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CS2468 Data Structures and Data Management

Lecturer: Lusheng Wang Office: B6422Phone: 2788 9820E-mail lwang@cs.cityu.edu.hk

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.htmlText Book: Michael T. Goodrich and Roberto Tamassia, Data

Structurea and Algorithms in Java, John Wiley & Sons, Inc.

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What We Cover

Analysis of Algorithms: worst case time and space complexityData 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.

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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 programsYou will be able to solve new problemsYou will be able to give non-trivial methods to solve problems.

(Your algorithm (program) will be faster than others.)

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Course Evaluations

Course work: 30%Final Exam: 70%Course Work: Three assignments (do them in groups of two people) +a mid term exam.

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Pre-requisites:

CS2360 Java Programming /orCS2362 Computer Programming for Engineers & Scientists /orCS2363 Computer Programming /orCS2372 Fundamentals of Programming /or equivalentIf 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…

Data Structures: A systematic way of organizing and accessing data. --No single data structure works well for ALL purposes.

AlgorithmInput Output

An algorithm is a step-by-step procedure forsolving a problem in a finite amount of time.

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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.

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An Example of an Algorithm

Algorithm sorting(X, n)Input array X of n integersOutput 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.

Part-B1 Stacks

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Abstract Data Types (ADTs)

An abstract data type (ADT) is an abstraction of a data structureAn 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

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The Stack ADT (§4.2)The Stack ADT stores arbitrary objectsInsertions and deletions follow the last-in first-out schemeThink of a spring-loaded plate dispenserMain stack operations:

push(object): inserts an element

object pop(): removes and returns the last inserted element

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

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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

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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: (

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Parentheses Matching Algorithm

Algorithm ParenMatch(X,n):Input: An array X of n tokens, each of which is either a grouping symbol, avariable, an arithmetic operator, or a numberOutput: true if and only if all the grouping symbols in X matchLet S be an empty stackfor i=0 to n-1 do

if X[i] is an opening grouping symbol thenS.push(X[i])

else if X[i] is a closing grouping symbol thenif 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}

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Parentheses Matching Example 1

Input: () (() [()])

i X[i]

Operation Stack Output

0 ( Push ( (

1 ) Pop ( Test if ( and X[i] match? YES

2 ( Push ( (

3 ( Push ( ((

4 ) Pop (Test if ( and X[i] match? YES

(

5 [ Push [ ([

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Parentheses Matching Example 1

Input: () (() [()])

i X[i]

Operation Stack

Output

6 ( Push ( ([(

7 ) Pop (Test if ( and X[i] match? YES

([

8 ] Pop [Test if [ and X[i] match? YES

(

9 ) Pop (Test if ( and X[i] match? YES

Test if stack is Empty? YES TRUE

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Parentheses Matching Example 2

Input: ( () [] ]()

i X[i]

Operation Stack Output

0 ( Push ( (

1 ( Push ( ((

2 ) Pop (Test if ( and X[i] match? YES

(

3 [ Push [ ([

4 ] Pop [Test if [ and X[i] match? YES

(

5 ] Pop (Test if ( and X[i] match ? NO FASLE

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Java Program VS C Program

Refer to the java program QuadraticSort.java and QuadraticSort.C.

Objects: items we deal withClass: type of objectsObjects 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.

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Stack Interface in Java

Java interface corresponding to our Stack ADTRequires 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;

}

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ExceptionsAttempting the execution of an operation of ADT may sometimes cause an error condition, called an exceptionExceptions are said to be “thrown” by an operation that cannot be executed

In the Stack ADT, operations pop and top cannot be performed if the stack is emptyAttempting the execution of pop or top on an empty stack throws an EmptyStackException

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Array-based Stack (Implementation)

A simple way of implementing the Stack ADT uses an arrayWe add elements from left to rightA variable t keeps track of the index of the top element

S0 1 2 t

…

Algorithm size()return t + 1

Algorithm pop()if isEmpty() then

throw EmptyStackException else

t t 1return S[t + 1]

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Array-based Stack (cont.)The array storing the stack elements may become fullA push operation will then throw a FullStackException

Limitation of the array-based implementation

Not intrinsic to the Stack ADT

S0 1 2 t

…

Algorithm push(o)if t = S.length 1 then

throw FullStackException else

t t + 1S[t] o

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Array-based Stack (Cont.)

A Stack might be emptytop returns the element at the top of the Stack, but does not remove the top element. When the Stack is empty, an error occurs.

S0 1 2 t

…

Algorithm isEmpty() if t<0 then return true else return false

Algorithm top()if isEmpty() then

throw EmptyStackException return S[t ]

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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

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Array-based Stack in Javapublic class ArrayStack implements Stack{// holds the stack elements

private Object S[ ];// index to top elementprivate int top = -1;// constructor

public ArrayStack(int capacity) { S = new Object[capacity]);

}

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Array-based Stack in Javapublic 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);

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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]; }

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Computing Spans (not in book)

We show how to use a stack as an auxiliary data structure in an algorithmGiven an array X, the span S[i] of X[i] is the maximum number of consecutive elements X[j] immediately preceding X[i] and such that X[j] X[j+1] Spans have applications to financial analysis

E.g., stock at 52-week high

6 3 4 5 2

1 1 2 3 1

X

S

01234567

0 1 2 3 4

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Quadratic AlgorithmAlgorithm spans1(X, n)

Input array X of n integersOutput array S of spans of X #S new array of n integers nfor i 0 to n 1 do n

s 1 nwhile 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 1X[]= 1,2,3, …, n-1, n.

Algorithm spans1 runs in O(n2) time

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Computing Spans with a Stack

We push the n elements in the stackWe 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

01234567

0 1 2 3 4 5 6 7

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Example:

i= 0, 1, 2, 3, 4, 5, 6, 7 X[i]= 6, 3, 4, 1, 2, 3, 5, 4S[i]= 1, 1, 2, 1, 2, 3, 4, 1.

4 5 53 32 21 1 4 4 43 3 36 6 6 6Stack Stack Stack Stack Stack

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Linear AlgorithmAlgorithm spans2(X, n) #

S new array of n integers nA new empty stack 1for i 0 to n 1 doA.push(i) n

i=n-1 ; j=n-1; 1 while (i<=0) do while (A.isEmpty()

X[A.top()] X[j] ) do n

j A.pop() n if A.isEmpty() then n

S[i] i 1 1 else

S[i] i j +1 n i=j-1; n

return S 1

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

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Summary of Stack

Understand the definition of Stack (basic)

Applications1. Parentheses (moderate)2. Computing Spans (not required)

Implementation of Stack is not required. You should be able to use stack to write

programs to solve problems

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Remarks

Emphasize the concept of ADT.More examples about ADTDelete the Span application examples. Add Queue part (perhaps to week 3)Teach slowly.