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Big O 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 (1704 words)

	Big O notation
		Big O notation (with a capital letter O, not a zero), also called Landau's symbol, is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functions.
		Obviously, this notation is abusing the equality symbol, since it violates the axiom of equality: "things equal to the same thing are equal to each other".
		In addition to the big O notations, another Landau symbol is used in mathematics: the little o.
	www.ebroadcast.com.au /lookup/encyclopedia/la/Landau_symbol.html (843 words)

	Talk:Big O notation - Wikipedia, the free encyclopedia
		I see no reason to change the title from "Big O notation", since this is what Google likes the most, perfectly correct and does not require us to fix any double redirects (the way I know Taku's moves, he probably won't do it himself).
		The page should only be moved to asymptotic notation (note singular) if it actually covers other types of this notation, not in the expectation that it will at some point.
		There is already an article at Asymptotic notation that duplicates much of this information as well, and a shorter one at asymptotic analysis that treats the basic concept without reference to the O-notation.
	en.wikipedia.org /wiki/Talk:Big_O_notation (3225 words)

	Zoom Astronomy Glossary: S
		Scientific notation is a mathematical format used to write very large and very small numbers; this system avoids using a lot of zeros.
		In scientific notation, there is a base number (a number between 1 and 10) multiplied by a power of ten.
		O and B stars are uncommon but very bright; M stars are common but dim.
	www.enchantedlearning.com /subjects/astronomy/glossary/indexs.shtml (8147 words)

	Cosmic Journeys
		An X-ray binary is a special case where one of the stars is a collapsed object such as a white dwarf, neutron star, or fl hole, and the separation between the stars is small enough so that matter is transferred from the normal star to the compact star star, producing X-rays in the process.
		The classes are: O, B, A, F, G, K, and M; O stars are the hottest; M the coolest.
		O and B stars are rare but very bright; M stars are numerous but dim.
	perry.sonoma.edu /journeys/glossary/Default.html (8934 words)

	[texhax] "big O" and "little O" notation in amsmath
		This big-O notation was introduced by P. Brachmann in 1894.
		This intuitively means that the error in using g(x) to approximate f(x) is negligible in comparison to h(x).
		The little-oh notation was first used by E. Landau in 1909.
	www.tug.org /mail-archives/texhax/2004-January/001491.html (501 words)

	What is "order of magnitude"?
		And there is a somewhat unrelated, "big-O" "little-o" notation for describing how fast a function's output grows as its input grows, but I won't go into that.
		In scientific notation, the "middle" of the range, multiplying-wize, is 3.16 (the square root of 10).
		It is written in scientific notation with 10^3, but 10^4 is sooo much closer.
	www.vendian.org /envelope/TemporaryURL/what_is_oom.html (672 words)

	Chess Move Record
		In order to record and communicate chess moves, several written notations are widely used in books and by the players themselves.
		With a little practice you should have no trouble identifying any square of a chess board from White or Black's side of the board.
		The formal notation of cxd5 means the pawn on c4 captures the piece on d5.
	www.logicalchess.com /info/reference/notation (408 words)

	Group theory
		He introduces the notation of powers, positive and negative, of permutations (with the power 0 giving the identity permutation), defines the order of a permutation, introduces cycle notation and used the term système des substitutions conjuguées for a group.
		Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.
		Cayley's papers of 1854 were so far ahead of their time that they had little impact.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Development_group_theory.html (1464 words)

	Common Errors in College Math
		notation still needs to be discussed, as it is used on nearly all handheld calculators.
		A formula or notation may work properly in one context, but some students try to apply it in a wider context, where it may not work properly at all.
		The letters x and k are used in different ways here, a little like the difference between bound and free variables in logic: Fix any constant k; then the equation states a relationship between two functions of the variable x.
	www.math.vanderbilt.edu /~schectex/commerrs (13320 words)

	More Notation-Theta and Little Oh
		Definition (Little Oh) Consider a function f(n) which is non-negative for all integers
		We say that ``f(n) is little oh g(n),'' which we write f(n)=o(g(n)), if and only if f(n) is O(g(n)) but f(n) is not
		Little oh notation represents a kind of loose asymptotic bound in the sense that if we are given that f(n)=o(g(n)), then we know that g(n) is an asymptotic upper bound since f(n)=O(g(n)), but g(n) is not an asymptotic lower bound since f(n)=O(g(n)) and
	www.brpreiss.com /books/opus8/html/page70.html (180 words)

	Reasoning About Resources
		This is a valid criticism of asymptotic analysis and big-O notation.
		Order notation is a useful tool, and should not be thought of as being just a theoretical exercise.
		Big-O notation provides a way of expressing rough bounds on the resources required in a form that is meaningful yet easy to work with.
	www.cs.dartmouth.edu /~brd/cs212/handouts/order.html (1905 words)

	Perl.com: Optimizing Your Perl
		There are other notations to represent the magnitude of minimum and actual runtimes.
		The linear search has little overhead, while the DB file has a large overhead.
		By using a little bit of complexity theory, it's possible to optimize your code and get huge speed gains.
	www.perl.com /lpt/a/2002/02/12/bigo.html (1447 words)

	Edinburgh Mathematics Programme (Site not responding. Last check: 2007-10-26)
		Upper estimates for growth rates of real functions, including big O and little o notation.
		Introduction to basic ideas and notation - stress on their relationship to the analysis of algorithms.
		Derive properties of growth rates and use appropriate notation for basic functions (powers of a variable, logarithms and exponentials).
	www.maths.ed.ac.uk /~derek/Syll/mi2.html (416 words)

	CS312 Lecture 11: Big-O notation; Red-black trees
		When we say that an algorithm runs in time T(n), we mean that T(n) is an upper bound on the running time that holds for all inputs of size n.
		Computer scientists have developed a convenient notation for hiding the constant factor.
		We write O(n) (read: ''order n'') instead of ''cn for some constant c.'' Thus an algorithm is said to be O(n) or linear time if there is a fixed constant c such that for all sufficiently large n, the algorithm takes time at most cn on inputs of size n.
	www.cs.cornell.edu /Courses/cs312/2002fa/lectures/lec11.htm (1855 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		It's basically an extension, and a little preview to Vector/Tensor Analysis, and most of the concepts are easily understandable, provided that you knew what you were doing in Calc 1..
		Any decent algorithmics course will require for you to analyze the "cost" of an algorithm, which is usually measured by orders of cost of the most expensive operation.
		Thus, the "Big Oh" notation, and its relatives, "Little Oh", "Big Omega", "Little Omega", and Theta notation.
	www.devhood.com /messages/message_view-2.aspx?thread_id=90349 (1582 words)

	Rolling Stone : Four Amendments & a Funeral
		Like a lot of people who have worked on the Hill a little too long, the aide had a strange look in his eyes -- the desperate look of a man who's been marooned on a remote island, subsisting on bugs and abalone for years on end.
		The Rules Committee meets in a squalid little space the size of a high school classroom, with poor lighting and nothing on the walls but lifeless landscapes and portraits of stern-looking congressmen of yore.
		One odd feature of the document is a notation at the top of the page that reads, "FYI -- this info has not been cleared." In government offices, documents must be cleared for public consumption before they can be distributed outside the agency.
	www.rollingstone.com /news/story/7539869/four_amendments__a_funeral (7176 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-26)
		The O is the capital letter O and is some kind of a function.
		My problem is, I couldn't find out what this big O notation means.
		It means that the rate of growth of f(n) is strictly less than any constant times the rate of growth of g(n).
	mathforum.org /library/drmath/view/54574.html (375 words)

	WebDeveloper.com - Big O Notation
		In learning about Big O notation, I have to identify which formula each method would use.
		The concept of Big O is quite confusing to me, but i've given a go at it, and was wondering if anyone could provide some correction/suggestions/advice.
		Normally you don't need to have too good an understanding of Big O since you just have to generate running time functions and find the Big O of them in lower level computer science courses.
	www.webdeveloper.com /forum/printthread.php?t=59835 (1023 words)

	[No title]
		But, convention is to write T(n) = O(f(n)), which is a little confusing.
		This notation gives us a language for talking about desired or achievable specifications.
		Can see "O()" since we have n terms, each of which is at most n^5.
	www.cs.cmu.edu /afs/cs/academic/class/15451-s99/www/lectures/lect0114 (893 words)

	Big O notation note
		Be sure to compare with the A7 line, noting the effect of a constant factor.
		It should now be a little clearer what we mean when we say that quicksort is better than bubblesort.
		The problem size is the number of elements to be sorted.
	www.csm.astate.edu /~rossa/cs3363/Onotation.html (354 words)

	An English-Persian Dictionary of Algorithms and Data Structures
		The notation is read, "f of n is big oh of g of n".
		The values of c and k must be fixed for the function f and must not depend on n.
		Other measures which may be important are compares, item moves, disk accesses, memory used, or elapsed ("wall clock") time.
	ce.sharif.edu /~dic-ads/d.php?r=O.1.2 (322 words)

	SBF Glossary: ON to ORY
		Notation in references (footnotes or endnotes): `in the work cited' [Latin opere citato.] This once-popular abbreviation was frequently a frustrating nuisance because it was often unclear which previously cited work was meant.
		As a civilian airport it was known as Chicago Orchard Airfield and Douglas Field (hence the O - R - D).
		A sequence of DNA base pairs (bp's) that is intelligible code for a peptide chain.
	www.plexoft.com /SBF/O02.html (5503 words)

	Earliest Uses of Symbols of Number Theory
		He kept to this notation in his later books on number theory.
		The notation µ(n) was introduced by Franz Mertens (1840-1927) in 1874 in "Über einige asymptotische Gesetze der Zahlentheorie," Crelle's Journal (Shapiro).
		The symbols O(·) and o(·) are usually called the Landau symbols.
	members.aol.com /jeff570/nth.html (1263 words)

	Ismail Home Page
		Explain the use of big O, omega, and theta notation to describe the amount of work done by an algorithm.
		Use big O, omega, and theta notation to give asymptotic upper, lower, and tight bounds on time and space complexity of algorithms.
		Convert among equivalently powerful notations for a language, including among DFAs, NFAs, and regular expressions, and between PDAs and CFGs.
	www.aabu.edu.jo /it/cs340.htm (1089 words)

	Two-dimensional Geometry and the Golden section
		On any circle (centre O), construct the 6 equally spaced points A, B, C, D, E and F on its circumference without altering your compasses, so they are the same distance apart as the radius of the circle.
		On two of the sides of that triangle (AE and AC), mark their mid-points P and Q by joining the centre O to two of the unused points of the hexagon (F and B).
		The third little geometrical gem is about a ring and an ellipse.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/phi2DGeomTrig.html (8440 words)

	Computational Complexity - BioInformatics Glossary (Site not responding. Last check: 2007-10-26)
		The set of all possible symbols in an application.
		(O) A theoretical measure of the execution of an algorithm, usually the time or memory needed, given the problem size n, which is usually the number of items.
		A complexity class defined by logarithmic, polynomial, and exponential bounds on time and space, for deterministic and nondeterministic machines.
	big.mcw.edu /topic.php/10.html (564 words)

	little-o notation (Site not responding. Last check: 2007-10-26)
		The notation is read, "f of n is little oh of g of n".
		HTML page formatted Wed Apr 19 12:22:48 2006.
		Paul E. Black, "little-o notation", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/littleOnotation.html (202 words)

	Math 225 Material Schedule Week 13 Notes (Site not responding. Last check: 2007-10-26)
		Go over the proof that is in the appendix using little "o" notation.
		In Math 233 the same ideas are used, so the students need to see it in this simpler setting.
		It is good excuse to review some more trigonometry.
	www.uwm.edu /People/ericskey/Math225-6/225notes12.html (82 words)

	Big O notation
		You got them all right; it is a little simpler that it seems at first.
		Technically, Big-O notation simply represents a set of functions -- the set of functions which grow at a slower or equal pace as the given function.
		When we say an algorithm runs in O(n^2) time, we mean that the amount of time it takes for the algorithm to run grows at a slower or equal pace as the function f(n) = n^2.
	www.ozzu.com /ftopic41683.html (1327 words)

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