asymptotic notation
- 1. ASYMPTOTIC NOTATIONS NADAR SARAWATHI COLLEGE OF ARTS AND SCIENCE.
- 2. Asymptotic notations are mathematical tools to represent time complexity of algorithms for asymptotic analysis. Asymptotic notations are the mathematical representations of space and time complexity of an algorithm. It’s used to measure speed and efficiency.
- 4. Big Oh notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity. This notation is known as the upper bound of the algorithm, or a worst case of an algorithm. It tells us that a certain function will never exceed a specified time for any value of input n.
- 5. If f(x)=anxn+an-1*xn-1+…..+a1x+a0 Where aieR,i=0,….,n; then f(x) is 0(xn). If x>1 we have |f(x)|=|anxn+an-1*xn-1+…..+a1x+a0| <|an|xn+|an-1|xn-1+…..+|a1|x+|a0| =xn(|an|+|an-1|/x+……+|a1|/xn-1+|a0|/xn) <xn(|an|+|an-1|+…..+|a1|+|a0|) So, |f(x)|<cxn where c=|an|+|an-1|+…..+|a1|+|a0|
- 7. Big Omega notation is used to define the lower bound of any algorithm or we can say the best case of any algorithm. It’s always indicates the minimum time required for any algorithm for all input values. It is used to find out the least time for an algorithm to takes an completion.
- 8. FeW(g) if f lim infn-> |f(n)/g(n)|>0. If lim inf |f(n)/g(n)|=c>0, then we have for each e>0 at most finitely many positive integers satisfying |f(n)/g(n)|<C-e.Thus, there exists an n0 such that. |f(n)|>(C-e)|g(n)| holds for all n>n0, proving that feW(g).
- 10. TheTheta notation bounds a functions from above and below, so it defines exact asymptotic behaviour. It’s used to average time for an algorithm it takes place an complete an or execution.
- 11. Let f(x)=anxn+an-1xn-1+…..+a1x+a0 Where a0,a1,…..,an are real numbers with an=0.Then f(x) is of order xn(or Q(xn)) The polynomial f(x)=8x5+5x2+10 is order of x5(or Q(x5)). The polynomial f(x)=8x199+7x100+x99+5x2+25 is order of x199(or Q(x199)).