Growth of Functions. Chapter 3. CPTR 430 Algorithms Growth of Functions 1
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1 Growth o Fuctios Chapter 3 CPTR 430 Alorithms Growth o Fuctios 1
2 Asymptotic Eiciecy o Alorithms Idea: Look at iput sizes lare eouh to make rui time order o rowth relevat How does the rui time o a alorithm icrease as the problem size icreases without boud? Geerally, a asymptotically more eiciet alorithm will be the better choice or all but very small problem sizes Other asymptotic measures are used besides Θ CPTR 430 Alorithms Growth o Fuctios 2
3 1 3 Fuctio Domai The domai is the set o atural umbers, : Sometimes the domai is exteded to iclude the real umbers ) ( or a subset o CPTR 430 Alorithms Growth o Fuctios 3
4 Θ-otatio Actually represets a set o uctios: Θ there exist positive costats c 1 such that 0 c1 c2 c2 ad 0 or all 0 While Θ is more correct, we usually say Θ Θ bouds a uctio rom above ad below CPTR 430 Alorithms Growth o Fuctios 4
5 O-otatio Also represets a set o uctios: O there exist positive costats c ad 0 or all such that 0 c 0 O bouds a uctio rom above CPTR 430 Alorithms Growth o Fuctios 5
6 Ω-otatio Also represets a set o uctios: Ω there exist positive costats c ad 0 or all such that 0 c 0 Ω bouds a uctio rom below For ay two uctios ad : Θ O ad Ω CPTR 430 Alorithms Growth o Fuctios 6
7 Asymptotic Equality i lim 1 Examples: x 2 x x 2 3x x 4 2x 2 Θ x 2, but 2x 2 x 2 CPTR 430 Alorithms Growth o Fuctios 7
8 3 3 Asymptotic Notatio o the Riht Side Whe asymptotic otatio appears o the riht side o a equatio, it elimiates oessetial detail: Θ meas where Θ CPTR 430 Alorithms Growth o Fuctios 8
9 o-otatio o or ay costat c 0 there exists a costat 0 such that 0 c or all 0 o is ot asymptotically tiht o lim 0 CPTR 430 Alorithms Growth o Fuctios 9
10 ω-otatio ω 0 0 c or ay costat c 0 such that or all 0 there exists a costat 0 Said aother way: ω o Or ω lim CPTR 430 Alorithms Growth o Fuctios 10
11 Properties o Asymptotic Fuctios Trasitivity Relexivity Symmetry Traspose symmetry CPTR 430 Alorithms Growth o Fuctios 11
12 Trasitivity Θ ad Θ h Θ h O ad O h O h Ω ad Ω h Ω h o ad o h o h ω ad ω h ω h CPTR 430 Alorithms Growth o Fuctios 12
13 Relexivity Θ O Ω CPTR 430 Alorithms Growth o Fuctios 13
14 Symmetry Θ Θ CPTR 430 Alorithms Growth o Fuctios 14
15 Traspose symmetry O o Ω ω CPTR 430 Alorithms Growth o Fuctios 15
16 Compariso to Where a b correspodi to uctios ad respectively: O Ω Θ o ω a a ab ab a b b b But the trichotomy property does ot hold: For ay a b ab, b, or a a b exactly oe o the ollowi must hold: CPTR 430 Alorithms Growth o Fuctios 16
17 Basic Mathematics Mootoicity Floor/ceili Modular arithmetic Polyomials Expoetials/loarithms Factorials Fuctioal iteratio Fiboacci umbers CPTR 430 Alorithms Growth o Fuctios 17
18 Mootoicity m A uctio is mootoically icreasi i: m m (For mootoically decreasi chae m to m ) A uctio is strictly icreasi i: m m (For mootoically decreasi chae m to ) CPTR 430 Alorithms Growth o Fuctios 18
19 Floors ad Ceilis For ay x loor o x, the reatest iteer less tha or equal to x is x, the For ay x ceili o x, the least iteer reater tha or equal to x is x, the The loor ad ceili uctios are mootoically icreasi For all : 2 2 CPTR 430 Alorithms Growth o Fuctios 19
20 mod Modular Arithmetic For ay a quotiet a, 0, the value a mod is the remaider o the a mod a a a mod b mod is writte a b a b mod is a divisor o b a CPTR 430 Alorithms Growth o Fuctios 20
21 Polyomials For d orm d where costats a 0 a d 0 0 a polyomial i o deree d is a uctio p a1 p d a i i i 0 o the ad are coeiciets o the polyomial, ad I a d 0, the the polyomial is asymptotically positive Asymptotically positive polyomial p a a Fuctio costat k a 0 a 0 o deree dp uctio a is mootoically icreasi uctio a is mootoically decreasi is polyomially bouded i O k Θ d or some CPTR 430 Alorithms Growth o Fuctios 21
22 m Expoetials For all a a 0: a 0 a 1 a 1 a m a m a m a 1 a 1 a am a m am For all ad a 1, a is mootoically icreasi i Whe coveiet we ll let CPTR 430 Alorithms Growth o Fuctios 22
23 Expoetial vs. Polyomial Growth For all costats a b such that a 1: b lim a 0 b o a Ay expoetial uctio with base reater tha 1 rows aster tha ay polyomial uctio CPTR 430 Alorithms Growth o Fuctios 23
24 Loarithms A loarithm is the iverse o a expoetial: y bx lob y x Notatios: l l lo l k ll lo2 biary loarithm loe atural loarithm lo10 commo loarithm k l expoetiatio l l compositio l k meas l k CPTR 430 Alorithms Growth o Fuctios 24
25 b Lo Relatioships For all a c, where a 0 b 0 c 0: a lo c ab lo b a lo b a lo b 1 a lo b a a lo b c blob a loc a loc b lob a lo c a lo c b lob a 1 lo a b clob a CPTR 430 Alorithms Growth o Fuctios 25
26 The chae o base equatio: Loarithm Bases lo b a lo c a lo c b is particularly useul: Chai the base o a alorithm rom oe costat to aother chaes the value o the loarithm by a costat actor I O-otatio we do t care about costat actors, so l is ote used i O-otatio For us, the most atural base o loarithms is 2 May alorithms split a problem ito two subproblems CPTR 430 Alorithms Growth o Fuctios 26
27 Polyloarithmic Boudi We say uctio is polyloarithmically bouded i: O l k or some costat k lim l b 2 a l l lim b a 0 Thus, l b o a, or ay costat a 0 Ay positive polyomial uctio rows aster tha ay polyloarithmic uctio CPTR 430 Alorithms Growth o Fuctios 27
28 Factorials Deied or o-eative iteers:! 1 i 1! i 0 0 Some relatioships:!!! o ω 2 l! Θ l CPTR 430 Alorithms Growth o Fuctios 28
29 Fuctioal Iteratio i meas the uctio iteratively applied i times to a iitial value Let be a uctio over ad i i 0: i i 1 i i i i 0 0 For example, i 2, the , so i 2i CPTR 430 Alorithms Growth o Fuctios 29
30 Iterated Loarithms Iterated loarithm: l (read lo star o ) l i is as deied i uctioal iteratio ( l) l i is deied oly i l i 1 0 l mi i 0 l i 1 CPTR 430 Alorithms Growth o Fuctios 30
31 Growth o Iterated Loarithms I a word, slow! l 2 l 4 l 16 l l Number o atoms i observable uiverse CPTR 430 Alorithms Growth o Fuctios 31
32 Fiboacci Numbers F 0 F 1 F i 0 1 Fi 1 Fi 2 or i 2 Deies the sequece Related to the olde ratio φ: φ φ F i φ i 5 φ i CPTR 430 Alorithms Growth o Fuctios 32
33 Fiboacci Numbers Grow Expoetially F i φ i 5 φ i φ i 5 φ i 5 φ1 5 φi Fi φ i 5 δ, where δ 1 2 Fi roud φ i 5 CPTR 430 Alorithms Growth o Fuctios 33
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