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o-notation

- For a given function g(n), we denote by o(g(n)) the set of functions:

o(g(n)) = {f(n): for any positive constant c > 0, there exists a constant n0 > 0 such that 0 f(n)<cg(n) for all n n0}

- f(n) becomes insignificant relative to g(n)as n approaches infinity: lim [f(n) / g(n)] = 0

n

- We say g(n) is an upper bound for f(n)that is not asymptotically tight.

O(*) versus o(*)

O(g(n)) = {f(n): there exist positive constants c and n0 such that 0 f(n)cg(n), for all n n0}.

o(g(n)) = {f(n): for any positive constant c > 0, there exists a constant n0 > 0 such that 0 f(n)<cg(n) for all n n0}.

Thus o(f(n)) is a weakened O(f(n)).

For example: n2 = O(n2)

n2 o(n2)

n2 = O(n3)

n2 = o(n3)

o-notation

- n1.9999 = o(n2)
- n2/ lg n = o(n2)
- n2o(n2) (just like 2< 2)
- n2/1000 o(n2)

w-notation

- For a given function g(n), we denote by w(g(n)) the set of functions

w(g(n)) = {f(n): for any positive constant c > 0, there exists a constant n0 > 0 such that 0 cg(n)<f(n) for all n n0}

- f(n) becomes arbitrarily large relative to g(n)as n approaches infinity: lim [f(n) / g(n)] =

n

- We say g(n) is a lower bound for f(n)that is not asymptotically tight.

w-notation

- n2.0001 = ω(n2)
- n2 lg n = ω(n2)
- n2ω(n2)

Comparison of Functions

f g a b

f (n) = O(g(n)) a b

f (n) = (g(n)) a b

f (n) = (g(n)) a = b

f (n) = o(g(n)) a < b

f (n) = w (g(n)) a > b

Properties

- Transitivity

f(n) = (g(n)) & g(n) = (h(n)) f(n) = (h(n))

f(n) = O(g(n)) & g(n) = O(h(n)) f(n) = O(h(n))

f(n) = (g(n)) & g(n) = (h(n)) f(n) = (h(n))

- Symmetry

f(n) = (g(n)) if and only if g(n) = (f(n))

- Transpose Symmetry

f(n) = O(g(n)) if and only if g(n) = (f(n))

Practical Complexities

- Is O(n2) too much time?
- Is the algorithm practical?

At CPU speed 109 instructions/second

Impractical Complexities

At CPU speed 109 instructions/second

2n

1ms x 2n

10

103

0.001 s

20

106

1 s

30

109

16.7 mins

40

1012

11.6 days

50

1015

31.7 years

60

1018

31710 years

Exponential Functions- Exponential functions increase rapidly, e.g., 2n will double whenever n is increased by 1.

Floors & Ceilings

- For any real number x, we denote the greatest integerless than or equal to x by x
- read “the floor of x”
- For any real number x, we denote the least integergreater than or equal to x by x
- read “the ceiling of x”
- For all real x, (example for x=4.2)
- x – 1 x x x x + 1
- For any integer n ,
- n/2 + n/2 = n

Polynomials

- Given a positive integer d, a polynomial in n of degree d is a function P(n) of the form
- P(n) =
- where a0, a1, …, ad are coefficient of the polynomial
- ad 0
- A polynomial is asymptotically positive iffad 0
- Also P(n) = (nd)

Exponents

- x0 = 1 x1 = x x-1 = 1/x
- xa . xb = xa+b
- xa / xb = xa-b
- (xa)b = (xb)a = xab
- xn + xn = 2xn x2n
- 2n + 2n = 2.2n = 2n+1

Logarithms (1)

- In computer science, all logarithms are to base 2 unless specified otherwise
- xa = b iff logx(b) = a
- lg(n) = log2(n)
- ln(n) = loge(n)
- lgk(n) = (lg(n))k
- loga(b) = logc(b) / logc(a) ; c 0
- lg(ab) = lg(a) + lg(b)
- lg(a/b) = lg(a) - lg(b)
- lg(ab) = b . lg(a)

Logarithms (2)

- a = blogb(a)
- alogb(n) = nlogb(a)
- lg (1/a) = - lg(a)
- logb(a) = 1/loga(b)
- lg(n) n for all n 0
- loga(a) = 1
- lg(1) = 0, lg(2) = 1, lg(1024=210) = 10
- lg(1048576=220) = 20

Summation

- Why do we need to know this?

We need it for computing the running time of a given algorithm.

- Example: Maximum Sub-vector

Given an array a[1…n] of numeric values (can be positive, zero and negative) determine the sub-vector a[i…j] (1 i j n) whose sum of elements is maximum over all sub-vectors.

Example: Max Sub-Vectors

MaxSubvector(a, n) {

maxsum = 0;

for i = 1 to n {

for j = i to n {

sum = 0;

for k = i to j { sum += a[k] }

maxsum = max(sum, maxsum);

}

}

return maxsum;

}

Summation

- Constant Series: For a, b 0,
- Quadratic Series: For n 0,
- Linear-Geometric Series: For n 0,

Proof of Geometric series

A Geometric series is one in which the sum approaches a given number as N tends to infinity.

Proofs for geometric series are done by cancellation, as demonstrated.

Factorials

- n! (“n factorial”) is defined for integers n 0 as
- n! =
- n! = 1 . 2 .3 … n
- n! < nn for n ≥ 2

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