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Growth of Functions

Growth of Functions. Asymptotic Notation. When we look at a large enough input n , we are studying the asymptotic efficiency Concerned with the running time as the input increases without bound

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Growth of Functions

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  1. Growth of Functions Jeff Chastine

  2. Asymptotic Notation • When we look at a large enough input n, we are studying the asymptotic efficiency • Concerned with the running time as the input increases without bound • Usually, an algorithm that is asymptotically more efficient is the best choice (except for small n) Jeff Chastine

  3. Θ-notation (‘Theta’) • Θ(g(n))={ƒ(n):there exist positive constants c1, c2, and n0 such that 0≤c1g(n) ≤ƒ(n) ≤c2g(n) for all n≥n0}. • This means that g(n) and f(n) have the same running time; are off by just a constant • "Sandwiched" for sufficiently large n • We say g(n) is an asymptotically tight bound for f(n) Jeff Chastine

  4. Big-Θ Jeff Chastine

  5. A Formal Definition ½n2 - 3n = Θ(n2) c1n2 ≤ ½n2 - 3n ≤ c2n2 for all n≥n0. Divide by n2 yields c1≤ ½ - 3/n ≤ c2 Jeff Chastine

  6. Why 6n3≠ Θ(n2) Suppose 6n3≤ c2n2 for all n≥n0 Divide each side by 6n2 n ≤ c2/6, which isn't true for all n! Divide each side by 6n3 1 ≤ c2/n, which isn't true for all n! Jeff Chastine

  7. Practice • Do this now! • Prove 3n2 + n = Θ (n2) Jeff Chastine

  8. Note • Now you can see: • Why constants (coefficients) don't matter • Why lesser terms don't matter d ∑aini = Θ(nd) i=0 Jeff Chastine

  9. Ο-notation (Big-Oh) • Ο(g(n))={ƒ(n):there exist positive constants c and n0 such that 0≤ƒ(n) ≤cg(n) for all n≥n0}. • This means that g(n) is always greater than f(n) at some point • We say g(n) is an asymptotic upper bound for f(n) • Associated with worst-case running time Jeff Chastine

  10. Big-Oh Jeff Chastine

  11. A Formal Definition 10n2 - 3n = Ο(n3) 10n2 - 3n ≤ c2n3 for all n≥n0. Divide by n2 yields 10 - 3/n ≤ c2n Jeff Chastine

  12. Ω-notation (‘Omega’) • Ω(g(n))={ƒ(n):there exist positive constants c and n0 such that 0≤cg(n) ≤ƒ(n) for all n≥n0}. • This means that g(n) is always less than f(n) at some point • We say g(n) is an asymptotic lower bound for f(n) • Associated with best-case running time Jeff Chastine

  13. Big-Ω Jeff Chastine

  14. Insertion Sort(revisited) • Running time falls between Ω(n) andΟ(n2) • Is the worst-case scenario of insertion sort Ω(n2) Jeff Chastine

  15. ο-notation (little-oh) • ο(g(n))={ƒ(n): for any positive constant c, there exists a constant n0 such that 0≤ƒ(n) ≤cg(n) for all n≥n0}. • Example: 2n = ο(n2), but 2n2 ≠ ο(n2) • This notation is not asymptotically tight Jeff Chastine

  16. ω-notation (little omega) • ω(g(n))={ƒ(n): for any positive constant c, there exists a constant n0 > 0 such that 0 ≤ cg(n) ≤ƒ(n) for all n≥n0}. • Example: n2/2 = ω(n), but n2/2 ≠ ω(n2) • This notation is not asymptotically tight Jeff Chastine

  17. Standard Notation • A function is monotonically increasing if m≤ n implies ƒ(m) ≤ ƒ(n) • There is also monotonically decreasing • The floor of a number rounds down x • The ceiling of a number rounds up x └ ┘ ┌ ┐ Jeff Chastine

  18. Jeff Chastine

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