By David M. Burton
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Writing g(N) = (N2 ) says not only that g(N) = O(N2 ), but also that the result is as good (tight) as possible. 1 Typical growth rates The important things to know are Rule 1. If T1 (N) = O(f(N)) and T2 (N) = O(g(N)), then (a) T1 (N) + T2 (N) = O(f(N) + g(N)) (intuitively and less formally it is O(max(f(N), g(N))) ), (b) T1 (N) ∗ T2 (N) = O(f(N) ∗ g(N)). Rule 2. If T(N) is a polynomial of degree k, then T(N) = (Nk ). Rule 3. logk N = O(N) for any constant k. This tells us that logarithms grow very slowly.
1. M. O. Albertson and J. P. Hutchinson, Discrete Mathematics with Algorithms, John Wiley & Sons, New York, 1988. 2. Z. , 1982. 3. R. A. Brualdi, Introductory Combinatorics, North-Holland, New York, 1977. 4. G. Cornell and C. S. Horstmann, Core Java, Vol. , 2009. 5. E. W. , 1976. 6. D. , 2005. 7. J. Gosling, B. Joy, G. Steele, and G. , 2005. 8. R. L. Graham, D. E. Knuth, and O. , 1989. 9. D. Gries, The Science of Programming, Springer-Verlag, New York, 1981. 10. B. W. Kernighan and P. J. , McGraw-Hill, New York, 1978.
Eventually, a runtime error results at the last line because the call to read tries to return a String but cannot. As a result, the typecast will generate a warning, and a corresponding instanceof test is illegal. Static Contexts In a generic class, static methods and ﬁelds cannot refer to the class’s type variables since, after erasure, there are no type variables. Further, since there is really only one raw class, static ﬁelds are shared among the class’s generic instantiations. Instantiation of Generic Types It is illegal to create an instance of a generic type.
An introduction to abstract mathematical systems by David M. Burton