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

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)=3 n – 2 n

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).

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, log k n –Polynomial: n, n 2, n 3, …, n k,.. Untractable problems: –Exponential: 2 n, 3 n, 2 n^2, … –Higher (super-exponential): 2 2^n, 2 n^n, …

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: log b 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(log 2 n)

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

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(log 2 n) time per update operation

Polynomial Time Most efficient: linear time O(n) –Depth-first and breadth-first search –Connectivity, cycles in graphs Good: O(n log n): –Sorting –Convex hulls (in Computational Geometry) Quadratic O(n 2 ) –Shortest paths in graphs –Minimum spanning tree Cubic O(n 3 ) –All pairs shortest path –Matrix multiplication (naïve) O(n 5 )? O(n 6 )? –Robot Motion Planning Problems with few (5, 6,…) degrees of freedom

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

Relative Growth logarithmic polynomial exponential

Largest instance solvable ComplexityLargest in one second Largest in one day Largest in one year n1,000,000 86,400,000,000 31,536,000,000,000 n log n62, 746 2,755,147,514798,160,978,500 n2n ,9385,615,692 n3n3 1004,42131,593 2n2n

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

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