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Lecture 3

Lecture 3. Analysis of Algorithms, Part II. Plan for today. Finish Big Oh , more motivation and examples, do some limit calculations. Little Oh, Theta notation Start Recurrences . Ex.1 ch.1 in textbook Tower of Hanoi. Binary search recurrence.

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Lecture 3

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  1. Lecture 3 Analysis of Algorithms, Part II

  2. Plan for today • Finish Big Oh, more motivation and examples, do some limit calculations. • Little Oh, Theta notation • Start Recurrences. • Ex.1 ch.1 in textbook • Tower of Hanoi. • Binary search recurrence

  3. An example of Proof by Induction for a Recursive Program • Ex 1 Ch 1 textbook • Function: int g(int n) if n<=1 return n else return 5 g(n-1) + 6 g(n-2) • Prove that g(n)=3n – 2n

  4. More about Big Oh • lim a(n)/b(n) = constant if and only if a(n) < c b(n) for all n sufficiently large • This allows to estimate, when we do not know how to compute limits. • Example: the Harmonic numbers H(n) = 1+1/2+1/3+1/4+…+1/n < 1+1+…+1=n Hence H(n)=O(n).

  5. Complexity Classes • Big Oh allows to talk about algorithms with asymptotically comparable running times • Most important in practice (tractable problems): • Logarithmic and poly-logarithmic: log n, logkn • Polynomial: n, n2, n3, …, nk, .. • Untractable problems: • Exponential: 2n, 3n, 2n^2, … • Higher (super-exponential): 22^n, 2n^n, …

  6. Most efficient: logarithmic and poly-logarithmic • Functions which are O(log n) or polynomials in log n. • Examples: • Base of the logarithm is irrelevant inside Big Oh: logb n (any base b) = O(log n) • Harmonic series: 1+1/2+1/3+…+ 1/n = O(log n) • Most efficient data structures have search time O(log n) or O(log2n)

  7. Upper bound for Harmonic series • Hn=1+1/2+1/3+…+1/n < 1+1+…+1 = n • So Hn = O(n), but this estimate is not good enough • To prove Hn = O(log n) need more advanced calculus (Riemann integrals) 1/x

  8. Examples • Binary search: main paradigm, search with O(log n) time per operation • Binary search trees: worst case time is linear, but on average search time is log n • B-trees (used in databases) and B+ trees • AVL-trees • Red-black trees • Splay trees, finger trees • Dynamic convex hull algorithms: O(log2n) time per update operation

  9. Polynomial Time • Most efficient: linear timeO(n) • Depth-first and breadth-first search • Connectivity, cycles in graphs • Good: O(n log n): • Sorting • Convex hulls (in Computational Geometry) • QuadraticO(n2) • Shortest paths in graphs • Minimum spanning tree • CubicO(n3) • All pairs shortest path • Matrix multiplication (naïve) • O(n5)? O(n6)? • Robot Motion Planning Problems with few (5, 6,…) degrees of freedom

  10. Exponential time • Towers of Hanoi O(2n) • Travelling Salesman • Satisfiability of boolean formulas and circuits Super-exponential time • Many Robot Motion Planning Problems with many (n) degrees of freedom O(22^n)

  11. Relative Growth exponential polynomial logarithmic

  12. Largest instance solvable

  13. Big Omega and Theta • Big Oh and Small Oh give Upper bounds on functions • Reverses the role in Big Oh: • f(n) = Omega(g(n)) iff g(n) = O(f(n)) lim g(n) / f(n) < constant • Big Oh gives upper bounds, Omega notation gives lower bounds • Theta: both Big Oh and Omega

  14. Examples • f(n) = n2+2n-10 and g(n)=3n2-23 are Theta(n2) and Theta of each other. Why? • f(n)=log2 n and g(n)=log3 n are Theta of each other (logarithm base doesn’t count) • f(n)=2n and g(n)=3n are NOT Theta of each other. f(n)=o(g(n)), grows much slower. Because lim 2n/3n=0.

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