Data Structures - NTUTcmliu/DS/NTUT_DS_S09u-GIT/ClassNotes/Ch1... · 1 CSIE, NTUT, TAIWAN 1 Applied Computing Lab Data Structures Chuan-Ming Liu Computer Science & Information Engineering

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Applied Computing LabApplied Computing LabApplied Computing LabApplied Computing Lab

Data Structures

Chuan-Ming Liu

Computer Science & Information Engineering

National Taipei University of Technology

Taiwan

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Instructor

Chuan-Ming Liu (劉傳銘)

Office: 1530 Technology Building

Computer Science and Information Engineering


TAIWAN

Phone: (02) 2771-2171 ext. 4251

Email: [email protected]

Office Hours: Mon: 11:10-12:00, 13:10-14:00 and Thu:10:10 - 12:00, OR by appointment.

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Teaching Assisant

Bill In-Chi Su (蘇英啟)

Office: 1226 Technology Building

Office Hours: Tue:10:00~12:00 and Wed: 10:00 ~ 12:00, OR by appointment

Email: [email protected]: 02-2771-2171 ext. 4262

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Text Books

Ellis Horowitz, Sartaj Sahni, and Susan Anderson-Frees, Fundamentals of Data

Structures in C, 2nd edition, Silicon Press, 2008.

Supplementary Texts

• Michael T. Goodrich and Roberto Tamassia, Data Structures and Algorithms in JAVA, 4th edition, John Wiley & Sons, 2006. ISBN: 0-471-73884-0.

• Sartaj Sahni, Data Structures, Algorithms, and Applications in JAVA, 2nd edition, Silicon Press, 2005. ISBN: 0-929306-33-3.

• Frank M. Carranno and Walter Savitch, Data Structures and Abstractions with Java, Prentice Hall, 2003. ISBN: 0-13-017489-0.

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Course Outline

• Introduction and Recursion

• Analysis Tools

• Arrays

• Stacks and Queues

• Linked Lists

• Sorting

• Hashing

• Trees

• Priority Queues

• Search Trees

• Graphs

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Course Work

• Assignments (50%): 6-8 homework sets

• Midterm (20%)

• Final exam (30%)

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Course Policy (1)

• No late homework is acceptable.

• For a regrade please contact me for the question within 10 days from the date when the quiz or exam was officially returned. No regrading after this period.

• Cheating directly affects the reputation of the Department and the University and lowers the morale of other students. Cheating in homework and exam will not be tolerated. An automatic grade of 0 will be assigned to any student caught cheating. Presenting another person's work as your own constitutes cheating. Everything you turn in must be your own doing.

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Course Policy (2)

• The following activities are specifically forbidden on

all graded course work:

– Theft or possession of another student's solution or partial

solution in any form (electronic, handwritten, or printed).

– Giving a solution or partial solution to another student,

even with the explicit understanding that it will not be

copied.

– Working together to develop a single solution and then

turning in copies of that solution (or modifications) under

multiple names.

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First Thing to Do

• Please visit the course web site

http://www.cc.ntut.edu.tw/~cmliu/DS/NTUT_

DS_S09u-GIT/

• Send an email to me using the email

address:[email protected]. I will make a

mailing list for this course. All the

announcements will be broadcast via this

mailing list.

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Introduction

Chuan-Ming Liu

Computer Science & Information Engineering


Taiwan

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Outline

• Data Structures and Algorithms

• Pseudo-code

• Recursion

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What is Data Structures

• A data structure * in computer science is a

way of storing data in a computer so that it can

be used efficiently.

– An organization of mathematical and logical

concepts of data

– Implementation using a programming language

– A proper data structure can make the algorithm or

solution more efficient in terms of time and space

*Wikipedia: http://en.wikipedia.org/wiki/Data_structure

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Why We Learn Data Structures

• Knowing data structures well can make our

programs or algorithms more efficient

• In this course, we will learn

– Some basic data structures

– How to tell if the data structures are good or bad

– The ability to create some new and advanced data

structures

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What is an Algorithm (1)

An algorithm is a finite set of instructions that, if followed, accomplishes a particular task. All the algorithms must satisfy the following criteria:

– Input

– Output

– Precision (Definiteness)

– Effectiveness

– Finiteness

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• Definiteness: each instruction is clear and

unambiguous

• Effectiveness: each instruction is

executable; in other words, feasibility

• Finiteness: the algorithm terminates after

a finite number of steps.

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Input OutputDefiniteness

Effectiveness

Finiteness

Computational Procedures

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Procedures vs. Algorithms

• Termination or not

• One example for procedure is OS

• Program, a way to express an algorithm

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Expressing Algorithms

• Ways to express an algorithm

– Graphic (flow chart)

– Programming languages (C/C++)

– Pseudo-code representation

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Outline


• Pseudo-code

• Recursion

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Example – Selection Sort

Suppose we must devise a program that sorts a

collection of n≥1 elements.

Idea: Among the unsorted elements, select the

smallest one and place it next in the sorted list.

for (int i=1; i<=n; i++) {

examine a[i] to a[n] and suppose

the smallest element is at a[j];

interchange a[i] and a[j];

}

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Pseudo-code for Selection Sort

SelectionSort(A)

/* Sort the array A[1:n] into nondecreasing order. */

for i ←1 to length[A]

do j ← i

for k← (i+1) to length[A]

do if A[k]<A[j]

then j ← k;

t← A[i]

A[i] ← A[j]

A[j] ← t

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Maximum of Three Numbers

This algorithm finds the largest of the numbers

a, b, and c.

Input Parameters: a, b, c

Output Parameter: x

max(a,b,c,x) {

x = a

if (b > x) // if b is larger than x, update x

x = b

if (c > x) // if c is larger than x, update x

x = c

}

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Pseudo-Code Conventions

• Indentation as block structure

• Loop and conditional constructs similar to

those in PASCAL, such as while, for,

repeat( do – while) , if-then-else

• // as the comment in a line

• Using = for the assignment operator

• Variables local to the given procedure

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Pseudo-Code Conventions

• Relational operators: ==, != , ≥, ≤.

• Logical operators: &&, ||, !.

• Array element accessed by A[i] and A[1..j] as

the subarray of A

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Outline


• Pseudo-code

• Recursion

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Recursion

• Recursion is the concept of defining a method

that makes a call to itself

• A method calling itself is making a recursive

call

• A method M is recursive if it calls itself (direct

recursion) or another method that ultimately

leads to a call back to M (indirect recursion)

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Repetition

• Repetition can be achieved by

– Loops (iterative) : for loops and while loops

– Recursion (recursive) : a function calls itself

• Factorial function

– General definition: n! = 1� 2� 3� �� (n-1)� n

– Recursive definition

−⋅

==

elsenfn

nnf

)1(

0 if1)(

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Method for Factorial Function

• recursive factorial function

public static public static public static public static intintintint recursiveFactorial(intintintint n) {if if if if (n == 0) return return return return 1;else return else return else return else return n * recursiveFactorial(n- 1);

}

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Content of a Recursive Method

• Base case(s)

– Values of the input variables for which we perform no

recursive calls are called base cases (there should be at

least one base case).

– Every possible chain of recursive calls must eventually

reach a base case.

• Recursive calls

– Calls to the current method.

– Each recursive call should be defined so that it makes

progress towards a base case.

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Visualizing Recursion

• Recursion trace

• A box for each

recursive call

• An arrow from each

caller to callee

• An arrow from each

callee to caller

showing return value

Example recursion trace:

recursiveFactorial(4)





return 1

return 1*1

return 2*1

return 3*2

return 4*6

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About Recursion

• Advantages

– Avoiding complex case analysis

– Avoiding nested loops

– Leading to a readable algorithm description

– Efficiency

• Examples

– File-system directories

– Syntax in modern programming languages

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Linear Recursion

• The simplest form of recursion

• A method M is defined as linear recursion if it

makes at most one recursive call

• Example: Summing the Elements of an Array

– Given: An integer array A of size m and an integer

n, where m≧n≧1.

– Problem: the sum of the first n integers in A.

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Summing the Elements of an Array

• Solutions:

– using (for) loop

– using recursion

Algorithm LinearSum(A, n):

Input: integer array A, an integer n ≧1, such that A has at least n elements

Output: The sum of the first n integers in A

if n = 1 then

return A[0]

else

return LinearSum(A, n - 1) + A[n - 1]

Note: the index starts from 0.

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Recursive Method

• An important property of a recursive method –

the method terminates

• An algorithm using linear recursion has the

following form:

– Test for base cases

– Recur

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Analyzing Recursive Algorithms

• Recursion trace

– Box for each instance of the method

– Label the box with parameters

– Arrows for calls and returnsLinearSum (A,5)

LinearSum (A,1)

LinearSum(A,2)

LinearSum (A,3)

LinearSum(A,4)

call

call

call

call return A[0] = 4

return 4 + A[1] = 4 + 3 = 7

return 7 + A[2] = 7 + 6 = 13

return 13 + A[3] = 13 + 2 = 15

call return 15 + A[4] = 15 + 5 = 20

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Example

• Reversing an Array by Recursion

– Given: An array A of size n

– Problem: Reverse the elements of A (the first

element becomes the last one, …)

• Solutions

– Nested loop ?

– Recursion

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Reversing an Array

Algorithm ReverseArray(A, i, j):

Input: An array A and nonnegative integer indices iand j

Output: The reversal of the elements in A starting at index i and ending at j

if i < j then

Swap A[i] and A[ j]

ReverseArray(A, i + 1, j - 1)

return

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Facilitating Recursion

• In creating recursive methods, it is important

to define the methods in ways that facilitate

recursion.

• This sometimes requires we define additional

parameters that are passed to the method.

– For example, we defined the array reversal method

as ReverseArray(A, i, j), not ReverseArray(A).

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Example – Computing Powers

• The power function, p(x, n)=xn, can be

defined recursively:

• Following the definition leads to an O(n) time

recursive algorithm (for we make n recursive

calls).

• We can do better than this, however.

−⋅

==

else)1,(

0 if1),(

nxpx

nnxp

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Recursive Squaring

• We can derive a more efficient linearly recursive algorithm by using repeated squaring:

• For example,

24 = 2(4/2)2 = (24/2)2 = (22)2 = 42 = 16

25 = 21+(4/2)2 = 2(24/2)2 = 2(22)2 = 2(42) = 32

26 = 2(6/2)2 = (26/2)2 = (23)2 = 82 = 64

27 = 21+(6/2)2 = 2(26/2)2 = 2(23)2 = 2(82) = 128

>

>

=

−⋅=

even is 0 if

odd is 0 if

0 if

)2/,(

)2/)1(,(

1

),(2

2

n

n

n

nxp

nxpxnxp

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A Recursive Squaring Method

Algorithm Power(x, n):

Input: A number x and integer n ≧ 0

Output: The value xn

if n = 0 then

return 1

if n is odd then

y = Power(x, (n - 1)/ 2)

return x · y ·y

else /* n is even */

y = Power(x, n/ 2)

return y · y

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Applied Computing LabApplied Computing LabApplied Computing LabApplied Computing LabAnalyzing the Recursive

Squaring Method

Algorithm Power(x, n):

Input: A number x and integer n ≧0

Output: The value xn

if n = 0 then

return 1

if n is odd then

y = Power(x, (n - 1)/ 2)

return x � y � y

else

y = Power(x, n/ 2)

return y � y

Each time we make a recursive call we halve the value of n; hence, we make log n recursive calls. That is, this method runs in O(log n) time.

It is important that we used a variable twice here rather than calling the method twice.

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Tail Recursion

• Tail recursion occurs when a linearly

recursive method makes its recursive call as

its last step.

• Such methods can be easily converted to

non-recursive methods (which saves on

some resources).

• The array reversal method is an example.

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Example – Using Iteration

Algorithm IterativeReverseArray(A, i, j ):

Input: An array A and nonnegative integer

indices i and j

Output: The reversal of the elements in A

starting at index i and ending at j

while i < j do

Swap A[i ] and A[ j ]

i = i + 1

j = j - 1

return

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Binary Recursion

• Binary recursion occurs whenever there are

two recursive calls for each non-base case.

• Example: BinaySum

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Applied Computing LabApplied Computing LabApplied Computing LabApplied Computing LabExample – Summing n

Elements in an Array

• Recall that this problem has been solved using

linear recursion

• Using binary recursion instead of linear recursion

Algorithm BinarySum(A, i, n):

Input: An array A and integers i and n

Output: The sum of the n integers in A starting at index i

if n = 1 then

return A[i ]

return BinarySum(A,i,n/2)+BinarySum(A,i+n/2,n/2)

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Recursion Trace

• Note the floor and ceiling used in the method

3, 1

2, 2

0, 4

2, 11, 10, 1

0, 8

0, 2

7, 1

6, 2

4, 4

6, 15, 1

4, 2

4, 1

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Fibonacci Numbers

• Fibonacci numbers are defined recursively:

F0 = 0

F1 = 1

Fi = Fi-1+ Fi-2 for i > 1.

• Example: 0, 1, 1, 2, 3, 5, 8, …

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Applied Computing LabApplied Computing LabApplied Computing LabApplied Computing LabFibonacci Numbers – Binary

Recursion

Algorithm BinaryFib(k):

Input: Nonnegative integer k

Output: The kth Fibonacci number Fk

if k≦ 1 then

return k

else

return BinaryFib(k-1) + BinaryFib(k-2)

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Analyzing the Binary Recursion

• Algorithm BinaryFib makes a number of calls

that are exponential in k

• By observation, there are many redundant

computations:

F0 = 0; F1 = 1; F2 = F1+ F0;

F3 = F2+ F1 =(F1

+ F0)+F1; …

• The above two results show the inefficiency of

the method using binary recursion

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A Better Fibonacci Algorithm

• Using linear recursion instead – avoid the redundant computation:Algorithm LinearFibonacci(k):

Input: A nonnegative integer k

Output: Pair of Fibonacci numbers (Fk, Fk-1)

if k = 1 then

return (k, 0)

else

(i, j) = LinearFibonacci(k - 1)

return (i +j, i)

• Runs in O(k) time.

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Multiple Recursion

• Motivating example: summation puzzles

• pot + pan = bib

• dog + cat = pig

• boy + girl = baby

• Multiple recursion: makes potentially many

recursive calls (not just one or two).

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Algorithm for Multiple Recursion

Algorithm PuzzleSolve(k,S,U):

Input: An integer k, sequence S, and set U (the universe of elements to test)

Output: An enumeration of all k-length extensions to S using elements in U

without repetitions

for all e in U do

Remove e from U {e is now being used}

Add e to the end of S

if k = 1 then

Test whether S is a configuration that solves the puzzle

if S solves the puzzle then

return “Solution found: ” S

else

PuzzleSolve(k - 1, S,U)

Add e back to U {e is now unused}

Remove e from the end of S

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Visualizing PuzzleSolve()

PuzzleSolve (3,() ,{a,b,c})

Initial call

PuzzleSolve (2,c,{a,b})PuzzleSolve (2,b,{a,c})PuzzleSolve (2,a,{b,c})

PuzzleSolve (1,ab ,{c} )

PuzzleSolve (1,ac ,{b}) PuzzleSolve (1,cb ,{a})

PuzzleSolve (1,ca ,{b})

PuzzleSolve (1,bc ,{a})

PuzzleSolve (1,ba ,{c})

abc

acb

bac

bca

cab

cba

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Data Structures - NTUTcmliu/DS/NTUT_DS_S09u-GIT/ClassNotes/Ch1... · 1 CSIE, NTUT, TAIWAN 1 Applied Computing Lab Data Structures Chuan-Ming Liu Computer Science & Information Engineering