
list of rows/columns
– A string is a list of characters
– Music is a list of sounds
– Stacks and queues can be implemented
using lists
• We will talk those later
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Contiguous list
• List is stored in a contiguous block of
memory cells (an array)
– Ex: list of names. Each name is occupied 8
bytes.
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Linked list
• List in which each entries are linked by
pointers
– Head pointer: Pointer to first entry in list
– NIL pointer: A “non-pointer” value used to
indicate end of list
Use customized data
type to define
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Static v.s. dynamic data structures
• Static data structures:
– Size and shape does not change
– Contiguous list
– Easily to locate elements. No need to store
address.
• Dynamic data structures:
– Size and shape can change
– Linked list
– Easily to delete/insert elements
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Linked list: delete/insert element
• Delete
• Insert
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Stack and Queue
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What is a stack?
• A list in which entries are removed and
inserted only at the head
– Top: The head of stack
– Bottom or base: The tail of stack
– Push: To insert an entry at the top
– Pop: To remove the entry at the top
– LIFO: Last-in-first-out
top
bottom
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When to use stacks?
• When the algorithm needs data LIFO?
– EX1: reverse a word, ABCCBA
• Push A
• Push B
• Push C
• Pop C
• Pop B
• Pop A
A
B
– EX2: check matching parentheses (3*[(1+1)*2]
• Push “(“
• Push “[“
• Push “(“
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• Find “)”, pop “(“, matched
• Find “]”, pop “[“, matched
• No more “)”, but still one “(“ in stack,
not matched
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C
Stack implementation
• Using a list + a pointer (head)
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Queue
• A list in which entries are removed at the
head and are inserted at the tail.
– Enqueue: insert an entry at the tail
– Dequeue: remove an entry at the head
– FIFO: First-in-first-out
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Tail
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Head
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Examples of using queues
• Ex1: the job queues
in operating system
• Ex2: simulation of the Josephus problem
– Dequeue 1
– Enqueue 1
– Dequeue 2
– Dequeue 3
– Enqueue 3
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5
4
3
2
1
Operation counts  2n
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Queue implementation
• A list + 2 pointers (head+tail)
– Enqueue A, B, C
– Dequeue A, enqueue D
– Dequeue B, enqueue E
Head pointer
Tail pointer
• If using a static list, the
queue crawls through
memory as entities are
inserted and removed.
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B
C
D
E
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Circular queue
• A technique that uses a fixed region of
memory space to implement queue.
head
tail
A
E
B
C
Enqueue A, B, C
Dequeue A, Enqueue D
Dequeue B, Enqueue E
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Trees
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What is a tree?
• A collection of nodes that are linked in a
hierarchical structure, in which every node
is linked by one parent, except the root.
– Node: An entry in a tree
– Parent: The node immediately
above a specified node
– Root: The node at the top
– Terminal or leaf node:
A node at the bottom
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Hierarchical relations
• Parent: The node immediately above a node
– The parent of F is B
• Child: A node immediately below a node
– The children of C are G and H.
A
• Ancestor: Parent, parent of parent, etc.
– The ancestor of K are F, B, and A.
B
C
D
• Descendent: Child, child of child, etc.
– The descendent of B are E, F, K, and L.
E
• Siblings: Nodes sharing a common
parent
F
K
G
H
I
J
L
– The siblings of C are B and D.
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Depth and height
• Textbook’s definition
A
– The depth of a tree is the longest
path from the root to a leaf node
• The length of a path is the
number of nodes on the path
• Ex: the depth of the tree is 4
B
E
• Conventional definition
C
F
K
G
D
H
I
J
L
• Use the word “height” instead of depth
• The length of a path is the number of links on the path
• Ex: The height of the tree is 3 (= 4 – 1)
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What are trees used for?
• Representing hierarchical data
– Organization chart
• Searching data
– Game tree
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Binary tree
• A tree in which each parent has at most
two children
Left child
Left subtree
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Right child
Right subtree
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Storing a binary tree in a list
• This is called a heap in some applications.
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Advantages of using heap
• Easily to find the index of parent & children
– Parent(B) = [index of B] / 2 = 1
– LeftChild(B) = [index of B]*2 = 4
– RightChild(B) = [index of B]*2 + 1= 5
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Problems for heap
• Heap is inefficient for storing the binary
tree that is sparse and unbalanced
– Sparse: most node has one or zero child
– Unbalanced: the right subtree is much larger
than the left
subtree, or
vice versa
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Storing a binary tree
using pointers
• Each node
Use customized
data type to
define
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Recursive structure
• Tree is a recursive structure
– The subtrees of a tree are trees
• The recursive algorithms for
a binary tree may look like this
procedure some_operation (root)
if (root is not NULL) then
( call some_operation(root.left_child)
do some operations on root
call some_operation(root.right_child))
– It is a depth first, in order algorithm for tree
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Hash Table
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Search
• Search is a common task in daily life
– Phone book: given a name, fine the phone
number
– Dictionary: given a word, find it’s definition
– Map: given an address, find the location or
route
– DNS: given an URL, find it’s IP address
• Tree can be used to speedup searches.
– How? And what is the operation count?
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Constant time search
• Something can be found in constant time
– EX: What is fifth element of the array A? A[4]
• An array is like a lookup table. One can
use the index to query and get the value
• Can we use this idea to organize data so
that searches can be done in the constant
time?
– Hash table (or hash map)
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Hash table
• Each record of data has a key field
– Key is like the index of an array.
– An unique identification of the data (ideally)
• The storage space is divided into buckets
– Each bucket is like an array cell.
– Each record is stored in the bucket
corresponding to its key, so it can be retrieved
in constant time
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How to define the mapping?
• Unique identification of a record is usually
too large to be the index for storage
– For example, the ASCII code for a string
We do not want to create such a large array!!
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Hash function
• A hash function computes a bucket
number for each key value
– EX: suppose there are only 41 buckets.
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Problem
• Collision: The case of two or more keys
hashing to the same bucket
– Major problem when table is over 75% full
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Solutions
• Use linked lists to store collided data
– The search time becomes linear to the
number of collided data
• Increase the number of buckets and
rehash all data
– Time/space tradeoff
• Design a better hash function/algorithm
– It’s a research problem
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References
• Textbook 8.1, 8.2, 8.3, 8.5, 9.5
• Wikipedia
• Thomas H. Cormen, Charles E. Leiserson,
Ronald L. Rivest, Clifford Stein,
“Introduction to Algorithms”
Related courses
• 資料結構，演算法，程式語言
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