ALG 2-2: Asymptotic Order of Growth (渐进分析)

1. Asymptotic Order of Growth

• Upper bounds. T(n) is O(f(n)) if there exist constants c > 0 and n0 ≥ 0 such that for all n ≥ n0 we have T(n) ≤c · f(n).
• 上界: T(n)为O(f(n))，如果存在常数c > 0 和 n0 ≥ 0，使得对所有 n ≥ n0 有 T(n) ≤ c·f(n)
• Lower bounds. T(n) is Ω(f(n)) if there exist constants c > 0 and n0 ≥ 0 such that for all n ≥n0 we have T(n) ≥c · f(n).
• 下界: T(n)为Ω(f(n))，如果存在常数c > 0 和 n0 ≥ 0，使得对所有 n ≥ n0 有 T(n) ≥ c·f(n)
• Tight bounds. T(n) is Θ(f(n)) if T(n) is both O(f(n)) and Ω(f(n)).

Ex: T(n) = 32n^2+ 17n + 32.

• T(n) is O(n^2), O(n^3), Ω(n^2), Ω(n), and Θ(n^2) .
• T(n) is not O(n), Ω(n^3), Θ(n), or Θ(n^3).

2. Properties （性质）

Transitivity. （交换性）

• If f = O(g) and g = O(h) then f = O(h).
• If f = Ω(g) and g = Ω(h) then f = Ω(h).
• If f = Θ(g) and g = Θ(h) then f = Θ(h).

Additivity. （相加性）

• If f = O(h) and g = O(h) then f + g = O(h).
• If f = Ω(h) and g = Ω(h) then f + g = Ω(h).
• If f = Θ(h) and g = O(h) then f + g = Θ(h).

3. Asymptotic Bounds for Some Common Functions （一些常见函数的渐近界）

1. Polynomials(多项式). a_0+ a_1*n + … + a_d*n^d is Θ(n^d) if a_d> 0.
   • Polynomial time. Running time is O(n^d) for some constant d independent of the input size n. (与输入大小n无关, 运行时间恒为O(n^d) )
2. Logarithms(对数). O(loga n) = O(logb n) for any constants a, b > 0.

• •  Logarithms. For every x > 0, log n = O(n^x). l（log grows slower than every polynomial )

　　  3. Exponentials(指数). For every r > 1 and every d > 0, n^d= O(r^n). （every exponential grows faster than every polynomial）