PPT on Analysis Of Algorithms.
The ppt includes Algorithms,notations,analysis,analysis of algorithms,theta notation, big oh notation, omega notation, notation graphs
PPT on Analysis Of Algorithms.
The ppt includes Algorithms,notations,analysis,analysis of algorithms,theta notation, big oh notation, omega notation, notation graphs
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
Describing basic networking concepts, topologies, the OSI model, and the media used to physically connect a network, for those interested in learning the fundamentals of computer networks.
Complete study notes.
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
Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...Sean Meyn
Many machine learning and optimization algorithms solve hidden root-finding problems through the magic of stochastic approximation (SA). Unfortunately, these algorithms are slow to converge: the optimal convergence rate for the mean squared error (MSE) is of order O(n⁻¹) at iteration n.
Far faster convergence rates are possible by reconsidering the design of exploration signals used in these algorithms. In this lecture the focus is on quasi-stochastic approximation (QSA), in which a multi-dimensional clock process defines exploration. It is found that algorithms can be designed to achieve a MSE convergence rate approaching O(n⁻⁴).
Although the framework is entirely deterministic, this new theory leans heavily on concepts from the theory of Markov processes. Most critical is Poisson’s equation to transform the QSA equations into a mean flow with additive “noise” with attractive properties. Existence of solutions to Poisson’s equation is based on Baker’s Theorem from number theory---to the best of our knowledge, this is the first time this theorem has been applied to any topic in engineering!
The theory is illustrated with applications to gradient free optimization.
Joint research with Caio Lauand, current graduate student at UF.
References
[1] C. Kalil Lauand and S. Meyn. Approaching quartic convergence rates for quasi-stochastic approximation with application to gradient-free optimization. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems, volume 35, pages 15743–15756. Curran Associates, Inc., 2022.
[2] C. K. Lauand and S. Meyn. Quasi-stochastic approximation: Design principles with applications to extremum seeking control. IEEE Control Systems Magazine, 43(5):111–136, Oct 2023.
[3] C. K. Lauand and S. Meyn. The curse of memory in stochastic approximation. In Proc. IEEE Conference on Decision and Control, pages 7803–7809, 2023. Extended version. arXiv 2309.02944, 2023.
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The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
1. Today’s Outline
• Admin: Assignment #1 due next thurs. at
11:59pm
• Asymptotic analysis
Asymptotic Analysis
CSE 373
Data Structures & Algorithms
Ruth Anderson
Spring 2007
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Linear Search vs Binary Search
Linear Search
Binary Search
Best Case
Asymptotic Analysis
Worst Case
So … which algorithm is better?
What tradeoffs can you make?
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Fast Computer vs. Slow Computer
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5
4
Fast Computer vs. Smart Programmer
(round 1)
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1
2. Fast Computer vs. Smart Programmer
(round 2)
Asymptotic Analysis
• Asymptotic analysis looks at the order of the
running time of the algorithm
– A valuable tool when the input gets “large”
– Ignores the effects of different machines or different
implementations of the same algorithm
• Intuitively, to find the asymptotic runtime, throw
away the constants and low-order terms
– Linear search is T(n) = 3n + 2 ∈ O(n)
– Binary search is T(n) = 4 log2n + 4 ∈ O(log n)
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Remember: the fastest algorithm has the
slowest growing function for its runtime
8
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Order Notation: Intuition
Asymptotic Analysis
• Eliminate low order terms
– 4n + 5 ⇒
– 0.5 n log n + 2n + 7 ⇒
– n3 + 2n + 3n ⇒
f(n) = n3 + 2n2
g(n) = 100n2 + 1000
• Eliminate coefficients
– 4n ⇒
– 0.5 n log n ⇒
– n log n2 =>
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Although not yet apparent, as n gets “sufficiently
large”, f(n) will be “greater than or equal to” g(n)10
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Order Notation: Definition
Definition of Order Notation
•
Upper bound: T(n) = O(f(n))
O( f(n) ) : a set or class of functions
Big-O
Exist constants c and n’ such that
T(n) ≤ c f(n) for all n ≥ n’
•
Lower bound: T(n) = Ω(g(n))
g(n) ∈ O( f(n) ) iff there exist consts c and n0 such that:
g(n) ≤ c f(n) for all n ≥ n0
Omega
Exist constants c and n’ such that
T(n) ≥ c g(n) for all n ≥ n’
•
Tight bound: T(n) = θ(f(n))
Example: g(n) =1000n vs. f(n) = n2
Is g(n) ∈ O( f(n) ) ?
Pick: n0 = 1000, c = 1
Theta
When both hold:
T(n) = O(f(n))
T(n) = Ω(f(n))
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2
3. Order Notation: Example
Notation Notes
Note: Sometimes, you’ll see the notation:
g(n) = O(f(n)).
This is equivalent to:
g(n) ∈ O(f(n)).
However: The notation
O(f(n)) = g(n)
is meaningless!
(in other words big-O is not symmetric)
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100n2 + 1000 ≤ 5 (n3 + 2n2) for all n ≥ 19
So f(n) ∈ O( g(n) )
Big-O: Common Names
–
–
–
–
–
–
–
–
constant:
logarithmic:
linear:
log-linear:
quadratic:
cubic:
polynomial:
exponential:
O(1)
O(log n)
O(n)
O(n log n)
O(n2)
O(n3)
O(nk)
O(cn)
14
Meet the Family
(logkn, log n2 ∈ O(log n))
• O( f(n) ) is the set of all functions asymptotically
less than or equal to f(n)
– o( f(n) ) is the set of all functions asymptotically
strictly less than f(n)
• Ω( f(n) ) is the set of all functions asymptotically
greater than or equal to f(n)
– ω( f(n) ) is the set of all functions asymptotically
strictly greater than f(n)
(k is a constant)
(c is a constant > 1)
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• θ( f(n) ) is the set of all functions asymptotically
equal to f(n)
15
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Big-Omega et al. Intuitively
Meet the Family, Formally
• g(n) ∈ O( f(n) ) iff
There exist c and n0 such that g(n) ≤ c f(n) for all n ≥ n0
– g(n) ∈ o( f(n) ) iff
There exists a n0 such that g(n) < c f(n) for all c and n ≥ n0
Equivalent to: limn→∞ g(n)/f(n) = 0
Asymptotic Notation
O
Ω
• g(n) ∈ Ω( f(n) ) iff
There exist c and n0 such that g(n) ≥ c f(n) for all n ≥ n0
θ
o
– g(n) ∈ ω( f(n) ) iff
There exists a n0 such that g(n) > c f(n) for all c and n ≥ n0
Equivalent to: limn→∞ g(n)/f(n) = ∞
ω
Mathematics Relation
≤
≥
=
<
>
• g(n) ∈ θ( f(n) ) iff
g(n) ∈ O( f(n) ) and g(n) ∈ Ω( f(n) )
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3
4. Pros and Cons of Asymptotic
Analysis
Types of Analysis
Two orthogonal axes:
– bound flavor
• upper bound (O, o)
• lower bound (Ω, ω)
• asymptotically tight (θ)
– analysis case
•
•
•
•
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worst case (adversary)
average case
best case
“amortized”
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Algorithm Analysis Examples
20
Analyzing the Loop
• Consider the following
program segment:
• Total number of times x is incremented is
executed =
x:= 0;
for i = 1 to N do
for j = 1 to i do
x := x + 1;
N
1+ 2 + 3 + ... = ∑ i =
• What is the value of x at
the end?
i =1
N(N + 1)
2
• Congratulations - You’ve just analyzed your first
program!
– Running time of the program is proportional to
N(N+1)/2 for all N
– Big-O ??
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Which Function Grows Faster?
Which Function Grows Faster?
n3 + 2n2 vs.
n3 + 2n2
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100n2 + 1000
23
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vs. 100n2 + 1000
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4
