DESIGN AND ANALYSIS OF ALGORITHMS

Apr 3, 2014Download as ppt, pdf18 likes32,536 views

This document discusses algorithms and their analysis. It defines an algorithm as a step-by-step procedure to solve a problem or calculate a quantity. Algorithm analysis involves evaluating memory usage and time complexity. Asymptotics, such as Big-O notation, are used to formalize the growth rates of algorithms. Common sorting algorithms like insertion sort and quicksort are analyzed using recurrence relations to determine their time complexities as O(n^2) and O(nlogn), respectively.

CSE 5311
DESIGN AND
ANALYSIS OF
ALGORITHMS

Definitions of Algorithm
 A mathematical relation between an observed quantity
and a variable used in a step-by-step mathematical
process to calculate a quantity
 Algorithm is any well defined computational procedure
that takes some value or set of values as input and
produces some value or set of values as output
 A procedure for solving a mathematical problem in a
finite number of steps that frequently involves repetition
of an operation; broadly : a step-by-step procedure for
solving a problem or accomplishing some end
(Webster‟s Dictionary)

Analysis of Algorithms
Involves evaluating the
following parameters
 Memory – Unit generalized as
“WORDS”
 Computer time – Unit generalized as
“CYCLES”
 Correctness – Producing the desired
output

Sample Algorithm
FINDING LARGEST NUMBER
INPUT: unsorted array „A[n]‟of n numbers
OUTPUT: largest number
----------------------------------------------------------
1 large ← A[j]
2 for j ← 2 to length[A]
3 if large < A[j]
4 large ← A[j]
5 end

Space and Time Analysis
(Largest Number Scan Algorithm)
SPACE S(n): One “word” is required to run the
algorithm (step 1…to store variable „large‟)
TIME T(n): n-1 comparisons are required to find
the largest (every comparison takes one cycle)
*Aim is to reduce both T(n) and S(n)

ASYMPTOTICS
Used to formalize that an algorithm has
running time or storage requirements that
are ``never more than,'' ``always greater
than,'' or ``exactly'' some amount

ASYMPTOTICS NOTATIONS
O-notation (Big Oh)
 Asymptotic Upper Bound
 For a given function g(n), we denote O(g(n)) as the set of
functions:
O(g(n)) = { f(n)| there exists positive
constants c and n0 such that
0 ≤ f(n) ≤ c g(n) for all n ≥ n0 }

ASYMPTOTICS NOTATIONS
Θ-notation
 Asymptotic tight bound
 Θ (g(n)) represents a set of functions such
that:
Θ (g(n)) = {f(n): there exist positive
constants c1, c2, and n0 such
that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n)
for all n≥ n0}

ASYMPTOTICS NOTATIONS
Ω-notation
 Asymptotic lower bound
 Ω (g(n)) represents a set of functions such
that:
Ω(g(n)) = {f(n): there exist positive
constants c and n0 such that
0 ≤ c g(n) ≤ f(n) for all n≥ n0}

 O-notation ------------------
 Θ-notation ------------------
 Ω-notation ------------------
Less than equal to (“≤”)
Equal to (“=“)
Greater than equal to
(“≥”)

Cntd…
 c1 , c2 & n0 -> constants
 T(n) exists between c1n & c2n
 Below n0 we do not plot T(n)
 T(n) becomes significant only above n0

Common plots of O( )
O(2n)
O(n3 )
O(n2)
O(nlogn)
O(n)
O(√n)
O(logn)
O(1)

Examples of algorithms for sorting
techniques and their complexities
 Insertion sort : O(n2)
 Selection sort : O(n2)
 Quick sort : O(n logn)
 Merge sort : O(n logn)

RECURRENCE RELATIONS
 A Recurrence is an equation or inequality
that describes a function in terms of its
value on smaller inputs
 Special techniques are required to analyze
the space and time required

RECURRENCE RELATIONS
EXAMPLE
EXAMPLE 1: QUICK SORT
T(n)= 2T(n/2) + O(n)
T(1)= O(1)
 In the above case the presence of function of T on both
sides of the equation signifies the presence of
recurrence relation
 (SUBSTITUTION MEATHOD used) The equations are
simplified to produce the final result:
……cntd

T(n) = 2T(n/2) + O(n)
= 2(2(n/22) + (n/2)) + n
= 22 T(n/22) + n + n
= 22 (T(n/23)+ (n/22)) + n + n
= 23 T(n/23) + n + n + n
= n log n
Cntd….

Cntd…
EXAMPLE 2: BINARY SEARCH
T(n)=O(1) + T(n/2)
T(1)=1
Above is another example of recurrence relation and the way to solve it
is by Substitution.
T(n)=T(n/2) +1
= T(n/22)+1+1
= T(n/23)+1+1+1
= logn
T(n)= O(logn)

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Computer Security Fundamentals Chapter 1remoteaimms

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DESIGN AND ANALYSIS OF ALGORITHMS

1. CSE 5311 DESIGN AND ANALYSIS OF ALGORITHMS

2. Definitions of Algorithm  A mathematical relation between an observed quantity and a variable used in a step-by-step mathematical process to calculate a quantity  Algorithm is any well defined computational procedure that takes some value or set of values as input and produces some value or set of values as output  A procedure for solving a mathematical problem in a finite number of steps that frequently involves repetition of an operation; broadly : a step-by-step procedure for solving a problem or accomplishing some end (Webster‟s Dictionary)

3. Analysis of Algorithms Involves evaluating the following parameters  Memory – Unit generalized as “WORDS”  Computer time – Unit generalized as “CYCLES”  Correctness – Producing the desired output

4. Sample Algorithm FINDING LARGEST NUMBER INPUT: unsorted array „A[n]‟of n numbers OUTPUT: largest number ---------------------------------------------------------- 1 large ← A[j] 2 for j ← 2 to length[A] 3 if large < A[j] 4 large ← A[j] 5 end

5. Space and Time Analysis (Largest Number Scan Algorithm) SPACE S(n): One “word” is required to run the algorithm (step 1…to store variable „large‟) TIME T(n): n-1 comparisons are required to find the largest (every comparison takes one cycle) *Aim is to reduce both T(n) and S(n)

6. ASYMPTOTICS Used to formalize that an algorithm has running time or storage requirements that are ``never more than,'' ``always greater than,'' or ``exactly'' some amount

7. ASYMPTOTICS NOTATIONS O-notation (Big Oh)  Asymptotic Upper Bound  For a given function g(n), we denote O(g(n)) as the set of functions: O(g(n)) = { f(n)| there exists positive constants c and n0 such that 0 ≤ f(n) ≤ c g(n) for all n ≥ n0 }

8. ASYMPTOTICS NOTATIONS Θ-notation  Asymptotic tight bound  Θ (g(n)) represents a set of functions such that: Θ (g(n)) = {f(n): there exist positive constants c1, c2, and n0 such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n≥ n0}

9. ASYMPTOTICS NOTATIONS Ω-notation  Asymptotic lower bound  Ω (g(n)) represents a set of functions such that: Ω(g(n)) = {f(n): there exist positive constants c and n0 such that 0 ≤ c g(n) ≤ f(n) for all n≥ n0}

10.  O-notation ------------------  Θ-notation ------------------  Ω-notation ------------------ Less than equal to (“≤”) Equal to (“=“) Greater than equal to (“≥”)

11. Mappings for n2 Ω (n2 ) O(n2 ) Θ(n2)

12. Bounds of a Function Cntd…

13. Cntd…  c1 , c2 & n0 -> constants  T(n) exists between c1n & c2n  Below n0 we do not plot T(n)  T(n) becomes significant only above n0

14. Common plots of O( ) O(2n) O(n3 ) O(n2) O(nlogn) O(n) O(√n) O(logn) O(1)

15. Examples of algorithms for sorting techniques and their complexities  Insertion sort : O(n2)  Selection sort : O(n2)  Quick sort : O(n logn)  Merge sort : O(n logn)

16. RECURRENCE RELATIONS  A Recurrence is an equation or inequality that describes a function in terms of its value on smaller inputs  Special techniques are required to analyze the space and time required

17. RECURRENCE RELATIONS EXAMPLE EXAMPLE 1: QUICK SORT T(n)= 2T(n/2) + O(n) T(1)= O(1)  In the above case the presence of function of T on both sides of the equation signifies the presence of recurrence relation  (SUBSTITUTION MEATHOD used) The equations are simplified to produce the final result: ……cntd

18. T(n) = 2T(n/2) + O(n) = 2(2(n/22) + (n/2)) + n = 22 T(n/22) + n + n = 22 (T(n/23)+ (n/22)) + n + n = 23 T(n/23) + n + n + n = n log n Cntd….

19. Cntd… EXAMPLE 2: BINARY SEARCH T(n)=O(1) + T(n/2) T(1)=1 Above is another example of recurrence relation and the way to solve it is by Substitution. T(n)=T(n/2) +1 = T(n/22)+1+1 = T(n/23)+1+1+1 = logn T(n)= O(logn)

DESIGN AND ANALYSIS OF ALGORITHMS

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