2 Fundamentals

In this chapter we discuss basic terminology and notation for graphs, some fundamental algorithms, and a few other mathematical preliminaries. We denote the set of integers by , the set of real numbers by , the set of complex numbers by , and the set of rationals by . For a set X of numbers, X denotes the subset of positive numbers in X , and X 0 the subset of nonnegative numbers. The set of positive integers is denoted by = + and the set of non-negative integers by 0 = 0 . We use n×m to denote the set of all real-valued matrices with n rows and m columns. If the entries of the matrix can be complex numbers, we write n×m. The n-dimensional identity matrix is denoted by In. The n-dimensional vector with all entries equal to 1 (equal to 0) is denoted by 1n (by 0n). For two functions f : → and g : → , we say that f is in O(g) if there are positive constants n0 ∈ and c ∈ + such that f(n) ≤ c · g(n) holds for all n ≥ n0. Furthermore, we say that f is in Ω(g) if g is in O(f). This notation is useful to estimate the asymptotic growth of functions. In particular, running-times of algorithms are usually specified using this notation.