The document discusses asymptotic analysis using limits. It defines big-O, Omega, Theta, little-o and little-omega notations using limits. Examples are provided to analyze the asymptotic behavior of functions using these notations. Summations, including arithmetic and geometric summations, are also analyzed asymptotically. For arithmetic summations, it is shown that the sum of first n terms is Theta of n^2. For geometric summations, the asymptotic behavior depends on the ratio r being greater than, equal to or less than 1, and limits are used to determine the asymptotic notation in each case.
Growth of Functions
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 6, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Growth of Functions
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 6, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
3. Asymptotic Analysis
Using Limits
Use of basic definition for determining the asymptotic behavior is often awkward. It
involves ad hoc approach or some kind of manipulation to prove algebraic relations.
Calculus provides an alternative method for the analysis. It depends on evaluating
the following limit.
f(n)
lim = c
n→∞ g(n)
where f(n) is a growth function for an algorithm and g(n) is a standard function
Depending upon the value c , the relation between f(n) and g(n) can be expressed in terms
of asymptotic notations. In most cases it is easier to use limits, compared to basic method, to
determine asymptotic behavior of growth functions.
It will be seen that the Calculus notation n→ ∞ is equivalent to the algebraic condition
for all n ≥ n0 . Either of these conditions implies large input
4. O-Notation Using Limit
Definition
If f(n) is running time of an algorithm and g(n) is some standard growth function such
that
f(n)
lim —— = c , where c is a positive constant such that 0 ≤ c < ∞
n → ∞ g(n)
then f(n) = O ( g(n) )
Note that infinity is excluded from the range of permissible values for the constant c
6. O-Notation
Examples
Example(3): lg n = O(n)
lim
n → ∞
lg n
n
= ∞ / ∞ ( Need to apply the L’ Hopital Rule )
In order to compute differential of lg n we first convert binary logarithm to natural logarithm.
Converting lg n (binary log ) to ln( n) ( natural log) , by the using formula lg n = ln n / ln 2
lim
lg n
= (ln n) (Converting to natural log)
n → ∞ n (ln 2) n
lim
n → ∞
1
ln 2. n
( Differentiating numerator and denominator)
= 0 ( Evaluating limits)
Therefore, lg n = O(n)
7. O-Notation
Examples
Example(4): n2 = O(2n)
lim
n → ∞
n2
=∞ / ∞ ( Need to apply the L’ Hopital Rule )
2n
Since d ( 2n ) = ln 2 . 2n, ( Calculus rule for differentiating the exponential functions)
dn
lim 2n (Differentiating numerator and denominator )
n → ∞ ln 2 2n
= ∞ / ∞ (Need again to apply the L’Hopital rule)
lim
n → ∞
2
(ln2)2. 2n
(Again differentiating numerator and denominator )
= 0 (Evaluating the limits )
Therefore, n2 = O(2n)
8. Ω-Notation Using Limit
Definition
If f(n) is running time of an algorithm and g(n) is some standard growth function such
that
f(n)
lim —— = c , where c is a constant, such that 0 < c ≤ ∞
n → ∞ g(n)
then f(n) = Ω( g(n) ).
Note that zero value for the constant c is excluded from the permissible range, but
infinity is included.
10. Ө-Notation Using Limit
Definition
If f(n) is running time of an algorithm and g(n) is some standard growth function such
that
f(n)
lim —— = c , where c is a constant, such that 0 < c < ∞
n → ∞ g(n)
then f(n) = Ө ( g(n) ).
Note that zero value and infinity are excluded from the permissible values for the
range of constant c
11. Θ-Notation
Examples
Example(1): 45n3 - 3n2 - 5n+ 20 = θ(n3)
lim
n → ∞
45n3 -3n2 - 5n + 20
n3
= 45-3 / n -5 / n2 +20/n3
= 45 - 0 – 0 + 0
=45 (non-zero constant)
Therefore, 45n3 - 3n2 - 5n+ 20 = θ(n3)
Example(2): n lg n + n + n2 = θ(n2)
lim
n → ∞
n lg n + n + n2
n2
= lg n / n +1/n +1
= 0 +0 + 1
=1 (non-zero constant)
Thus, n lg n + n + n2 = θ(n2)
12. Θ-Notation
Examples
Example(3): 45n3 - 3n2 - 5n+ 20 ≠ θ(n4)
lim
n → ∞
45n3 -3n2 - 5n + 20
n4
= 45 / n - 3 / n2 - 5 / n3 + 20 / n4
= 0 - 0 - 0 + 0
= 0 (zero is excluded from the permissible range for Ө-notation)
Therefore, 45n3 - 3n2 - 5n+ 20 ≠ θ(n3)
Example(4): n lg n + n ≠ θ(n2)
lim
n → ∞
n lg n + n
n2
= lg n / n + 1/n ( Using the L’Hopital Rule to evaluate the limit of lgn /n)
= 0 +0
=0
Thus, n lg n + n ≠ θ(n2)
13. o-Notation Using Limit
Definition
If f(n) is running time and g(n) is some standard growth function such that
f(n)
lim —— = 0 ( zero )
n → ∞ g(n)
then f(n) = o( g(n) ) (Read f (n) is small-oh of g(n))
o( g(n) ) is referred to as the loose upper bound for f(n)
15. o-Notation
Examples
Example(3): lg n = o (n)
lim lg n
=
∞ ( Need to use L’Hopital Rule)
n → ∞ n ∞
lim ln n
( Converting to natural log ,lg n = log2 n =log e n / loge 2 )
n → ∞ n ln 2
lim 1
( Differentiating numerator and denominator )
n → ∞ n ln2
= 0
Therefore, lg n = o (n)
16. ω-Notation
Definition
If f(n) is running time and g(n) is some standard growth function such that
f(n)
lim —— = ∞
n → ∞ g(n)
then f(n) = ω(g(n)) (Read f(n) is small-omega of g(n))
ω(g(n)) is referred to as the loose lower bound for f(n)
18. ω-Notation
Examples
Example(3): n! =ω(2n)
lim
n → ∞
n!
= ∞ / ∞ (Need to use the LHpoital Rule
2n
The function n! cannot be differentiated directly. We first use Stirling’s approximation:
lim
n! = √ 2πn ( n /e ) n for large n
n!
n → ∞
=
2n
√ 2πn ( n / e ) n
2n
= √ 2πn ( n /2 e ) n
= ∞
Thus, n! = ω( 2n )
19. Asymptotic Notation
Summary
Let f(n) be time complexity and g(n) standard function, such that lim
n→∞
f(n)
——— = α
g(n)
Table below summarizes the asymptotic behavior of f(n) in terms of g(n)
Notation Using Basic Definition Using Limits Asymptotic Bound
f(n)=O( g(n) ) f(n) ≤ c.g(n) for some c>0, and n ≥ n0 0 < α < ∞ upper tight
f(n)=o( g(n) ) f(n) <c.g(n) for all c>0, and n ≥ n0 α = 0 upper loose
f(n)=Ω( g(n) ) f(n) ≥ c.g(n) for some c>0 and n ≥ n0 0 < α ≤ ∞ lower tight
f(n)=ω( g(n) ) f(n) > c.g(n) for all c>0 and n ≥ n0 α = ∞ lower loose
f(n)=θ( g(n) ) c1.g ≤ f(n) ≤ c2.g(n) for some c1>0, c2>0
and n ≥ n0
0 < α < ∞ tight
20. Asymptotic Set Notations
Definitions
The asymptotic notation can also be expressed in Set notations by using the limits
f(n)
Let f(n) be time complexity and g(n) standard function, such that lim ——— = c
where c is zero, positive constant or infinity.
n→∞ g(n)
The set notations for the asymptotic behavior is are defined as follows
(i) O(g(n) ) = { f(n): 0 ≤ c < ∞ }
(ii) Ω( g(n) ) = { f(n) : 0 < c ≤ ∞ }
(iii) Ө( g(n) ) = { f(n): 0 < c < ∞ }
(iv) o( g(n) ) = { f(n): c = 0 }
(v) ω( g(n) ) = { f(n): c = ∞ }
21. Asymptotic Set Notations
Examples
Example (1): Let f(n)O ( g(n) ), f(n) Ω( f(n) ) . Then
O(g(n) ) = { f(n): 0 ≤ c < ∞ } (definition of O-notation)
Ω( g(n) ) = { f(n) : 0 < c ≤ ∞ } (definition of Ω-notation )
O(g(n) ) ∩ Ω ( g(n) ) = { f(n) : 0 < c < ∞ } ( Performing Set intersection operation )
= Ө (g(n)) (definition of Ө-notation }
Thus, O(g(n) ) ∩ Ω ( g(n) ) = Ө (g(n))
Example (2):Suppose f(n) o(g(n)), and f(n)ω(g(n)). Then
o(g(n)) = { f(n): c = 0 } (definition of o-notation)
Ω(g(n))= { f(n): c = ∞ } (definition of ω-notation)
Therefore, o(g(n)) ∩ ω( g(n)) =φ ( Empty set )
23. Arithmetic Summation
Asymptotic Behavior
The sum of first of n terms of arithmetic series is :
1 + 2 + 3………..+n = n(n+1)/2
Let f(n)= n(n+1)/2
and g(n)= n2
f(n)
lim
n(n+1)/2 n2 /2 + n/2
= = = 1 /2 + 1/2n =1/2 + 0=1/2
n→∞ g(n) n2 n2
Since the limit is non-zero and finite it follows
f(n) = θ( g(n)) = θ(n2)
Or, 1 + 2 + ………..+ n = θ(n2)
24. Geometric Summation
Asymptotic Behavior
The asymptotic behavior of geometric series
1 +r + r2+……..+rn
depends on geometric ratio r. Three cases need to be considered
Case r > 1: It can be shown that sum f(n) of first n terms is as follows
rn+1 - 1
f(n) = 1 +r + r2+……..+rn = —————
r - 1
Case r = 1: This is trivial
f(n)= 1 + 1 + 1+ ……+1 = n =Ө(n)
Case r < 1: It can be shown that
2 n
1 - rn+1
f(n) = 1 +r + r +……..+r = —————
1 - r
The asymptotic behavior in first case and third case is explored by computing limits.
25. Geometric Summation
Case r > 1
2 n
rn+1 - 1
Let f(n) = 1 +r + r
Let g(n)= rn
+……..+r = —————
r - 1
Consider, the limit
lim
n→∞
f(n)
g(n)
rn+1 - 1
=
(r - 1).rn
r - 1/rn
=
(r - 1)
Since r>1 , 1/rn →0 as n→ ∞
lim
n→∞
f(n)
g(n)
=
r
(r - 1)
>0, since r > 1
Therefore, f(n) = θ(rn) for r>1
Or, 1 +r + r2+……..+rn = θ(rn) for r > 1
26. Geometric Summation
Case r < 1
2 n
rn+1 - 1
Consider 1 +r + r +……..+r = —————
r - 1
Let g(n)=c where c is some positive constant
Taking the limit
lim
f(n) rn+1 - 1
=
1- rn+1
=
n→∞ g(n) (r - 1).c (1- r )c
Since r<1 , rn+1 →0 as n→ ∞
lim
n→∞
f(n)
g(n)
=
1
(1 - r). c
>0, since r < 1
Therefore, f(n)= θ(g(n))=θ(c) = θ(1) for r <1
Or, 1 +r + r2+……..+rn
=θ(1) for r <1
27. Geometric Summation
Asymptotic Behavior
From the preceding analysis it follows that the asymptotic behavior of geometric summation is
S(n) =1+r+r2+.........+ rn =
Ө(rn) when r>1
Ө(1) when r<1
When r <1, the largest term in the geometric summation would be 1 ( the first term) , and
S(n) =Ө(1) . On the other hand if r>1, the largest term would be rn ( the last term), and
S(n)=Ө(rn) In the light of this observation it can said that asymptotic behavior of geometric
summation is determined by the largest term i.e S(n)=Ө (largest term).
Example(1): The geometric series
1+21+ 22 +….+ 2n
has geometric ratio r=2>1 ,
Thus, 1+21+ 22 +….+ 2n =Ө(2n) , ► Ө(largest term)
Example(2): The geometric series
1+(2/3)1+ (2/3)2 +….+ (2/3)n
has geometric ratio r=2/3 <1 ,
Thus, 1+(2/3)1+ (2/3)2 +….+ (2/3)n =Ө(1) , ►Ө(largest term)
28. Logarithm Summation
Asymptotic Behavior
The logarithmic series has the summation
lg(1)+lg(2)+ lg(3)+….. +lg(n)
Let f(n) = lg(2)+ lg(3)+….+.lg(n) = lg(2.3……..n) = lg( n!)
and g(n) = n lg n
lim
n→∞
f(n)
g(n) =
lg n!
n lg n =
lg ( √(2πn)(n/e)n )
n lg n
(Using Stirling’s approximation)
Now, lg ( √(2πn)(n/e)n ) = (1+lg π + lg n)/2 + n lg n - n lg e , therefore
lim lg ( √(2πn)(n/e)n ) = ( 1+lg π + lg n ) /(2 nlg n) + 1 - lg e / lg n = (0+ 1- 0)=1
n→∞ n lg n
Since limit is non-zero and finite, it follows
f(n)= θ(g(n))
Or, lg(1)+lg(2)+ lg(3)+….. +lg(n) = θ(n lg n) …
29. Harmonic Summation
Asymptotic Behavior
The sum of first n terms of Harmonic series is
1+ 1/2 + 1/3+…..+1/n
Let f(n) = 1+ 1/2 + 1/3+…..+1/n
and g(n) = lg n
lim
n→∞
f(n)
g(n)
1+1/2+1/3+ …..+1/n
=
lg n
It can be shown that 1+ 1/2+ 1/3+…+1/n = lg (n) + γ + 1/2n – 1/12n2+..where γ ≈ 0.5772
lim
n→∞
f(n)
g(n)
lg n + γ + 1/2n – 1/12n2 +…
=
lg n
= 1+0+0-0+0+…=1
Since limit is finite and non-zero, it follows
f(n) = θ( g(n))
Or, 1+ 1/2 + 1/3+…..+1/n = θ (lg n )