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O -Notation

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

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  1. O-Notation April 23, 2003 Prepared by Doug Hogan CSE 260

  2. 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.

  3. 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

  4. 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).

  5. O-notation: A graphical view

  6. 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)|

  7. 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

  8. 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

  9. 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)

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

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

  12. Graphically… M = 45; x0 = 6

  13. Another graphical example f(x) = 7x3 - 2x + 3 12x3 x3 M = 12, x0 = 1 7x3 - 2x + 3 is O(x3)

  14. 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

  15. 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”

  16. 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)

  17. 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). □

  18. Generalization • If a0, a1,…, anare real numbers and an ≠ 0 anxn+an-1xn-1 +… + a1x + a0is NOTO(xm) for all m<n

  19. 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)).

  20. 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.

  21. 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)] (Headington 546)

  22. 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 (Headington 547)

  23. Works Cited Epp, Susanna. Discrete Mathematics with Applications. 2nd Ed. Belmont, CA: Brooks, 1995. Headington, Mark A., and David Riley. Data Abstraction and Structures using C++. Lexington, MA: Heath, 1994.

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