Download Data Structures

Applied Computing Lab
Data Structures
Chuan-Ming Liu
Computer Science & Information Engineering
National Taipei University of Technology
Taiwan
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Applied Computing Lab
Instructor
Chuan-Ming Liu (劉傳銘)
Office: 1530 Technology Building
Computer Science and Information Engineering
National Taipei University of Technology
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]
Phone: 02-2771-2171 ext. 4262
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Applied Computing Lab
Text Books
Ellis Horowitz, Sartaj Sahni, and Susan
Anderson-Frees, Fundamentals of Data
Structures in C, 2nd edition, Silicon Press,
2008.
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Supplementary Texts
Michael T. Goodrich and Roberto Tamassia, Data Structures and
Algorithms in JAVA, 4th edition, John Wiley & Sons, 2006. ISBN: 0471-73884-0.
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
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Introduction and Recursion
Analysis Tools
Arrays
Stacks and Queues
Linked Lists
Sorting
Hashing
Trees
Priority Queues
Search Trees
Graphs
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Applied Computing Lab
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
National Taipei University of Technology
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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What is an Algorithm (2)
• 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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What is an Algorithm (3)
Input
Definiteness
Output
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
• Data Structures and Algorithms
• 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
• Data Structures and Algorithms
• 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
1
if n = 0

f ( n) = 
else
n ⋅ f (n − 1)
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Method for Factorial Function
• recursive factorial function
public static int recursiveFactorial(int
int n) {
if (n == 0) return 1;
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
Example recursion trace:
• 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
return 4*6
recursiveFactorial(4)
return 3*2
recursiveFactorial(3)
return 2*1
recursiveFactorial(2)
return 1*1
recursiveFactorial(1)
return 1
recursiveFactorial(0)
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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 Algorithm LinearSum(A, n):
Input: integer array A, an integer n ≧
1, such that A has at least n elements
– using recursion
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 returns
call
return 15 + A[4] = 15 + 5 = 20
LinearSum (A,5)
call
return 13 + A[3] = 13 + 2 = 15
LinearSum (A,4)
call
return 7 + A[2] = 7 + 6 = 13
LinearSum (A,3)
call
return 4 + A[1] = 4 + 3 = 7
LinearSum (A,2)
call
return A[0] = 4
LinearSum (A,1)
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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 i
and 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:
1
if n = 0

p( x, n) = 
else
 x ⋅ p( x, n − 1)
• Following the definition leads to an O(n) time
recursive algorithm (for we make n recursive
calls).
• We can do better than this, however.
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Recursive Squaring
• We can derive a more efficient linearly
recursive algorithm by using repeated squaring:
1
if n = 0


p( x, n) = x ⋅ p( x, (n − 1) / 2) 2 if n > 0 is odd
 p( x, n / 2) 2
if n > 0 is even

• For example,
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25
26
27
=
=
=
=
2(4/2)2 = (24/2)2 = (22)2 = 42 = 16
21+(4/2)2 = 2(24/2)2 = 2(22)2 = 2(42) = 32
2(6/2)2 = (26/2)2 = (23)2 = 82 = 64
21+(6/2)2 = 2(26/2)2 = 2(23)2 = 2(82) = 128
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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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Analyzing the RecursiveApplied Computing Lab
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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Applied Computing Lab
Binary Recursion
• Binary recursion occurs whenever there are
two recursive calls for each non-base case.
• Example: BinaySum
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Example – Summing n
Elements in an Array
Applied Computing Lab
• 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
0, 8
0, 4
4, 4
0, 2
0, 1
2, 2
1, 1
2, 1
4, 2
3, 1
4, 1
6, 2
5, 1
6, 1
7, 1
• Note the floor and ceiling used in the method
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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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Computing Lab
Fibonacci Numbers – Applied
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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Applied Computing Lab
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()
Initial call
PuzzleSolve (3,() ,{a,b,c})
PuzzleSolve (2,b,{a, c})
PuzzleSolve (2,a,{b,c})
PuzzleSolve (1,ab ,{c} )
PuzzleSolve (1,ba ,{c})
abc
bac
PuzzleSolve (2,c,{ a,b})
PuzzleSolve (1,ca ,{b})
cab
PuzzleSolve (1,ac ,{b})
PuzzleSolve (1,bc ,{a} )
PuzzleSolve (1,cb ,{a})
acb
bca
cba
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