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Q. Example of asymptotic notations?
The function f (n) belongs to the set (g(n)) if there exists positive constants c such that for satisfactorily large values of n we have 0<= c*g(n) <=f(n). In another words,
O(g(n)) ={ 0<= c*g(n) <=f(n) for all n >= no }.
Assume we have a function f (n) = 4n2 + n then order of function is O(n2). The asymptotic notations give information about the upper and lower bounds on complexity of an algorithm with help of O and ? notations. E.g. in the sorting algorithm the upper bound is O (n ln n) and lower bound is ? (n ln n). But problems such as matrix multiplication have complexities such as O (n3) to O (n2.38).
Algorithms that have similar lower and upper bounds are called optimal algorithms. So a few sorting algorithms are optimal whereas matrix multiplication based algorithms are not.
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