In this chapter, we review standard asymptotic notation, introduce the formal computational model we shall use throughout the rest of the text, and discuss basic algorithms for computing with large integers.

Asymptotic notation

We review some standard notation for relating the rate of growth of functions. This notation will be useful in discussing the running times of algorithms, and in a number of other contexts as well.

Suppose that x is a variable taking non-negative integer or real values, and let g denote a real-valued function in x that is positive for all sufficiently large x; also, let f denote any real-valued function in x. Then

• f = O(g) means that |f(x)| ≤ cg(x) for some positive constant c and all sufficiently large x (read, “f is big-O of g”),

• f = Ω(g) means that f(x) ≥ cg(x) for some positive constant c and all sufficiently large x (read, “f is big-Omega of g”),

• f = ⊗(g) means that cg(x) ≤ f(x) ≤ dg(x), for some positive constants c and d and all sufficiently large x (read, “f is big-Theta of g”),

• f = o(g) means that f/g → 0 as x → ∞ (read, “f is little-o of g”), and

• f ∼ g means that f/g → 1 as x → ∞ (read, “f is asymptotically equal to g”).

Example 3.1. Let f(x) ≔ x2 and g(x) ≔ 2x2 - x+1. Then f = O(g) and f = Ω(g). Indeed, f = ⊗(g). ▪