By Thomas H. Cormen
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Extra resources for Introduction to Algorithms (MIT Electrical Engineering and Computer Science)
Moreover, these bounds are asymptotically as tight as possible: for instance, the running time of insertion sort is not Q(n 2 ) , since insertion sort runs in 8( n) time when the input is already sorted. It is not contradictory, however, to say that the worst-case running time of insertion sort is Q(n 2 ) , since there exists an input that causes the algorithm to take Q(n 2 ) time. When we say that the running time (no modifier) of an algorithm is Q(g(n)), we mean that no matter what particular input of size n is chosen for each value of n, the running time on that set of inputs is at least a constant times g(n), for sufficiently large n.
We might also write 2n 2 + 3n + 1 = 2n 2 + 8(n). How do we interpret such formulas? When the asymptotic notation stands alone on the right-hand side of an equation, as in n = O(n 2 ) , we have already defined the equal sign to mean set membership: n E O(n 2 ) . In general, however, when asymptotic notation appears in a formula, we interpret it as standing for some anonymous function that we do not care to name. For example, the formula 2n 2 + 3n + 1 = 2n 2 + 8(n) means that 2n 2 + 3n + 1 = 2n 2 + j(n), where f(n) is some function in the set 8(n).
Comparison of functions Many of the relational properties of real numbers apply to asymptotic comparisons as well. For the following, assume that j(n) and g(n) are asymptotically positive. Transitivity: j(n) = 8(g(n» and g(n) = 8(h(n» imply j(n) = 8(h(n» , j(n) = O(g(n» and g(n) O(h(n» imply j(n) = O(h(n» , j(n) = Q(g(n» and g(n) = Q(h(n» imply j(n) = Q(h(n» , j(n) = o(g(n» and g(n) = o(h(n» imply j(n) = o(h(n» , j(n) w(g(n» and g(n) w(h(n» imply j(n) = w(h(n» . Reflexivity: j(n) = 8(j(n» , j(n) = O(f(n» , j(n) = Q(f(n».
Introduction to Algorithms (MIT Electrical Engineering and Computer Science) by Thomas H. Cormen