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Note that in the while loop two comparisons and one addition are performed. Thus one could use 3n as an estimate just as well. Also note that the very first line and the last two lines are not counted in. The reasons for those are firstly that differences in implementation details such as languages, commands, compilers and machines make differences in constant factors meaningless, and secondly that for large values of n, the highest degree term in n dominates the estimate. Since we are mostly interested in the behavior of algorithms for large values of n , lower terms can be ignored compared with the highest term. The concept that is used to address these issues is something called big-oh, and that is what we are going to study here.

Big - oh

The following example gives the idea of one function growing more rapidly than another. We will use this example to introduce the concept the big-Oh.

Example: f(n) = 100 n2, g(n) = n4, the following table and Figure 2 show that g(n) grows faster than f(n) when n>10. We say f is big-Oh of g.

n f(n) g(n)
10 10,000 10,000
50 250,000 6,250,000
100 1,000,000 100,000,000
150 2,250,000 506,250,000

Definition (big-oh): Let f and g be functions from the set of integers (or the set of real numbers) to the set of real numbers. Then f(x) is said to be O( g(x) ) , which is read as f(x) is big-oh of g(x) , if and only if there are constants C and n0 such that

     | f(x) | ≤ C | g(x) |

whenever x>n0 .

Note that big-oh is a binary relation on a set of functions (What kinds of properties does it have ? reflexive ? symmetric ? transitive ?).

The relationship between f and g can be illustrated as follows when f is big-oh of g.

For example, 5 x + 10  is big-oh of  x2,   because 5 x + 10<5 x2 + 10 x2 = 15 x2   for   x>1 .  

Hence for C = 15 and n0 = 1 ,   | 5x + 10 | ≤ C | x2 | . Similarly it can be seen that  3 x2 + 2 x + 4<9 x2  for   x>1 .   Hence 3 x2 + 2 x + 4 is O( x2 ) . In general, we have the following theorem:

Theorem 1: an xn + ... + a1 x + a0   is   O( xn )   for any real numbers an , ..., a0 and any nonnegative number n .

Note: Let f(x) = 3 x2 + 2 x + 4, g(x) = x2, from the above illustration, we have that f(x) is O(g(x)). Also, since x2<3 x2 + 2 x + 4, we can also get g(x) is O(f(x)). In this case, we say these two functions are of the same order.

Growth of combinations of functions

Big-oh has some useful properties. Some of them are listed as theorems here. Let use start with the definition of max function.

Definition (max function): Let f1(x) and f2(x) be functions from a set A to a set of real numbers B. Then max( f1(x) , f2(x) ) is the function from A to B that takes as its value at each point x the larger of f1(x) and f2(x).

Theorem 2: If  f1(x) is O( g1(x) ) , and   f2(x) is O( g2(x) ) , then  (f1 + f2)( x )  is   O( max( g1(x) , g2(x) ) ) .

From this theorem it follows that if  f1(x)  and  f2(x) are O( g(x) ) , then  (f1 + f2)( x )  is O( g(x) ) , and

(f1 + f2)( x )  is  O( max( f1(x) , f2(x) ) ) .

Theorem 3: If  f1(x) is O( g1(x) ) , and   f2(x) is O( g2(x) ) , then  (f1 * f2)( x )  is   O( g1(x) * g2(x) ) .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
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I'm not sure why it wrote it the other way
I got X =-6
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oops. ignore that.
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Commplementary angles
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
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I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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