Introducction to Algorithm
- 1. INTRODUCTION TO ALGORITHM Prepared by R. Gomthijayam Asst. Prof. of CA
- 2. Algorithms • The word algorithm comes from the name of a Persian mathematician Abu Ja’far Mohammed ibn-i Musa al Khowarizmi. • In computer science, this word refers to a special method useable by a computer for solution of a problem. The statement of the problem specifies in general terms the desired input/output relationship. • For example, sorting a given sequence of numbers into nondecreasing order provides fertile ground for introducing many standard design techniques and analysis tools.
- 3. The problem of sorting
- 4. Asymptotic Notation The notation we use to describe the asymptotic running time of an algorithm are defined in terms of functions whose domains are the set of natural numbers ...,2,1,0N
- 5. O-notation • For a given function , we denote by the set of functions • We use O-notation to give an asymptotic upper bound of a function, to within a constant factor. • means that there existes some constant c s.t. is always for large enough n. )(ng ))(( ngO 0 0 allfor)()(0 s.t.andconstantspositiveexistthere:)( ))(( nnncgnf ncnf ngO ))(()( ngOnf )(ncg)(nf
- 6. Ω-Omega notation • For a given function , we denote by the set of functions • We use Ω-notation to give an asymptotic lower bound on a function, to within a constant factor. • means that there exists some constant c s.t. is always for large enough n. )(ng ))(( ng 0 0 allfor)()(0 s.t.andconstantspositiveexistthere:)( ))(( nnnfncg ncnf ng ))(()( ngnf )(nf )(ncg
- 7. -Theta notation • For a given function , we denote by the set of functions • A function belongs to the set if there exist positive constants and such that it can be “sand- wiched” between and or sufficienly large n. • means that there exists some constant c1 and c2 s.t. for large enough n. )(ng ))(( ng 021 021 allfor)()()(c0 s.t.and,,constantspositiveexistthere:)( ))(( nnngcnfng nccnf ng )(nf ))(( ng 1c 2c )(1 ngc )(2 ngc Θ ))(()( ngnf )()()( 21 ngcnfngc
- 8. Standard notations and common functions • Logarithms: )lg(lglglg )(loglog logln loglg 2 nn nn nn nn kk e
- 9. Standard notations and common functions • Factorials For the Stirling approximation: ne n nn n 1 12! 0n )lg()!lg( )2(! )(! nnn n non n n
- 10. Performance Measurement Needs programming language working program computer compiler and options to use javac -o