if x 1 2345 x ln1 x x for x 1 1 x a function

Unformatted text preview: ⎦ ≥ ( a − ( b − 1 )) / b NCKU IIM NCKU 資料結構 Chapter 3 資料結構 14 Modular Arithmetic • For any integer a and any positive integer n, the value a mod n is the remainder (or residue) of the quotient a/n : remainder residue a/n a mod n = a -⎣a/n⎦n • If(a mod n) = (b mod n). We write a ≡ b (mod n) and say mod that a is equivalent to b, modulo n. equivalent • We write a ≢ b (mod n) if a is not equivalent to b modulo n. NCKU IIM NCKU 資料結構 Chapter 3 資料結構 15 Polynomials v.s. Exponentials d ∑an – A function f(n) is polynomial bounded if polynomial f (n) = O(n k ) = O(nO (1) ) for some constant k • Polynomials: P (n) = i i i=0 . • Exponentials: – Any positive exponential function with a base strictly greater than 1 grows faster than any polynomial. n = o(a ) (a > 1) b n ∞ xi ex = ∑ i =0 i ! 1 + x ≤ e x ≤ 1 + x + x 2 if | x |< 1 xn lim n →∞ (1 + ) = e x n NCKU IIM NCKU 資料結構 Chapter 3 資料結構 16 Logarithms x 2 x3 x 4 x5 ln(1 + x) = x − + − + − ... if | x |< 1 2345 x ≤ ln(1 + x) < x for x > −1 1+ x • A function f(n) is polylogarithmically bounded if polylogarithmically f (n) = O(log k n) = O(log O (1) n) for any postive constant k • Any positive polynomial function grows faster than any polylogarithmic function. log b n = o( n a ) NCKU IIM NCKU 資料結構 Chapter 3 資料結構 17 Factorials • Stirling’s approximation 1 nn n!= 2πn ( ) (1 + Θ( )) n ne n ! = o( n...
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