The main idea of asymptotic analysis is to have a measure of the efficiency of algorithms that don’t depend on machine-specific constants, mainly because this analysis doesn’t require algorithms to be implemented and time taken by programs to be compared. We have already discussed Three main asymptotic notations. The following 2 more asymptotic notations are used to represent the time complexity of algorithms.
Little o Asymptotic Notation:
Big O is used as a tight upper bound on the growth of an algorithm’s effort (this effort is described by the function f(n)), even though, as written, it can also be a loose upper bound. “Little o” (o) notation is used to describe an upper bound that cannot be tight.
In the domain of algorithm analysis, the little o notation is a valuable tool used to describe the behavior of functions as they approach certain limits. When we say that a function f(n) is o(g(n)), we are essentially stating that f(n) grows slower than g(n) as n approaches infinity. In simpler terms, if f(n) = o(g(n)), it means that g(n) grows faster than f(n).
Thus, Little o means loose upper-bound of f(n). Little o is a rough estimate of the maximum order of growth whereas Big-O may be the actual order of growth.
Mathematical Relation:
f(n) = o(g(n)) means lim f(n)/g(n) = 0 n→∞Examples:
Is 7n + 8 ∈ o(n2)
In order for that to be true, for any c, we have to be able to find an n0 that makes f(n) < c * g(n) asymptotically true.
lets took some example,
If c = 100,we check the inequality is clearly true. If c = 1/100 , we’ll have to use a little more imagination, but we’ll be able to find an n0. (Try n0 = 1000.) From these examples, the conjecture appears to be correct.
then check limits,
lim f(n)/g(n) = lim (7n + 8)/(n2) = lim 7/2n = 0 (l’hospital)
n→∞ n→∞ n→∞hence 7n+8 ∈ o(n2)
Little ω Asymptotic Notation:
On the other hand, little ω notation is used to describe the relationship between two functions when one grows strictly faster than the other. If a function f(n) is ω(g(n)), it means that g(n) grows slower than f(n) as n approaches infinity.
f(n) has a higher growth rate than g(n) so main difference between Big Omega (Ω) and Little omega (ω) lies in their definitions.In the case of Big Omega f(n)=ω(g(n)) and the bound is 0<=cg(n)<=f(n), but in case of little omega, it is true for 0<=c*g(n)<f(n).
The relationship between Big Omega (Ω) and Little Omega (ω) is similar to that of Big-O and Little o except that now we are looking at the lower bounds. Little Omega (ω) is a rough estimate of the order of the growth whereas Big Omega (Ω) may represent exact order of growth. We use notation to denote a lower bound that is not asymptotically tight, and f(n) ∈ ω(g(n)) if and only if g(n) ∈ ο((f(n)).
Mathmatical Relation:
if f(n) ∈ ω(g(n)) then, lim f(n)/g(n) = ∞ n→∞ Example:
Prove that 4n+6 ∈ ο(1);
The little omega(ο) running time can be proven by applying limit formula given below.
if lim f(n)/g(n) = ∞ then functions f(n) is ο(g(n))
n→∞
here,we have functions f(n)=4n+6 and g(n)=1
lim (4n+6)/(1) = ∞
n→∞ and,also for any c we can get n0 for this inequality 0<=c*g(n)<f(n) ,
0<=c*1<4n+6. So,hence proved.
Conclusion:
In conclusion, little o and little ω notations are essential tools in algorithm analysis that allow us to compare the growth rates of functions in a precise manner. By understanding these notations, we can better analyze and predict the performance of algorithms as their input sizes grow.