O -Notation

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O -Notation. April 23, 2003 Prepared by Doug Hogan CSE 260. O -notation: The Idea. Big-O notation is a way of ranking about how much time it takes for an algorithm to execute How many operations will be done when the program is executed?

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O-Notation

April 23, 2003

Prepared by Doug Hogan

CSE 260

O-notation: The Idea
• Big-O notation is a way of ranking about how much time it takes for an algorithm to execute
• How many operations will be done when the program is executed?
• Find a bound on the running time, i.e. functions that are on the same order.
• We care about what happens for large amounts of data  asymptotic order.
O-notation: The Idea
• Use mathematical tools to find asymptotic order.
• Real functions to approximate integer functions.
• Depends on some variable, like n or X, which is usually the size of an array or how much data is going to be processed
O-notationThe complicated math behind it all…
• Given f and g, real functions of variable x…
• First form:
• g provides an upper bound for f ≡ graph of f lies closer to the x axis than g
• More general form:
• g provides an upper bound for f ≡ graph of f lies closer to the x axis than some positive multiple (M) of g after some minimum value of x(x0).
So what does “closer to the x-axis” MEAN?
• -M ∙ g(x) ≤ f(x) ≤ M ∙ g(x)
• But that’s absolute value…
• |f(x)| ≤ M ∙ |g(x)|
Another graphical view

y

M ∙g

f

g

After x0, |f(x)| ≤ M ∙ |g(x)|

Before x0, nothing claimed about f’s growth

x

x0

Formal Definition
• Let f and g be real-valued functions defined on the same set of reals.
• f is of order g, written f(x) = O(g(x)), iff there exists
• a positive real number M (multiple)
• a real number x0 (starting point)

such that for all x in the domain of f and g, |f(x)| ≤ M ∙ |g(x)| when x > x0

Example
• Use the definition of O-notationto express|17x6 – 3x3 + 2x + 8| ≤ 30|x6| for all x > 1
• M = 30
• x0 = 1
• 17x6 – 3x3 + 2x + 8 is O(x6)
Graphically…

17x6 – 3x3 + 2x + 8 is O(x6); M = 30; x0 = 1

30x6

17x6 – 3x3 + 2x + 8

x6

Problem
• Use the definition of O-notationto express for all x > 6
• M = 45
• x0 = 6
Graphically…

M = 45; x0 = 6

Another graphical example

f(x) = 7x3 - 2x + 3

12x3

x3

M = 12, x0 = 1 7x3 - 2x + 3 is O(x3)

Using O-notation…
• Order of Power Functions:
• For any rational numbers r and s, if r < s,

xr is O(xs)

• Order of Polynomial Functions:
• If a0, a1,…, anare real numbers and an ≠ 0

anxn+an-1xn-1 +… + a1x + a0is O(xm) for all m ≥ n

Examples
• Example:
• Find an order for
• f(x) = 7x5 + 5x3 – x + 4 (all reals x)
• O(x5)
• Is that the only answer?
• No…
• But it’s the “best”
Showing that a function is NOT Big-O of another…
• Show that x2 is not O(x).
• [Arguing by contradiction.] Suppose not, that x2 is O(x).
• By definition of O(…), then there exist
• a positive real number M
• a real number x0

such that |x2| ≤ M ∙ |x| for all x > x0 (1)

Showing that a function is NOT Big-O of another…, ctd.
• Let x be a positive real number greater than both M and x0, i.e. x>M and x>x0.
• Then by multiplying both sides of x>M by x, x∙x>M∙x.
• Since x is positive, |x2|>M∙|x|.
• So there is a real number x>x0 s.t. |x2|>M∙|x|.
• This contradicts (1) above. So, the supposition is false and thus x2 is not O(x). □
Generalization
• If a0, a1,…, anare real numbers and an ≠ 0

anxn+an-1xn-1 +… + a1x + a0is NOTO(xm) for all m<n

Best approximation

Definition

• Suppose S is a set of functions from a subset of RtoR and fis a R->R function.
• Function gis a best big-Oapproximation for f in Siff
• f(x) is O(g(x))
• for any h in S, if f(x) is O(h(x)), then g(x) is O(h(x)).
Problem
• Find best big-O approx for f(x) = 5x3 – 2x + 1
• By thm. on polynomial orders,
• f(x) is O(xn) for all n ≥ 3
• By previous property,
• f(x) is NOT O(xm) for all m < 3
• So O(x3) is the best approximation.
O-Arithmetic

Let f and g be functions and k be a constant.

• O(k*f) = O(f)
• O(f *g) = O(f) * O(g)
• O(f/g) = O(f) / O(g)
• O(f) ≥ O(g) iff f dominates* g
• O(f + g) = Max[O(f), O(g)]

Dominance

Let f and g be functions and a,b,m,n be constants.

• xxdominates x!
• x!dominates ax
• axdominates bxif a > b
• ax dominates xn if n > m
• x dominates logax if a > 1
• logax dominateslogbx if b > a > 1
• logax dominates1 if a > 1