Algorithms is A sequence of computational steps that transform the input into output. solving a well-specified computational problem

1. Design and Analysis of Algorithms
Qilong Feng

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Course Outline
 Instructor
 Qilong Feng
 Office number:Computer building 413
 e-mail : csufeng@mail.csu.edu.cn
 Textbook
 Introduction to The Design & Analysis of Algorithms
(ISBN 01-2003-2170)
 References:
 T. H. Cormen, C. E. Leiserson, R. L. Rivest, Introduction to
Algorithms, The MIT Press. http://theory.lcs.mit.edu/~clr/

3. Copyright Li Zimao
@ 2007-2008-1 SCUEC
Textbook Cover
What does the cover include?
Puzzles, Notions, and Stories
of Algorithms.

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Algorithms
What is algorithm?

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Algorithm and Philosophy
Example: Given a jar with unlimited capacity, and unlimited
number of balls. The balls are labeled from 1.
When 1minutes close to 12:00, do: put the 1~10 balls into the jar,
and take the ball with label 10 out of the jar.
When 1/2 minutes close to 12:00, do: put the 11~20 balls into the
jar, and take the ball with label 20 out of the jar.
When 1/4 minutes close to 12:00, do: put the 21~30 balls into the
jar, and take the ball with label 30 out of the jar.
……

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Algorithm and Philosophy
Assume that putting and getting balls takes no time.
Question: When it is 12:00, how many balls are in the jar?
Answer:
Go on the above process untill 12:00
Why?
Infinity.

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Algorithm and Philosophy
Are you sure? Let us change some rules:
When 1minutes close to 12:00, do: put the 1~10 balls into the jar, and
take the ball with label 1 out of the jar.
When 1/2 minutes close to 12:00, do: put the 11~20 balls into the jar,
and take the ball with label 2 out of the jar.
When 1/4 minutes close to 12:00, do: put the 21~30 balls into the jar,
and take the ball with label 3 out of the jar.
……
Question: When it is 12:00, how many balls are in the jar?
Answer: 0 Why?

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Algorithm and Philosophy
The order of taking balls changes everything!
I don’t think it is right!!!
When 1minutes close to 12:00, do: put the 1~10 balls into the jar, and
take the ball randomly out of the jar.
When 1/2 minutes close to 12:00, do: put the 11~20 balls into the jar,
and take the ball randomly out of the jar.
When 1/4 minutes close to 12:00, do: put the 21~30 balls into the jar,
and take the ball randomly out of the jar.
…… Answer: 0

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Algorithm and Philosophy
Too amazing!
The really different part for the above three methods: the label of ball
when taking it out of jar.
Labels change the results!!!!
Labels change the way of our thinking.
Nothing is infinity, and infinity is nothing!
∞
The differences between “nothing” and “infinity” only exist in
human’s mind!

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Algorithm and Philosophy
In some sense, algorithm is a way of thinking, or
algorithm is one kind of Philosophy.

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Algorithms
 A sequence of computational steps that transform
the input into output.
 solving a well-specified computational problem.
What is algorithm?

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Algorithms
 A sequence of computational steps that transform
the input into output.
 solving a well-specified computational problem.
What is algorithm?

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Example of well-specified problem: Sorting
 Input: a sequence of numbers: 1, 100, 8, 25, 11, 9, 2, 1,
200.
 Output: a sorted (increasing order or decreasing order)
sequence of numbers
 1, 2, 8, 9, 11, 25, 100, 200.
Another example:
Create web page (your homepage) using HTML.
-No computational steps.

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Algorithms
An algorithm is a sequence of unambiguous instructions for
solving a computational problem, i.e., for obtaining a required
output for any input in a finite amount of time.
“computer”
problem
algorithm
input output

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Properties of Algorithms
 Finiteness
terminates after a finite number of steps
 Definiteness
Each step must be rigorously and unambiguously specified.
 Input
Valid inputs must be clearly specified.
 Output
can be proved to produce the correct output given a valid input.
 Effectiveness
Steps must be sufficiently simple and basic.
-e.g., check if 2 is the largest integer n for which there is a solution
to the equation xn + yn = zn in positive integers x, y, and z

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Examples
 Is the following a legitimate algorithm?
i 1
While (i <= 10) do
a  i + 1
Print the value of a
End of loop
Stop

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Algorithms
Why do we study algorithm?

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Algorithms
 Algorithm is the soul of computer.
 Algorithm is everywhere.
 Algorithm can train our way of thinking.
 This course will make you different.

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Computing the Greatest Common Divisor of
Two Integers
 Gcd(m, n): the largest integer that divides both m and n.

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Computing the Greatest Common Divisor of
Two Integers
 Gcd(m, n): the largest integer that divides both m and n.
 First try -- Euclid’s algorithm:

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Computing the Greatest Common Divisor of
Two Integers
 Gcd(m, n): the largest integer that divides both m and n.
 First try -- Euclid’s algorithm:
gcd(m, n) = gcd(n, m mod n)
 Step1: If n = 0, return the value of m as the answer and
stop; otherwise, proceed to Step 2.
 Step2: Divide m by n and assign the value of the
remainder to r.
 Step 3: Assign the value of n to m and the value of r to n.
Go to Step 1.

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Descriptions of Algorithms
 Flow chart
 Programs
 Natural languages :Ambiguous
 Pseudo-code
 A mixture of a natural language and programming
language-like structures
 Precise and succinct.
 Pseudocode in this course
 omits declarations of variables
 use indentation to show the scope of such statements as
for, if, and while.
 use  for assignment

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Pseudocode of Euclid’s Algorithm
Algorithm Euclid(m, n)
while n ≠ 0 do
r  m mod n
m  n
n  r
return m
Questions:
 Finiteness: how do we know that Euclid’s algorithm actually comes
to a stop?
 Definiteness: non ambiguity
 Effectiveness: effectively computable.

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Second Try for gcd(m, n)
 Consecutive Integer Algorithm
 Step1: Assign the value of min{m, n} to t.
 Step2: Divide m by t. If the remainder of this division is 0, go
to Step3;otherwise, go to Step 4.
 Step3: Divide n by t. If the remainder of this division is 0,
return the value of t as the answer and stop; otherwise,
proceed to Step4.
 Step4: Decrease the value of t by 1. Go to Step2.
 Questions
 Which algorithm is faster, the Euclid’s or this one?

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Third try for gcd(m, n)
 Middle-school procedure (brute force)
 Step1: Find the prime factors of m.
 Step2: Find the prime factors of n.
 Step3: Identify all the common factors in the two
prime found in Step1 and Step2.
 Step4: Compute the product of all the common factors
and return it as the gcd of the numbers given.
 Question
 Is this a legitimate algorithm?

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What can we learn from the previous 3
examples?
 The same algorithm can be represented in several
different ways. (different pseudocode)
 There might exists more than one algorithm for a
certain problem!
 Algorithms for the same problem can be based on
very different ideas and can solve the problem with
dramatically different speeds.!

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The Process of Design an Algorithm
 Understanding the problem
 Asking questions, do a few examples by hand, think
about special cases, etc.
 Design an algorithm
 Proving correctness
 Analyzing an algorithm
 Time efficiency : how fast the algorithm runs
 Space efficiency: how much extra memory the
algorithm needs.
 Coding an algorithm