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Topic: Asymptotic notation

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	Big O notation - Wikipedia, the free encyclopedia
		Big O notation is a mathematical notation used to describe the asymptotic behavior of functions.
		It has two main areas of application: in mathematics, it is usually used to characterize the residual term of a truncated infinite series, especially an asymptotic series, and in computer science, it is useful in the analysis of the complexity of algorithms.
		Inside asymptotic notation (and only there), it is common to use the symbols V and E, when someone really means V and E.
	en.wikipedia.org /wiki/Big_O_notation (1424 words)

	Asymptotic notation - Wikipedia, the free encyclopedia
		In mathematical analysis, and in particular in the analysis of algorithms, to classify the growth of functions one has recourse to asymptotic notations.
		The notation is an equality between the function at hand, and the known function it behaves like or is compared to in the asymptotic limit.
		These notations are widely used in algorithmics to estimate the complexity of an algorithm.
	en.wikipedia.org /wiki/Asymptotic_notation (679 words)

	hetland.org: Algorithmic Complexity (Site not responding. Last check: 2007-10-09)
		In many ways, asymptotic notation can seem very imprecise and intuitive, but it important to know that it is precisely defined; it is also crucial to understand exactly what it means.
		Using asymptotic notation also means that counting the characters of a string, the digits of an integer or the edges of a graph are all fair ways of finding the problem size, as these are all asymptotically equivalent to the number of bits.
		As we use asymptotic notation, it doesn't matter whether this function returns the number of basic operations in the processor, the number of seconds or the number of centuries required.
	www.hetland.org /algorithms/complexity.php?printable=on (3122 words)

	Computer Science:Data Structures:Asymptotic Notation - Wikibooks, collection of open-content textbooks
		So if you know you are working with only a small amount of data and you have concerns for speed and code space, a trade off can be made for a function that does not behave well for large amounts of data.
		Generally, we use asymptotic notation as an convenient way to examine what can happen in a function in the worst case or in the best case.
		This rule is general for the various asymptotic notations.
	en.wikibooks.org /wiki/Computer_Science:Data_Structures:Asymptotic_Notation (852 words)

	Asymptotic Analysis
		Asymptotic analysis is based on the idea that as the problem size grows, the complexity will eventually settle down to a simple proportionality to some known function.
		These notations are useful for expressing the complexity of an algorithm without getting lost in unnecessary detail.
		To be mathematically precise about the notation, we should point out that it's the absolute value of the functions that is being compared.
	userpages.umbc.edu /~anastasi/Courses/341/Spr00/Lectures/Asymptotic/asymptotic/asymptotic.html (1442 words)

	Recitation 16 (Site not responding. Last check: 2007-10-09)
		On Thursday you were introduced to asymptotic complexity as a notation for expressing the resource usage of programs in terms of the size of their inputs.
		Last Thursday we used this definition to show that all constant functions are in the same family O(1), all degree-1 polynomials are in the family O(n), all degree-2 polynomials are in the family O(n^2), and so on.
		We've become a little casual with big-O notation; O(n) is a class of functions, not any function in particular, so strictly we can't substitute it in an equation.
	www.cs.cornell.edu /courses/cs312/2000fa/recitation/recitation_16.html (1109 words)

	Lecture 1, Comp Sc 341s
		We will need the language and notation of asymptotic bounds on functions to state all of our resource constraints.
		We use this notation to say, for example, that a particular algorithm uses "time O(n)" to solve a problem.
		We will say more about asymptotic bounds in the next lecture, and discuss the particular bounds that we will use to define complexity classes.
	www.mtholyoke.edu /courses/barring/341/lecture/1.htm (1301 words)

	Lecture 2 - asymptotic notation
		This is where the dreaded big O notation comes in!
		Since running our algorithm on a machine which is twice as fast will effect the running times by a multiplicative constant of 2 - we are going to have to ignore constant factors anyway.
		Asymptotically, the base of the log does not matter:
	www.cs.sunysb.edu /~algorith/lectures-good/node2.html (1146 words)

	Computer Science 3711 / Engineering 5891, Winter '01: Course Diary
		Simplifications of recurrences in context of asymptotic worst-case bounds.
		Though it is in a sense an abuse of notation, you will frequently see O(1) or THETA(1) used to indicate an unspecified constant in various expressions.
		Application of sophisticated tree generation and search techniques does not change asymptotic behavior of the algorithms (still exponential in worst case) but may dramatically lower time required to solve instances in practice.
	www.cs.mun.ca /~harold/Courses/Old/CS3711.W01/Diary (4204 words)

	big-O notation
		The notation is read, "f of n is big oh of g of n".
		Strictly speaking, 3n + 4 is O(n²), too, but big-O notation is often misused to mean equal to rather than less than.
		Paul E. Black, "big-O notation", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/bigOnotation.html (462 words)

	asymptotic upper bound (Site not responding. Last check: 2007-10-09)
		Definition: An asymptotic bound, as function of the size of the input, on the worst (slowest, most amount of space used, etc.) an algorithm will do to solve a problem.
		See also big-O notation, asymptotic lower bound, asymptotically tight bound.
		Sandeep Kumar Shukla, "asymptotic upper bound", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/asymptoticUpperBound.html (107 words)

	Trees
		For large input sizes only the order of growth of the running time - the asymptotic running time - is relevant.
		When we talk about the running time of an algorithm, we use three different notations to give upper, lower or tight bounds of the asymptotic running time.
		We review the notations, definitions and elementary properties of the elements of discrete mathematics that are most relevant for the Algorithms course.
	aduni.org /courses/algorithms/courseware/handouts/Reciation_01.html (1121 words)

	CS251 First Week Notes (Site not responding. Last check: 2007-10-09)
		In order to prevent this, a method called Asymptotic Analysis was developed.
		Asymptotic Analysis counts number of instructions in an algorithm.
		Asymptotic Analysis is also called 'Big O' Notation.
	www.cs.purdue.edu /homes/cs251/LectureNotes/Spring2002/week1-2 (423 words)

	Ernest Schimmerling (Site not responding. Last check: 2007-10-09)
		Asymptotic notation is a way of making mathematically precise the notion of one function approximating another.
		Although used extensively in computer science, especially algorithms, asymptotic notation is also useful for other fields of mathematics.
		In this talk I will discuss Isabelle's type theory and our formalization of Big-O notation in it, as well as showing some examples of its use in number theory towards a proof of the PNT.
	www.math.cmu.edu /~eschimme/seminar/donnelly.html (121 words)

	Asymptotic Notation. Part II: Examples and Problems
		The widely used textbook by Brassard and Bratley [4] includes a chapter devoted to asymptotic notation (Chapter 3, pp.
		We have attempted to test how suitable the current version of Mizar is for recording this type of material in its entirety.
		The terminology and notation used in this paper have been introduced in the following articles [19] [25] [2] [21] [8] [5] [6] [20] [23] [1] [11] [9] [26] [14] [16] [17] [12] [15] [22] [10] [18] [3] [7] [24] [13]
	mizar.org /JFM/Vol11/asympt_1.html (201 words)

	[No title]
		Asymptotic Notation: * We judge data structures on the resources they use to perform the desired tasks.
		The main resources we consider are time and space (and often just time).
		Next time: We will express the running time of algorithms recursively and then bound them with asymptotic notation.
	www.cs.cornell.edu /Courses/cs410/1998su/Lectures/lect1.txt (498 words)

	Computer Science 3711 / Engineering 5891, Winter '02: Course Diary
		How you interpret asymptotic notation depends on how the function T(n) was derived.
		Such tricks can drastically reduce actual running time for CST traversal; however, all of these algorithms are still asymptotically exponential time in the worst case (you still might have to visit all the nodes in the tree).
		Asymptotic behavior may only matter at absurdly large input sizes.
	www.cs.mun.ca /~harold/Courses/Old/CS3711.W02/Diary (10818 words)

	[No title]
		Logarithms A. Recall that y = log2 n means that 2**y = n, so log2 n is the power to which 2 can be raised to get n 1.
		log2 (2n) = n Notation**: log n == log2 n loge n == ln n 3.
		B. We need to determine the asymptotic behavior of logarithms.
	www.ship.edu /~jcthom/schedule/csc210/Notes/AsymNot.html (1541 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Comparing the asymptotic running time -an algorithm that runs in O(n) time is better than one that runs in O(n2) time -similarly, O(log n) is better than O(n) -hierarchy of functions: -log n
		Finding the average case can be very difficult, so typically algorithms are measured by the worst-case time complexity.
		Comparing the asymptotic running time -an algorithm that runs in O(n) time is better than one that runs in O(n2) time -similarly, O(log n) is better than O(n) -hierarchy of functions: log n
	www.cse.iitd.ernet.in /~amitk/cs201/Analysis.ppt (549 words)

	Computer Science:Data Structures:Arrays - Wikibooks, collection of open-content textbooks
		An array is a particular method of storing elements of indexed data.
		In this way, we can go through the indexes from 0 to 30 and get the related tasks for each day in
		This page was last modified 05:10, 15 February 2005.
	en.wikibooks.org /wiki/Programming:Data_Structures:Arrays (1172 words)

	Geneseo CSci 141 Asymptotic Notation (Site not responding. Last check: 2007-10-09)
		Why does it not really matter what you count as operations?
		As long as you count something proportional to the total number of constant-time chunks of work that an algorithm does, you'll come up with asymptotically the same answer any other such count would have.
		What can you say about the best-case execution time of the list search algorithm?
	www.cs.geneseo.edu /~baldwin/csci141/Spring2002/theta.html (223 words)

	CS332 - Analysis of Algorithms (Site not responding. Last check: 2007-10-09)
		A page of lecture notes (with audio clips) on asymptotic notation from Suny Stony Brook Univerity.
		A cool part of this page uses asymptotic notation to analyze the brain space needed to remember songs!
		This gives a little insight into why "100 Bottles of Beer" is so easy to remember!
	www.dickinson.edu /~braught/courses/cs332f03/classes/class04.html (126 words)

	[No title]
		Asymptotic notation (courtesy of Steven Skiena, developed for “The Algorithm Design Manual”)
		Because it is usually very hard to compute the average running time, since we must somehow average over all the instances, we usually strive to analyze the worst case running time.
		We know nothing about the lower bound of this!
	www.cs.unca.edu /~bruce/Fall04/csci333/asymptoticNotation.html (327 words)

	University of Calgary - Department of Computer Science - CPSC 413 Schedule
		Lecture #2: Asymptotic Notation and Growth of Functions
		Quiz #1: Asymptotic Notation and Growth of Functions
		Lecture #9: Divide and Conquer Algorithms, Part 1
	pages.cpsc.ucalgary.ca /~eberly/Courses/CPSC413/schedule.html (53 words)

	CPSC 413: Material Organized by Topic
		This is based on material presented in Chapter 2 of the textbook.
		A Proof of the Same Asymptotic Relationship, Using Constructive Induction
		Lecture Notes from 1998 - Not Required Reading!
	pages.cpsc.ucalgary.ca /~eberly/Courses/CPSC413/topics.html (497 words)

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