Lecture4 - Mathematical Background and Running Time...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Mathematical Background and Running Time Analysis EECS 233-2- Previous Lecture: Mathematical Defnitions T ( N ) = O ( f ( N )) if there are positive constants c and n such that T ( N )<= cf ( N ) when N >= n T ( N ) = ( f ( N )) if there are positive constants c and n such that T ( N )>= cf ( N ) when N >= n T ( N ) = ( f ( N )) iff T ( N ) = O ( f ( N )) and T ( N ) = ( f ( N )) T ( N ) = o ( f ( N )) if for all constants c there exists an n such that T ( N )< cf ( N ) when N > n-3- How to Determine the Relative Growth Rate? If lim N->infinity T ( N ) / f ( N ) = 0: T ( N ) = o ( f ( N )) = c != 0: T ( N ) = ( f ( N )) = infinity: f ( N ) = o ( T ( N )) If no convergence <= c : T ( N ) = O ( f ( N )) T ( N ) = N f ( N ) = N if N is odd, or N 2 if N is even >= c : f ( N ) = O ( T ( N )) Otherwise (could be > c and < c for any c ): no relation T ( N ) = N f ( N ) = 1 if N is odd, or N 2 if N is even T ( N ) = o ( f ( N )) if for all constants c there exists an n such that T ( N )< cf ( N ) when N > n . T ( N ) = ( f ( N )) if there are positive constants c and n such that T ( N )<= cf ( N ) when N >= n AND there are positive constants c* and n* such that T ( N )>= cf ( N ) when N >= n* T ( N ) = O ( f ( N )) if there are positive constants c and n such that T ( N )<= cf ( N ) when N >= n-4- Logarithm Properties Non-negative, monotonic function for N 1 Sub linear growth: Basic equivalencies: Corollary: logarithm grows slower than any polynomial! log( N ) = o ( N x ) " log(log( N ) = o (log( N x ) = o ( x log( N )) " log Y = o ( xY ) = o ( Y ) log( X ) = o ( X )-5- Relative Growth Rates Examples (which grows faster?) 1000000 versus 0.01*sqrt( N ) log( N ) versus sqrt( N ) N log( N ) versus N 1.001 N 3 versus 10000* N 2 log 2 ( N ) versus 10*log( N 5 ) 2*log 2 ( N ) versus log 3 ( N ) N *2 N versus 3 N 10*log( N 5 ) = 50*log( N ) 3 N = 1.5 N * 2 N N 1.001 = N * N 0.001 -6-...
View Full Document

Page1 / 22

Lecture4 - Mathematical Background and Running Time...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online