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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- Winter '12
- bob
- Big O notation, LG, Order theory, Monotonic function, NCKU IIM
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