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Big-O

Big-O. Algorithm Analysis. Exact analysis: produce a function f ( n) measuring how many basic steps are needed for a given inputs n On any input of size n, the algorithm runs for at most 7n 2 + 3n + 8 steps Do we really need such a precise measurement?

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Big-O

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  1. Big-O

  2. Algorithm Analysis • Exact analysis: produce a function f(n) measuring how many basic steps are needed for a given inputs n On any input of size n, the algorithm runs for at most 7n2+ 3n + 8 steps • Do we really need such a precise measurement? • To compare two algorithm, we are concerned with how well the algorithm scales, • Too much details of f(n) is meaningless! • (asymptotic) analysis

  3. Asymptotic Analysis • Asymptotic analysis of an algorithm describes the relative efficiency of an algorithm when n gets very large

  4. Asymptotic Analysis (cont’d) N For large values of n, Algorithm 1 grows faster than Algorithm 2

  5. Why does Complexity Matter? • Assume that you have 3 algorithms to sort a list • f(n) = nlog2n • g(n) = n2 • h(n) = n3 • Assume that each step takes 1 microsecond (i.e., 10-6 seconds)

  6. Asymptotic Analysis (cont’d)

  7. Bound • f(n) is contributed from N^2 • The grow rate is bounded • with such bound, we only need to refer to well known functions and do not need to be concerned with the details of the function • Big-O notion: upper bound on the growth rate of a function, for sufficiently large values of n.

  8. Big-O Notation • Definition: f(n)= O(g(n))if there exist positive constants c and N such that f(n)  cg(n) for all n  N • Big-O expresses an upper bound on the growth rate of a function, for sufficiently large values of n. • Graph to understand Big-O

  9. An Example • Prove that 2n2 + 1 = O(n2)

  10. Proof with Induction

  11. Typical functions

  12. Properties of Big-O Notation • Fact 1 (Transitivity) • If f(n)=O(g(n)) and g(n)=O(h(n)), then f(n)=O(h(n)) • Prove it (done in class) • Fact 2 • If f(n)=O(h(n)) and g(n)=O(h(n)), then f(n)+g(n)=O(h(n)) • Fact 3 • The function ank=O(nk) • Fact 4 • The function nk=O(nk+j)for any positive j

  13. Properties of Big-O Notation (cont’d) • It follows from those facts that every polynomial is big-O of n raised to the largest power f(n) = aknk + ak-1nk-1 + . . . + a1n1 + a0 = O(nk)

  14. Properties of Big-O Notation (cont’d) • Fact 5 • If f(n)=cg(n), then f(n)=O(g(n)) • Fact 6 • The function logan = O(logbn)for any positive numbers a and b 1 • WHY? • Fact 7 • logan = O(lgn)for any positive a  1, where lg n = log2n

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