1 / 23

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?

bernadine-wolf
Télécharger la présentation

O -Notation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related