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Algorithms
1. Notion of an algorithm
2. Properties of an algorithm
3. The GCD algorithm
4. Correctness of the GCD algorithm
5. Termination of the GCD algorithm
6. Performance
7. Data structures
Need for studying algorithms:
• The study of algorithms is the cornerstone of computer
science. It can be recognized as the core of computer
science.
• Computer programs would not exist without algorithms.
With computers becoming an essential part of our
professional & personal life‘s, studying algorithms
becomes a necessity, more so for computer science
engineers.
• Another reason for studying algorithms is that if we know a
standard set of important algorithms, They enhance our
analytical skills & help us in developing new algorithms for
required applications
Problem
Datastructure
Algorithm
Program
Computer
+
Input Output
1-Notion of an algorithm
Definition
An algorithm is a clearly specified set of instructions
describing the solution to a specific problem.
• An algorithm is finite set of instructions that is followed,
accomplishes a particular task. (Sahani)
An algorithm
• takes the input and transforms it into an adequate output,
• must be independent from any programming language,
• is written in a level of detail that allows to reproduce it in
any programming language,
• has to be designed so it can be reused and understood by
others.
Algorithms must satisfy the following
criteria
1.Input: Zero or more quantities are externally supplied.
2.Output : At least one quantity is produced.
3.Definiteness:Each instruction is clear and unambiguous.
4.Finiteness: If we trace out the instructions of an
algorithm, then for all cases, the algorithm terminates after
a finite number of steps.
5.Effectiveness: Every instruction must be very basic so
that it can be carried out, in principle,by a person using only
pencil and paper. It is not enough that each operation be
definite as in criterion3; it also must be feasible.
2-Properties of an algorithm
Some properties must be satisfied by an algorithm in
order to allow a successful execution of the
corresponding program:
1. Correctness: if the input conditions are satisfied and the
algorithm instructions executed, then the correct output is
produced.
2. Termination: the algorithm must terminate after a finite number
of steps. Thus, it has to be composed by a finite number of steps.
This can be ensured if the algorithm avoids an infinite loop.
3. Performance: Quantification of the space and time complexities.
• Algorithms that are definite and effective
are also called computational procedures.
• The same algorithm can be represented in
several ways
• Several algorithms to solve the same
problem
• Different ideas different speed
The four distinct areas of
studying algorithms
1. How to devise algorithms
2. How to validate algorithms
3. How to analyze algorithms
4. How to test a program
The GCD algorithm:
Given two positive integers m and n, find the greatest common divisor,
gcd(m,n).
A view of the problem:
All numbers from 1 up to the smallest of m and n, say n.
Naive algorithm:
Go through search space (from 1 to n)
Keep track of largest number that divides both m and n.
Is there a more efficient way of doing this?
Yes, the Euclidean Algorithm (Euclid – c.350 B.C.E.)
Pseudocode of the GCD algorithm :
Algorithm GreatestCommonDivisor
Input: Two positive integers, m and n
Ouptut: The gcd of m and n
repeat
r  m mod n
m  n
n  r
until (r == 0)
Output m and STOP.
Example:
m = 24, n = 9
r  24 mod 9 = 6
m  9
n  6
r  9 mod 6 = 3
m  6
n  3
r  6 mod 3 = 0
m  3
n  0
Output 3
STOP
Euclids algorithm
Step1:if n=0 return val of m & stop else proceed step 2
Step 2:Divide m by n & assign the value of remainder to r
Step 3:Assign the value of n to m,r to n,Go to step1.
Another algorithm to solve the same problem
Euclids algorithm
Step1:Assign the value of min(m,n) to t
Step 2:Divide m by t.if remainder is 0,go to step3 else goto step4
Step 3: Divide n by t.if the remainder 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 step 2
Thus, ri will become 0 in at most 2 log n iterations
Example: m = 1989 and n = 1590
Remainder sequence:
399 = 1989 mod 1590  399 < 1989/2
393 = 1590 mod 399  393 < 1590/2
6 = 399 mod 393  6 < 399/2 < 1989/4
3 = 393 mod 6  3 < 393/2 < 1590/4
0 = 6 mod 3  0 < 6/2 < 399/4 < 1989/8
We stop here, but if we continue: < 1590/8
< 1989/16
< 1590/16
Number of Steps:
In 2 * 3 steps, ri reduced by factor of 8
2 * 4 ri 16
2 * 5 ri 32
2 * 6 ri 64
……
2 * log n ri n
Thus, ri becomes 0 in at most 2 log n iterations.
4-Correctness of the GCD algorithm.
4-Correctness of the GCD algorithm.
• Correctness is an important issue in algorithm design. This means
Correctness is an important issue in algorithm design. This means
proving that the algorithm works for all legal inputs.
proving that the algorithm works for all legal inputs.
It is analogous to proving the correctness of a theorem in
It is analogous to proving the correctness of a theorem in
mathematics!
mathematics!
• The correctness of the GCD algorithm depends on the following loop
The correctness of the GCD algorithm depends on the following loop
invariant:
invariant:
gcd(m,n)=gcd(n,r) where r=m%n
gcd(m,n)=gcd(n,r) where r=m%n
Claim 1:
gcd(m,n) = gcd(n,r)
Proof:
The gcd of the new pair is equal to the gcd of the previous pair.
How is this correct?
Let us write:
m = q * n + r
where 0  r  n.
This implies that:
a common divisor of m and n is also
a common divisor of n and r,
and vice versa. q.e.d.
5-Termination of the GCD algorithm.
5-Termination of the GCD algorithm.
• Show that algorithm terminates in a finite number of steps.
Show that algorithm terminates in a finite number of steps.
• This must be true for every valid input.
This must be true for every valid input.
• Can we show this for Algorithm GCD?
Can we show this for Algorithm GCD?
• We must show that
We must show that ri
ri goes to 0 in a finite number of steps.
goes to 0 in a finite number of steps.
Observe:
Observe:
• The sequence of remainders strictly decreases.
The sequence of remainders strictly decreases.
• They are all non-negative.
They are all non-negative.
• Thus
Thus ri
ri will become 0 in at most 2 log
will become 0 in at most 2 log n
n steps.
steps.
6-Performance of the GCD algorithm.
6-Performance of the GCD algorithm.
• Quantification of performance of the algorithm.
Quantification of performance of the algorithm.
• Crucial parameters: time and space.
Crucial parameters: time and space.
• Called time and space complexity of the algorithm.
Called time and space complexity of the algorithm.
• Will be discussed latter in the course.
Will be discussed latter in the course.
For example:
For example:
Time complexity of GCD:
Time complexity of GCD:
Takes at most 2 log
Takes at most 2 log n
n steps, where
steps, where n
n <
< m
m
Thus, worst-case time complexity:
Thus, worst-case time complexity: O
O(log
(log n
n)
)
7-Data structures
7-Data structures
• The study of different ways of organizing data.
The study of different ways of organizing data.
• Why?
Why?
• Efficiency of algorithm depends on how data is organized.
Efficiency of algorithm depends on how data is organized.
• Reason for studying data structures and algorithms together.
Reason for studying data structures and algorithms together.
• Organic connection between the two areas.
Organic connection between the two areas.
• In 60-212 the programming language JAVA is studied.
In 60-212 the programming language JAVA is studied.
• Here: An algorithmic perspective.
Here: An algorithmic perspective.
• Independent of programming language (eg. C, C++, JAVA).
Independent of programming language (eg. C, C++, JAVA).
For example:
• The median of a list of n numbers is a number m such that:
n/2 numbers in the list are  m, and
n/2 … are  m.
• Many definitions of median, we take:
If n is even:
Two medians:
lower median and upper median,
Then, median is average of lower and upper medians.
If n is odd:
Both medians (lower and upper) are the same.
• Consider this problem:
Given a sorted list of n numbers, find the median.
• A crucial question:
How should we store the list?
• We store it in an array, A, then
The median is found in constant time, O(1) !!
• Median = (A[5]+A[6])/2 = (19+22)/2 = 20.5
2 5 9 16 19 22 26 27 30 31
1 2 3 4 5 6 7 8 9 10
• Whereas in a linked list:
Traverse half of the list in n/2 steps, which is O(n) !
• Quite simple stated:
The way in which data is organized is crucial in complexity.
More examples like this will be seen later.
2 5 9 16 19 22 ….
first
Example of problems:
• Design an efficient algorithm to determine if a list has repeated
elements.
• Given a list of n elements find their minimum (or maximum).
• Given n points in the plane, find the pair(s) of points which are closest
to each other.
• Given n points in the plane determine if any three are contained in a
straight line.
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Notion of Algorithms.pdf

  • 1. Algorithms 1. Notion of an algorithm 2. Properties of an algorithm 3. The GCD algorithm 4. Correctness of the GCD algorithm 5. Termination of the GCD algorithm 6. Performance 7. Data structures
  • 2. Need for studying algorithms: • The study of algorithms is the cornerstone of computer science. It can be recognized as the core of computer science. • Computer programs would not exist without algorithms. With computers becoming an essential part of our professional & personal life‘s, studying algorithms becomes a necessity, more so for computer science engineers. • Another reason for studying algorithms is that if we know a standard set of important algorithms, They enhance our analytical skills & help us in developing new algorithms for required applications
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  • 9. 1-Notion of an algorithm Definition An algorithm is a clearly specified set of instructions describing the solution to a specific problem. • An algorithm is finite set of instructions that is followed, accomplishes a particular task. (Sahani) An algorithm • takes the input and transforms it into an adequate output, • must be independent from any programming language, • is written in a level of detail that allows to reproduce it in any programming language, • has to be designed so it can be reused and understood by others.
  • 10. Algorithms must satisfy the following criteria 1.Input: Zero or more quantities are externally supplied. 2.Output : At least one quantity is produced. 3.Definiteness:Each instruction is clear and unambiguous. 4.Finiteness: If we trace out the instructions of an algorithm, then for all cases, the algorithm terminates after a finite number of steps. 5.Effectiveness: Every instruction must be very basic so that it can be carried out, in principle,by a person using only pencil and paper. It is not enough that each operation be definite as in criterion3; it also must be feasible.
  • 11. 2-Properties of an algorithm Some properties must be satisfied by an algorithm in order to allow a successful execution of the corresponding program: 1. Correctness: if the input conditions are satisfied and the algorithm instructions executed, then the correct output is produced. 2. Termination: the algorithm must terminate after a finite number of steps. Thus, it has to be composed by a finite number of steps. This can be ensured if the algorithm avoids an infinite loop. 3. Performance: Quantification of the space and time complexities.
  • 12. • Algorithms that are definite and effective are also called computational procedures. • The same algorithm can be represented in several ways • Several algorithms to solve the same problem • Different ideas different speed
  • 13. The four distinct areas of studying algorithms 1. How to devise algorithms 2. How to validate algorithms 3. How to analyze algorithms 4. How to test a program
  • 14. The GCD algorithm: Given two positive integers m and n, find the greatest common divisor, gcd(m,n). A view of the problem: All numbers from 1 up to the smallest of m and n, say n. Naive algorithm: Go through search space (from 1 to n) Keep track of largest number that divides both m and n. Is there a more efficient way of doing this? Yes, the Euclidean Algorithm (Euclid – c.350 B.C.E.)
  • 15. Pseudocode of the GCD algorithm : Algorithm GreatestCommonDivisor Input: Two positive integers, m and n Ouptut: The gcd of m and n repeat r  m mod n m  n n  r until (r == 0) Output m and STOP.
  • 16. Example: m = 24, n = 9 r  24 mod 9 = 6 m  9 n  6 r  9 mod 6 = 3 m  6 n  3 r  6 mod 3 = 0 m  3 n  0 Output 3 STOP
  • 17. Euclids algorithm Step1:if n=0 return val of m & stop else proceed step 2 Step 2:Divide m by n & assign the value of remainder to r Step 3:Assign the value of n to m,r to n,Go to step1.
  • 18. Another algorithm to solve the same problem Euclids algorithm Step1:Assign the value of min(m,n) to t Step 2:Divide m by t.if remainder is 0,go to step3 else goto step4 Step 3: Divide n by t.if the remainder 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 step 2
  • 19. Thus, ri will become 0 in at most 2 log n iterations Example: m = 1989 and n = 1590 Remainder sequence: 399 = 1989 mod 1590  399 < 1989/2 393 = 1590 mod 399  393 < 1590/2 6 = 399 mod 393  6 < 399/2 < 1989/4 3 = 393 mod 6  3 < 393/2 < 1590/4 0 = 6 mod 3  0 < 6/2 < 399/4 < 1989/8 We stop here, but if we continue: < 1590/8 < 1989/16 < 1590/16
  • 20. Number of Steps: In 2 * 3 steps, ri reduced by factor of 8 2 * 4 ri 16 2 * 5 ri 32 2 * 6 ri 64 …… 2 * log n ri n Thus, ri becomes 0 in at most 2 log n iterations.
  • 21. 4-Correctness of the GCD algorithm. 4-Correctness of the GCD algorithm. • Correctness is an important issue in algorithm design. This means Correctness is an important issue in algorithm design. This means proving that the algorithm works for all legal inputs. proving that the algorithm works for all legal inputs. It is analogous to proving the correctness of a theorem in It is analogous to proving the correctness of a theorem in mathematics! mathematics! • The correctness of the GCD algorithm depends on the following loop The correctness of the GCD algorithm depends on the following loop invariant: invariant: gcd(m,n)=gcd(n,r) where r=m%n gcd(m,n)=gcd(n,r) where r=m%n
  • 22. Claim 1: gcd(m,n) = gcd(n,r) Proof: The gcd of the new pair is equal to the gcd of the previous pair. How is this correct? Let us write: m = q * n + r where 0  r  n. This implies that: a common divisor of m and n is also a common divisor of n and r, and vice versa. q.e.d.
  • 23. 5-Termination of the GCD algorithm. 5-Termination of the GCD algorithm. • Show that algorithm terminates in a finite number of steps. Show that algorithm terminates in a finite number of steps. • This must be true for every valid input. This must be true for every valid input. • Can we show this for Algorithm GCD? Can we show this for Algorithm GCD? • We must show that We must show that ri ri goes to 0 in a finite number of steps. goes to 0 in a finite number of steps. Observe: Observe: • The sequence of remainders strictly decreases. The sequence of remainders strictly decreases. • They are all non-negative. They are all non-negative. • Thus Thus ri ri will become 0 in at most 2 log will become 0 in at most 2 log n n steps. steps.
  • 24. 6-Performance of the GCD algorithm. 6-Performance of the GCD algorithm. • Quantification of performance of the algorithm. Quantification of performance of the algorithm. • Crucial parameters: time and space. Crucial parameters: time and space. • Called time and space complexity of the algorithm. Called time and space complexity of the algorithm. • Will be discussed latter in the course. Will be discussed latter in the course. For example: For example: Time complexity of GCD: Time complexity of GCD: Takes at most 2 log Takes at most 2 log n n steps, where steps, where n n < < m m Thus, worst-case time complexity: Thus, worst-case time complexity: O O(log (log n n) )
  • 25. 7-Data structures 7-Data structures • The study of different ways of organizing data. The study of different ways of organizing data. • Why? Why? • Efficiency of algorithm depends on how data is organized. Efficiency of algorithm depends on how data is organized. • Reason for studying data structures and algorithms together. Reason for studying data structures and algorithms together. • Organic connection between the two areas. Organic connection between the two areas. • In 60-212 the programming language JAVA is studied. In 60-212 the programming language JAVA is studied. • Here: An algorithmic perspective. Here: An algorithmic perspective. • Independent of programming language (eg. C, C++, JAVA). Independent of programming language (eg. C, C++, JAVA).
  • 26. For example: • The median of a list of n numbers is a number m such that: n/2 numbers in the list are  m, and n/2 … are  m. • Many definitions of median, we take: If n is even: Two medians: lower median and upper median, Then, median is average of lower and upper medians. If n is odd: Both medians (lower and upper) are the same.
  • 27. • Consider this problem: Given a sorted list of n numbers, find the median. • A crucial question: How should we store the list? • We store it in an array, A, then The median is found in constant time, O(1) !! • Median = (A[5]+A[6])/2 = (19+22)/2 = 20.5 2 5 9 16 19 22 26 27 30 31 1 2 3 4 5 6 7 8 9 10
  • 28. • Whereas in a linked list: Traverse half of the list in n/2 steps, which is O(n) ! • Quite simple stated: The way in which data is organized is crucial in complexity. More examples like this will be seen later. 2 5 9 16 19 22 …. first
  • 29. Example of problems: • Design an efficient algorithm to determine if a list has repeated elements. • Given a list of n elements find their minimum (or maximum). • Given n points in the plane, find the pair(s) of points which are closest to each other. • Given n points in the plane determine if any three are contained in a straight line.