applied computing lab csie, ntut, taiwan 1 data structures chuan-ming liu computer science &...
Post on 15-Jan-2016
229 views
TRANSCRIPT
![Page 1: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/1.jpg)
CSIE, NTUT, TAIWAN1
Applied Computing Lab
Data Structures
Chuan-Ming LiuComputer Science & Information Engineering
National Taipei University of Technology
Taiwan
![Page 2: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/2.jpg)
CSIE, NTUT, TAIWAN2
Applied Computing Lab
Instructor
Chuan-Ming Liu (劉傳銘 )Office: 1530 Technology Building
Computer Science and Information EngineeringNational Taipei University of Technology
TAIWAN
Phone: (02) 2771-2171 ext. 4251Email: [email protected] Office Hours: Mon: 11:10-12:00, 13:10-14:00 and
Thu:10:10 - 12:00, OR by appointment.
![Page 3: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/3.jpg)
CSIE, NTUT, TAIWAN3
Applied Computing Lab
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
![Page 4: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/4.jpg)
CSIE, NTUT, TAIWAN4
Applied Computing Lab
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 Algorit
hms 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.
![Page 5: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/5.jpg)
CSIE, NTUT, TAIWAN5
Applied Computing Lab
Course Outline• Introduction and Recursion • Analysis Tools • Arrays • Stacks and Queues • Linked Lists • Sorting • Hashing • Trees • Priority Queues • Search Trees • Graphs
![Page 6: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/6.jpg)
CSIE, NTUT, TAIWAN6
Applied Computing Lab
Course Work
• Assignments (50%): 6-8 homework sets
• Midterm (20%)
• Final exam (30%)
![Page 7: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/7.jpg)
CSIE, NTUT, TAIWAN7
Applied Computing Lab
Course Policy (1)
• No late homework is acceptable. • For a regrade please contact me for the question withi
n 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.
![Page 8: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/8.jpg)
CSIE, NTUT, TAIWAN8
Applied Computing Lab
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.
![Page 9: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/9.jpg)
CSIE, NTUT, TAIWAN9
Applied Computing Lab
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.
![Page 10: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/10.jpg)
CSIE, NTUT, TAIWAN10
Applied Computing Lab
Introduction
Chuan-Ming LiuComputer Science & Information Engineering
National Taipei University of TechnologyTaiwan
![Page 11: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/11.jpg)
CSIE, NTUT, TAIWAN11
Applied Computing Lab
Outline
• Data Structures and Algorithms
• Pseudo-code
• Recursion
![Page 12: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/12.jpg)
CSIE, NTUT, TAIWAN12
Applied Computing Lab
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
![Page 13: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/13.jpg)
CSIE, NTUT, TAIWAN13
Applied Computing Lab
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
![Page 14: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/14.jpg)
CSIE, NTUT, TAIWAN14
Applied Computing Lab
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
![Page 15: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/15.jpg)
CSIE, NTUT, TAIWAN15
Applied Computing Lab
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.
![Page 16: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/16.jpg)
CSIE, NTUT, TAIWAN16
Applied Computing Lab
What is an Algorithm (3)
Input OutputDefiniteness
Effectiveness
Finiteness
Computational Procedures
![Page 17: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/17.jpg)
CSIE, NTUT, TAIWAN17
Applied Computing Lab
Procedures vs. Algorithms
• Termination or not
• One example for procedure is OS
• Program, a way to express an algorithm
![Page 18: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/18.jpg)
CSIE, NTUT, TAIWAN18
Applied Computing Lab
Expressing Algorithms
• Ways to express an algorithm
– Graphic (flow chart)
– Programming languages (C/C++)
– Pseudo-code representation
![Page 19: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/19.jpg)
CSIE, NTUT, TAIWAN19
Applied Computing Lab
Outline
• Data Structures and Algorithms
• Pseudo-code
• Recursion
![Page 20: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/20.jpg)
CSIE, NTUT, TAIWAN20
Applied Computing Lab
Example – Selection Sort
Suppose we must devise a program that sorts a collection of n1 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 supposethe smallest element is at a[j];interchange a[i] and a[j];
}
![Page 21: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/21.jpg)
CSIE, NTUT, TAIWAN21
Applied Computing Lab
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
![Page 22: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/22.jpg)
CSIE, NTUT, TAIWAN22
Applied Computing Lab
Maximum of Three Numbers
This algorithm finds the largest of the numbers a, b, and c.
Input Parameters: a, b, cOutput Parameter: xmax(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}
![Page 23: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/23.jpg)
CSIE, NTUT, TAIWAN23
Applied Computing Lab
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
![Page 24: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/24.jpg)
CSIE, NTUT, TAIWAN24
Applied Computing Lab
Pseudo-Code Conventions
• Relational operators: ==, != , , .
• Logical operators: &&, ||, !.
• Array element accessed by A[i] and A[1..j] as the subarray of A
![Page 25: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/25.jpg)
CSIE, NTUT, TAIWAN25
Applied Computing Lab
Outline
• Data Structures and Algorithms
• Pseudo-code
• Recursion
![Page 26: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/26.jpg)
CSIE, NTUT, TAIWAN26
Applied Computing Lab
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)
![Page 27: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/27.jpg)
CSIE, NTUT, TAIWAN27
Applied Computing Lab
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)(
![Page 28: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/28.jpg)
CSIE, NTUT, TAIWAN28
Applied Computing Lab
Method for Factorial Function
• recursive factorial function public static int recursiveFactorial(int n) { if (n == 0) return 1; else return n * recursiveFactorial(n- 1);
}
![Page 29: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/29.jpg)
CSIE, NTUT, TAIWAN29
Applied Computing Lab
Content of a Recursive Method
• Base case(s)– Values of the input variables for which we perform no r
ecursive 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.
![Page 30: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/30.jpg)
CSIE, NTUT, TAIWAN30
Applied Computing Lab
Visualizing Recursion
• Recursion trace• A box for each recur
sive call• An arrow from each
caller to callee• An arrow from each
callee to caller showing return value
Example recursion trace:
recursiveFactorial(4)
recursiveFactorial(3)
recursiveFactorial(2)
recursiveFactorial(1)
recursiveFactorial(0)return 1
return 1*1
return 2*1
return 3*2
return 4*6
![Page 31: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/31.jpg)
CSIE, NTUT, TAIWAN31
Applied Computing Lab
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
![Page 32: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/32.jpg)
CSIE, NTUT, TAIWAN32
Applied Computing Lab
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.
![Page 33: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/33.jpg)
CSIE, NTUT, TAIWAN33
Applied Computing Lab
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 elementsOutput: 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.
![Page 34: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/34.jpg)
CSIE, NTUT, TAIWAN34
Applied Computing Lab
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
![Page 35: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/35.jpg)
CSIE, NTUT, TAIWAN35
Applied Computing Lab
Analyzing Recursive Algorithms
• Recursion trace– Box for each instance of the method– Label the box with parameters– Arrows for calls and returns
LinearSum(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
![Page 36: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/36.jpg)
CSIE, NTUT, TAIWAN36
Applied Computing Lab
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
![Page 37: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/37.jpg)
CSIE, NTUT, TAIWAN37
Applied Computing Lab
Reversing an Array
Algorithm ReverseArray(A, i, j): Input: An array A and nonnegative integer indices i a
nd j Output: The reversal of the elements in A starting at i
ndex i and ending at j if i < j then
Swap A[i] and A[ j]ReverseArray(A, i + 1, j - 1)
return
![Page 38: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/38.jpg)
CSIE, NTUT, TAIWAN38
Applied Computing Lab
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).
![Page 39: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/39.jpg)
CSIE, NTUT, TAIWAN39
Applied Computing Lab
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
![Page 40: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/40.jpg)
CSIE, NTUT, TAIWAN40
Applied Computing Lab
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 = 1625= 21+(4/2)2 = 2(24/2)2 = 2(22)2 = 2(42) = 3226= 2(6/2)2 = (26/2)2 = (23)2 = 82 = 6427= 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
![Page 41: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/41.jpg)
CSIE, NTUT, TAIWAN41
Applied Computing Lab
A Recursive Squaring Method
Algorithm Power(x, n): Input: A number x and integer n ≧ 0 Output: The value xn
if n = 0 thenreturn 1
if n is odd theny = Power(x, (n - 1)/ 2)return x · y ·y
else /* n is even */y = Power(x, n/ 2)return y · y
![Page 42: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/42.jpg)
CSIE, NTUT, TAIWAN42
Applied 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 thenreturn 1
if n is odd theny = Power(x, (n - 1)/ 2)return x · y · y
elsey = 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.
![Page 43: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/43.jpg)
CSIE, NTUT, TAIWAN43
Applied Computing Lab
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.
![Page 44: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/44.jpg)
CSIE, NTUT, TAIWAN44
Applied Computing Lab
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 doSwap A[i ] and A[ j ]
i = i + 1
j = j - 1
return
![Page 45: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/45.jpg)
CSIE, NTUT, TAIWAN45
Applied Computing Lab
Binary Recursion
• Binary recursion occurs whenever there are two recursive calls for each non-base case.
• Example: BinaySum
![Page 46: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/46.jpg)
CSIE, NTUT, TAIWAN46
Applied Computing LabExample – Summing n
Elements in an Array
• Recall that this problem has been solved using linear recursion
• Using binary recursion instead of linear recursionAlgorithm 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)
![Page 47: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/47.jpg)
CSIE, NTUT, TAIWAN47
Applied Computing Lab
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
![Page 48: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/48.jpg)
CSIE, NTUT, TAIWAN48
Applied Computing Lab
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, …
![Page 49: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/49.jpg)
CSIE, NTUT, TAIWAN49
Applied Computing LabFibonacci Numbers – Binary
Recursion
Algorithm BinaryFib(k):Input: Nonnegative integer kOutput: The kth Fibonacci number Fk
if k ≦ 1 thenreturn k
elsereturn BinaryFib(k-1) + BinaryFib
(k-2)
![Page 50: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/50.jpg)
CSIE, NTUT, TAIWAN50
Applied Computing Lab
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
![Page 51: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/51.jpg)
CSIE, NTUT, TAIWAN51
Applied Computing Lab
A Better Fibonacci Algorithm
• Using linear recursion instead – avoid the redundant computation:Algorithm LinearFibonacci(k):
Input: A nonnegative integer kOutput: 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.
![Page 52: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/52.jpg)
CSIE, NTUT, TAIWAN52
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).
![Page 53: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/53.jpg)
CSIE, NTUT, TAIWAN53
Applied Computing Lab
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 Sif k = 1 then
Test whether S is a configuration that solves the puzzleif S solves the puzzle then
return “Solution found: ” Selse
PuzzleSolve(k - 1, S,U)Add e back to U {e is now unused}Remove e from the end of S
![Page 54: Applied Computing Lab CSIE, NTUT, TAIWAN 1 Data Structures Chuan-Ming Liu Computer Science & Information Engineering National Taipei University of Technology](https://reader036.vdocuments.us/reader036/viewer/2022081504/56649d3f5503460f94a18937/html5/thumbnails/54.jpg)
CSIE, NTUT, TAIWAN54
Applied Computing Lab
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