Ch3. Asymptotic Notation
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1 Ch. Asymptotic Notatio copyright 006
2 Preview of Chapters Chapter How to aalyze the space ad time complexities of program Chapter Review asymptotic otatios such as O, Ω, Θ, o for simplifyig the aalysis Chapter 4 Show how to measure the actual ru time of a program by usig a clockig method
3 Bird s eye view I this chapter We review asymptotic otatios: O, Ω, Θ, o The otatios are for makig statemet about program performace whe the iput data is large Big-Oh O is the most popular asymptotic otatio Asymptotic otatios will be itroduced i both iformal ad rigorous maer
4 Table of Cotets Itroductio Asymptotic Notatio & Mathematics Complexity Aalysis Example Practical Complexities 4
5 Itroductio (1/ Reasos to determie operatio cout ad step cout To predict the growth i ru time To compare the time complexities of two programs Facts of the previous two approaches The operatio cout method igores all others except key operatios The step cout method overcome the above shortage, but the otio of step is iexact xy ad xy+z+(y/z treated as a same step? Two aalysts may arrive at 4 + ad for the same program Asymptotic aalysis focuses o determiig the biggest terms (but ot their coefficiet i the complexity fuctio. 5
6 Itroductio (/ c c c If the step cout is, coefficiets ad term caot give ay particular meaigs whe the istace size is large c c c c c ( + 0 c c lim c is importat whe is c1 is domiat factor whe is large very large! Let 1 ad be two large values of the istace size t( t( 1 c11 1 ( c1 We ca coclude that if the istace size is doubled, the rutime icreases by a factor of 4 6
7 Itroductio (/ Program A + or + Program B 8 or 4 Whe is large, program B is faster tha program A 7
8 Table of Cotets Itroductio Asymptotic Notatio & Mathematics Complexity Aalysis Example Practical Complexities 8
9 Asymptotic Notatio: cocepts Defiitio q( p( lim 0 p( is asymptotically bigger tha q( q( is asymptotically smaller tha p( p( ad q( is asymptotically equal iff either is asymptotically bigger tha the other 9
10 Asymptotic Notatio: terms Commoly occurrig terms 1 < log < < log < < < <! 10
11 Asymptotic Notatio: Big Oh f ( O( lim 0 or Costat C f( is big oh of f ( The above otatio meas that f( is asymptotically smaller tha or equal to, multiplied by some costat C gives a asymptotic upper boud for f( The otatio gives o clue to the value of this costat C, it oly states that it exists 11
12 Big Oh arithmetic Defiitio f ( O( iff exist such that f ( Cosider f( + c 4, 0 positive costats c ad c for all, Whe the f( < 4 f( O(, therefore 0 0 f( is bouded above by some fuctio at all poits to the right of 0 1
13 f ( O( 0 is ay iteger greater tha m 1
14 14 Big Oh example , ( 4 ( c f + + Ο + c f , ( 4 ( Ο + Θ Big O gives us a upper boud, but does ot promise a careful (tight upper boud!!!
15 Asymptotic Notatio: Big Theta Theta(Θ Notatios f ( Θ( f ( c1 lim c f( is theta of f( is asymptotically equal to is a asymptotic tight boud for f( 15
16 Big Theta Defiitio f ( Θ( iff positive costats such that c 1 c 1 ad c f ( c ad a 0 exist for all, 0 Example : Proof we have :choose c Θ(, c ad for all, 0 f( is bouded above ad below by some fuctio at all poits to the right of 0 16
17 f ( Θ( iff c1g ( f ( c 17
18 18 Big Theta arithmetic Example 1 that such 1 0 ( ,, ( ( 1, ( > Θ + + c c whe f f
19 Asymptotic Notatio: Big Omega Omega(Ω Notatios f f ( lim or Cost. ( Ω( f( is omega of f( is asymptotically bigger tha or equal to is a asymptotic lower boud for f( It is the reverse of big-o otatio 19
20 Defiitio Big Omega arithmetic f ( Ω( iff exist such that f positive costats c ad ( c for all, 0 0 f(+ > for all, So f ( Ω( f( is bouded below by a fuctio at all poits to the right of 0 0
21 f ( Ω( iff f ( c 1
22 Little Oh Notatio (o Defiitio f ( o( iff f ( O( ad f ( Ω( Upper boud that is ot asymptotically tight Example + o( as + O( ad + Ω(
23 Big oh ad Little oh Big O otatio may or may ot be asymptotically tight O( : tight vs. O( : ot tight We use little o otatio to deote a upper boud that is ot asymptotically tight Ex : o( o(
24 Leged i Asymptotic Notatio Θ Ο Roughly f( meas f( Roughly f( meas f( < I geeral we use eve though we get! Ο Ο I fact, is a kid of Aother reaso: I geeral, fidig is difficult! Ο Θ Our textbook will use ad iterchageably! Θ Θ Θ 4
25 Table of Cotets Itroductio Asymptotic Notatio & Mathematics Complexity Aalysis Example Practical Complexities 5
26 Sequetial Search Expressio of step cout as a asymptotic otatio (program.1 Igore terms without! Best case t Worst case t sequetialsearch sequetialsearch ( Ω(1 ( O( > > lower boud is1 upper boud is I fact, the worst case is Big-Theta( : Remember Big O is a kid of Big-Theta 6
27 Example.4 Biary Search public static it biarysearch(comparable [] a, Comparable x {// Search a[0] < a[1] <... < a[a.legth-1] for x. it left 0; it right a.legth - 1; while (left < right { it middle (left + right/; if (x.equals(a[middle] retur middle; if (x.compareto(a[middle] > 0 left middle + 1; else right middle - 1; } retur -1; // x ot foud } Each iteratio of the while loop Decrease i search space by a factor about Best case complexity Ω(1 Worst case complexity Θ(log a.legth // because we have to go dow to leaf odes! 7
28 Table of Cotets Itroductio Asymptotic Notatio & Mathematics Complexity Aalysis Example Practical Complexities 8
29 Practical Complexities 1,000,000,00 istructios per secod computer To execute a program of complexity f( 9
30 Summary Big-O upper boud Big-theta tight (upper & lower boud Big-omega lower boud Ο( Ο( Ο( Θ( Θ( Θ( Ω( Ω( Ω( 0
31 Summary I this chapter We reviewed asymptotic otatio O, Ω, Θ, o Asymptotic otatio is for makig statemet about program performace whe the iput data is large Big O otatio is the most popular asymptotic otatio Asymptotic otatios were itroduced i both iformal ad rigorous maer 1
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