lecture3 - IE170: Algorithms in Systems Engineering:...

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Unformatted text preview: IE170: Algorithms in Systems Engineering: Lecture 3 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University January 19, 2007 Jeff Linderoth IE170:Lecture 3 Taking Stock Last Time Lots of funky math Playing with summations: Formulae and Bounds Sets A brief introduction to our friend This Time Questions on Homework? , O and Recursion Analyzing Recurrences Jeff Linderoth IE170:Lecture 3 Comparing Algorithms Consider algorithm A with running time given by f and algorithm B with running time given by g . We are interested in knowing L = lim n f ( n ) g ( n ) What are the four possibilities? L = 0 : g grows faster than f L = : f grows faster than g L = c : f and g grow at the same rate. The limit doesnt exist. Jeff Linderoth IE170:Lecture 3 Notation We now define the set ( g ) = { f : c 1 ,c 2 ,n > 0 such that c 1 g ( n ) f ( n ) c 2 g ( n ) n n } If f ( g ) , then we say that f and g grow at the same rate or that they are of the same order . Note that f ( g ) g ( f ) We also know that if lim n f ( n ) g ( n ) = c for some constant c , then f ( g ) . Jeff Linderoth IE170:Lecture 3 Big- O Notation O ( g ) = { f | constants c,n > 0 s.t. f ( n ) cg ( n ) n n } If f O ( g ) , then we say that f is big-O of g or that g grows at least as fast as f If we say 2 n 2 + 3 n + 1 = 2 n 2 + O ( n ) this means that 2 n 2 + 3 n + 1 = 2 n 2 + f ( n ) for some f O ( n ) (e.g. f ( n ) = 3 n + 1 ). Jeff Linderoth IE170:Lecture 3 Big- Notation ( g ) = { f | constants c,n > s.t. cg ( n ) f ( n ) n n } f ( g ) means that g is an asymptotic lower bound on f f grows faster than g Note f ( g ) f O ( g ) and f ( g ) . f ( g ) g O ( f ) . Jeff Linderoth IE170:Lecture 3 Strict Asymptotic Bounds. Little oh f o ( g ) lim n f ( n ) g ( n ) = 0 f ( g ) g o ( f ) lim n f ( n ) g ( n ) = Note f o ( g ) f O ( g ) \ ( g ) ....
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This note was uploaded on 08/06/2008 for the course IE 170 taught by Professor Ralphs during the Spring '07 term at Lehigh University .

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lecture3 - IE170: Algorithms in Systems Engineering:...

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