Transcript Notes for Lecture 1

CS2468
Data Structures and Data Management
Lecturer: Lusheng Wang
Office: B6422
Phone: 2788 9820
E-mail [email protected]
Welcome to ask questions at ANY time.
Java Source code:
http://net3.datastructures.net/download.html
The course Website:
http://www.cs.cityu.edu.hk/~lwang/cs2468.html
Text Book: Michael T. Goodrich and Roberto Tamassia, Data
Structurea and Algorithms in Java, John Wiley & Sons, Inc.
Stacks
1
What We Cover
Analysis of Algorithms: worst case time and space
complexity
Data Structures: stack, queue, linked list, tree, priority
queue, heap, and hash;
Searching algorithms: binary and AVL search trees;
Sorting algorithms: merge sort, quick sort, bucket sort
and radix sort; (Reduce some contents)
Graph: data structure, depth first search and breadth
first search. (add some interesting contents).
LCS and shortest path.
Stacks
2
Why This Course?
You will be able to evaluate the quality of a
program (Analysis of Algorithms: Running time
and memory space )
You will be able to write fast programs
You will be able to solve new problems
You will be able to give non-trivial methods to
solve problems.
(Your algorithm (program) will be faster than others.)
Stacks
3
Course Evaluations
Course work: 30%
Final Exam: 70%
Course Work:
Three assignments (do them in groups of
two people) +a mid term exam.
Stacks
4
Pre-requisites:
CS2360 Java Programming /or
CS2362 Computer Programming for Engineers &
Scientists /or
CS2363 Computer Programming /or
CS2372 Fundamentals of Programming /or equivalent
If you are not familiar with Java, it is difficult to take
this course (but still possible).
Spend the rest of 8-10 days to study Java.
Test: Design a class with two integers C1 and C2 and a
method f() to calculate the average value of c1 and c2.
If you cannot do that, you should worry…
Stacks
5
Data Structures: A systematic way of organizing
and accessing data.
--No single data structure works well for ALL purposes.
Input
Algorithm
Output
An algorithm is a step-by-step procedure for
solving a problem in a finite amount of time.
Algorithm Descriptions
Nature languages: Chinese, English, etc.
Pseudo-code: codes very close to computer languages,
e.g., C programming language.
Programs: C programs, C++ programs, Java programs.
Goal:
Allow a well-trained programmer to be able to
implement.
Allow an expert to be able to analyze the running time.
Stacks
7
An Example of an Algorithm
Algorithm sorting(X, n)
Input array X of n integers
Output array X sorted in a non-decreasing order
for i  0 to n  1 do
for j  i+1 to n-1 do
if (X[i]>X[j]) then
{ s=X[i];
X[i]=X[j];
X[j]=s;
}
return X
// after i-th round, the i-th smallest number is at i-th position.
Stacks
8
Part-B1
Stacks
Abstract Data Types (ADTs)
An abstract data
type (ADT) is an
abstraction of a
data structure
An ADT specifies:



Data stored
Operations on the
data
Error conditions
associated with
operations
Example: ADT modeling a
students record

The data stored are
 Student name, id No., as1,
as2,as3, exam

The operations supported are
 int averageAs(as1,as2,as3)
 Int finalMark(as1, as2,as3, exam) )

Error conditions:
 Calculate the final mark for
absent student
Stacks
10
The Stack ADT (§4.2)
The Stack ADT stores
arbitrary objects
Insertions and deletions
follow the last-in first-out
scheme
Think of a spring-loaded
plate dispenser
Main stack operations:


push(object): inserts an
element
object pop(): removes and
returns the last inserted
element
Stacks
Auxiliary stack
operations:



object top(): returns the
last inserted element
without removing it
integer size(): returns the
number of elements
stored
boolean isEmpty():
indicates whether no
elements are stored
11
Applications of Stacks
Direct applications


Undo sequence in a text editor
Chain of method calls in the Java Virtual
Machine
Indirect applications


Auxiliary data structure for algorithms
Component of other data structures
Stacks
12
Parentheses Matching
An expression, i.e.,[(2+3)*x+5]*2.
Each “(”, “{”, or “[” must be paired with
a matching “)”, “}”, or “[”





correct: ( )(( )){([( )])}
correct: ((( )(( )){([( )])}))
incorrect: )(( )){([( )])}
incorrect: ({[ ])}
incorrect: (
Stacks
13
Parentheses Matching Algorithm
Algorithm ParenMatch(X,n):
Input: An array X of n tokens, each of which is either a grouping symbol, a
variable, an arithmetic operator, or a number
Output: true if and only if all the grouping symbols in X match
Let S be an empty stack
for i=0 to n-1 do
if X[i] is an opening grouping symbol then
S.push(X[i])
else if X[i] is a closing grouping symbol then
if S.isEmpty() then
return false {nothing to match with}
if S.pop() does not match the type of X[i] then
return false {wrong type}
if S.isEmpty() then
return true {every symbol matched}
else
return false {some symbols were never matched}
Stacks
14
Parentheses Matching Example 1
Input: () (() [()])
i
X[i]
Operation
Stack
0
(
Push (
1
)
Pop (
Test if ( and X[i] match?
(
YES
2
(
Push (
(
3
(
Push (
((
4
)
Pop (
Test if ( and X[i] match?
(
5
[
Output
Push [
YES
([
Stacks
15
Parentheses Matching Example 1
Input: () (() [()])
i
X[i] Operation
Stack Output
6
(
Push (
([(
7
)
Pop (
Test if ( and X[i] match?
([
8
9
]
)
YES
Pop [
Test if [ and X[i] match?
YES
Pop (
Test if ( and X[i] match?
YES
Test if stack is Empty?
YES
Stacks
(
TRUE
16
Parentheses Matching Example 2
Input: ( () [] ]()
i
X[i]
Operation
Stack
0
(
Push (
(
1
(
Push (
((
2
)
Pop (
Test if ( and X[i] match?
(
YES
3
[
Push [
([
4
]
Pop [
Test if [ and X[i] match?
(
YES
Pop (
Test if ( and X[i] match ?
NO
5
]
Stacks
Output
FASLE
17
Java Program VS C Program
Refer to the java program QuadraticSort.java
and QuadraticSort.C.
Objects: items we deal with
Class: type of objects
Objects store data and provides methods.

Structure of data (components of data)
 Forexamlpe: A stack holding: (([.

Methods (functions) to do things
 For example: push(), pop(), Top(), etc.
Stacks
18
Stack Interface in Java
Java interface
corresponding to
our Stack ADT
Requires the
definition of class
EmptyStackException
public interface Stack {
public int size();
public boolean isEmpty();
public Object top()
throws EmptyStackException;
public void push(Object o);
public Object pop()
throws EmptyStackException;
}
Stacks
19
Exceptions
Attempting the
execution of an
operation of ADT may
sometimes cause an
error condition, called
an exception
Exceptions are said to
be “thrown” by an
operation that cannot
be executed
Stacks
In the Stack ADT,
operations pop and
top cannot be
performed if the
stack is empty
Attempting the
execution of pop or
top on an empty
stack throws an
EmptyStackException
20
Array-based Stack (Implementation)
A simple way of
implementing the
Stack ADT uses an
array
We add elements
from left to right
A variable t keeps
track of the index of
the top element
Algorithm size()
return t + 1
Algorithm pop()
if isEmpty() then
throw EmptyStackException
else
tt1
return S[t + 1]
…
S
0 1 2
t
Stacks
21
Array-based Stack (cont.)
The array storing the
stack elements may
become full
A push operation will
then throw a
FullStackException


Algorithm push(o)
if t = S.length  1 then
throw FullStackException
else
tt+1
Limitation of the arrayS[t]  o
based implementation
Not intrinsic to the
Stack ADT
…
S
0 1 2
t
Stacks
22
Array-based Stack (Cont.)
A Stack might be
empty
top returns the
element at the top
of the Stack, but
does not remove the
top element. When
the Stack is empty,
an error occurs.
Algorithm isEmpty()
if t<0 then return true
else return false
Algorithm top()
if isEmpty() then
throw EmptyStackException
return S[t ]
…
S
0 1 2
t
Stacks
23
Performance and Limitations for
array-based Stack
Performance



Let n be the number of elements in the stack
The space used is O(n)
Each operation runs in time O(1)
Limitations


The maximum size of the stack must be defined a
priori and cannot be changed
Trying to push a new element into a full stack
causes an implementation-specific exception
Stacks
24
Array-based Stack in Java
public class ArrayStack implements Stack{
// holds the stack elements
private Object S[ ];
// index to top element
private int top = -1;
// constructor
public ArrayStack(int capacity) {
S = new Object[capacity]);
}
Stacks
25
Array-based Stack in Java
public Object pop() throws EmptyStackException{
if isEmpty()
throw new EmptyStackException
(“Empty stack: cannot pop”);
Object temp = S[top];
// facilitates garbage collection
S[top] = null;
top = top – 1;
return temp;
}
public int size() {
return (top + 1);
Stacks
26
Array-based Stack in Java
public boolean isEmpty() {
return (top < 0);
}
public void push(Object obj) throws FullStackException {
if (size() == capacity)
throw new FullStackException("Stack overflow.");
S[++top] = obj;
}
public Object top() throws EmptyStackException {
if (isEmpty())
throw new EmptyStackException("Stack is empty.");
return S[top];
}
Stacks
27
Computing Spans (not in book)
7
We show how to use a stack 6
as an auxiliary data structure 5
in an algorithm
4
Given an array X, the span
3
S[i] of X[i] is the maximum
2
number of consecutive
elements X[j] immediately
1
preceding X[i] and such that 0
X[j]  X[j+1]
Spans have applications to
financial analysis

E.g., stock at 52-week high
Stacks
0
X
S
1
6
1
3
1
2
3
4
2
5
3
4
2
1
28
Quadratic Algorithm
Algorithm spans1(X, n)
Input array X of n integers
Output array S of spans of X
#
S  new array of n integers
n
for i  0 to n  1 do
n
s1
n
while s  i  X[i  s]  X[i-s+1] 1 + 2 + …+ (n  1)
ss+1
1 + 2 + …+ (n  1)
S[i]  s
n
return S
1
X[]= 1,2,3, …, n-1, n.
Algorithm spans1 runs in O(n2) time
Stacks
29
Computing Spans with a Stack
We push the n elements in the
stack
We sacn the elements in the
stack in the reverse order.


We find the last element x[j]
with x[j]<=x[i] and x[j-1]>x[i]
and set s[i]=i-j+1.
If no such a j exists, s[i]=i+1.
Compute the s[i] for the
remaining i’s as follows:



for (i=n-2; i>=0; i--)
if (s[i+1]>1 & s[i]==0) then
s[i]=s[i+1]-1
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7
Stacks
30
Example:
i= 0, 1, 2, 3, 4, 5, 6, 7
X[i]= 6, 3, 4, 1, 2, 3, 5, 4
S[i]= 1, 1, 2, 1, 2, 3, 4, 1.
4
5
3
2
1
4
3
6
Stack
5
3
2
1
4
3
6
Stack
4
3
6
Stack
6
Stack
Stack
Stacks
31
Linear Algorithm
Each index of the array
 Is pushed into the
stack exactly one
 Is popped from the
stack at most once
The statements in the
two while loops are
executed at most n
times
Algorithm spans2 runs in
O(n) time
for (i=n-2; i>=0; i--)
if (s[i+1]>1 &
s[i]==0) then
s[i]=s[i+1]-1
Algorithm spans2(X, n)
S  new array of n integers
A  new empty stack
for i  0 to n  1 doA.push(i)
i=n-1 ; j=n-1;
while (i<=0) do
while (A.isEmpty() 
X[A.top()]  X[j] ) do
j  A.pop()
if A.isEmpty() then
S[i]  i + 1
else
S[i]  i  j +1
i=j-1;
return S
Stacks
#
n
1
n
1
n
n
n
1
n
n
1
32
Summary of Stack
Understand the definition of Stack (basic)
Applications
1.
2.
Parentheses (moderate)
Computing Spans (not required)
 Implementation of Stack is not required.
 You should be able to use stack to write
programs to solve problems
Stacks
33
Remarks
Emphasize the concept of ADT.
More examples about ADT
Delete the Span application examples.
Add Queue part (perhaps to week 3)
Teach slowly.
Stacks
34