CS473-Algorithms I. Lecture 2. Asymptotic Notation. CS 473 Lecture 2 1

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1 CS473-Algorithms I Lecture Asymptotic Notatio CS 473 Lecture 1

2 O-otatio (upper bouds) f() = O(g()) if positive costats c, 0 such that e.g., = O( 3 ) 0 f() cg(), 0 c 3 c c = 1 & 0 = or c = & 0 = 1 Asymptotic ruig times of algorithms are usually defied by fuctios whose domai are N={0, 1,, } (atural umbers) CS 473 Lecture

3 O-otatio (upper bouds) = is fuy; oe-way equality O-otatio is sloppy, but coveiet though sloppy, must uderstad what really meas thik of O(g()) as a set of fuctios: O(g()) = {f(): positive costats c, 0 such that 0 f() cg(), 0 } hece, = O( 3 ) meas that O( 3 ) CS 473 Lecture 3

4 O-otatio O-otatio is a upper-boud otatio e.g., makes o sese to say ruig time of a algorithm is at least O( ). Why? - let ruig time be T() - T() O( ) meas T() h() for some h() O( ) - however, this is true for ay T() sice h() = 0 O( ), & ruig time > 0, so stmt tells othig about ruig time CS 473 Lecture 4

5 O-otatio (upper bouds) f() = O(g()) cg() f() 0 CS 473 Lecture 5

6 -otatio (lower bouds) f() =(g()) if positive costats c, 0 such that 0 cg() f(), 0 e.g., = (lg ) (c = 1, 0 = 16) i.e., 1 lg 16 (g()) = {f(): positive costats c, 0 such that 0 cg() f(), 0 } CS 473 Lecture 6

7 -otatio (lower bouds) f() = (g()) f() cg() 0 CS 473 Lecture 7

8 CS 473 Lecture 8 -otatio (tight bouds) f()=(g()) if positive costats c 1, c, 0 such that 0 c 1 g() f() c g(), 0 example: ) ( 1 c c c c

9 -otatio: example (0 c 1 h() c ) 1/ h() =1/-/ 1/ ( 0 ) CS 473 Lecture 9

10 CS 473 Lecture 10 5, 1, 10 1 therefore 5, ) ( 0, 1 1 ) ( c c c h c h -otatio: example (0 c 1 h() c )

11 -otatio (tight bouds) (g())={f(): positive costats c 1, c, 0 such that 0 c 1 g() f() c g(), 0 } c g() f() c 1 g() 0 CS 473 Lecture 11

12 -otatio (tight bouds) Prove that 10-8 () - suppose c, 0 exist such that 10-8 c, 0 - but the c cotradictio sice c is a costat Theorem: leadig costats & low-order terms do t matter Justificatio: ca choose the leadig costat large eough to make high-order term domiate other terms CS 473 Lecture 1

13 -otatio (tight bouds) Theorem: (O ad ) - is stroger tha both O ad - i.e., (g()) O(g()) ad (g()) (g()) CS 473 Lecture 13

14 Usig asymptotic otatio for describig ruig times O-otatio used to boud worst-case ruig times also bouds ruig time o arbitrary iputs as well e.g., O( ) boud o worst-case ruig time of isertio sort also applies to its ruig time o every iput CS 473 Lecture 14

15 Usig O-otatio for describig ruig times Abuse to say ruig time of isertio sort is O( ) - for a give, actual ruig time depeds o particular iput of size - i.e., ruig time is ot oly a fuctio of - however, worst-case ruig time is oly a fuctio of CS 473 Lecture 15

16 Usig O-otatio for describig ruig times What we really mea by ruig time of isertio sort is O( ) - worst-case ruig time of isertio sort is O( ) or equivaletly - o matter what particular iput of size is chose (for each value of) ruig time o that set of iputs is O( ) CS 473 Lecture 16

17 Usig -otatio for describig ruig times used to boud the best-case ruig times also bouds the ruig time o arbitrary iputs as well e.g., () boud o best-case ruig time of isertio sort ruig time of isertio sort is () CS 473 Lecture 17

18 Usig -otatio for describig ruig times ruig time of a algorithm is (g()) meas - o matter what particular iput of size is chose (for ay ), ruig time o that set of iputs is at least a costat times g(), for sufficietly large - however, it is ot cotradictory to say worst-case ruig time of isertio sort is ( ) sice there exists a iput that causes algorithm to take ( ) time CS 473 Lecture 18

19 Usig -otatio for describig ruig times 1) used to boud worst-case & best-case ruig times of a algorithm if they are ot asymptotically equal ) used to boud ruig time of a algorithm if its worst & best case ruig times are asymptotically equal CS 473 Lecture 19

20 Usig -otatio for describig ruig times Case (1): a -boud o worst-/best-case ruig time does ot apply to its ruig time o arbitrary iputs e.g., ( ) boud o worst-case ruig time of isertio sort does ot imply a ( ) boud o ruig time of isertio sort o every iput sice T() = O( ) & T() = () for isertio sort CS 473 Lecture 0

21 Usig -otatio for describig ruig times Case (): implies a -boud o every iput - e.g., merge sort T() = O(lg) T() = (lg) T() = (lg) CS 473 Lecture 1

22 Asymptotic otatio i equatios Asymptotic otatio appears aloe o RHS of a equatio - meas set membership - e.g., = O( ) meas O( ) Asymptotic otatio appears o RHS of a equatio - stads for some aoymous fuctio i the set - e.g., = + () meas that = + h(), for some h() () i.e., h() = CS 473 Lecture

23 Asymptotic otatio appears o LHS of a equatio stads for ay aoymous fuctio i the set - e.g., + () = ( ) meas that for ay fuctio g() () some fuctio h() ( ) such that +g() = h(), RHS provides coarser level of detail tha LHS CS 473 Lecture 3

24 Other asymptotic otatios o-otatio upper boud provided by O-otatio may or may ot be tight - e.g., boud = O( ) is asymptotically tight boud = O( ) is ot asymptotically tight o-otatio deotes a upper boud that is ot asymptotically tight CS 473 Lecture 4

25 o-otatio o(g()) = {f(): for ay costat c 0, a costat 0 0 such that 0 f() < cg(), 0 } Ituitively, lim f ( ) g( ) 0 - e.g., = o( ), ay positive c satisfies - but o( ), c does ot satisfy CS 473 Lecture 5

26 -otatio deotes a lower boud that is ot asymptotically tight (g()) = {f(): for ay costat c 0, Ituitively lim a costat 0 0 such that 0 cg() < f(), 0 } f ( ) g ( ) - e.g., / = (), ay c satisfies - but / ( ), c1/ does ot satisfy CS 473 Lecture 6

27 Asymptotic compariso of fuctios similar to the relatioal properties of real umbers - Trasitivity: (holds for all) e.g., f() = (g()) & g() = (h()) f() = (h()) - Reflexivity: (holds for, O, ) e.g., f() = O(f()) - Symmetry: (holds oly for ) e.g., f() = (g()) g() = (f()) - Traspose symmetry: ((O ) ad (o )) e.g., f() = O(g()) g() = (f()) CS 473 Lecture 7

28 Aalogy to the compariso of two real umbers f() = O(g()) a b f() = (g()) a b f() = (g()) a = b f() = o(g()) a < b f() = (g()) a > b CS 473 Lecture 8

29 Aalogy to the compariso of two real umbers Trichotomy property of real umbers does ot hold for asymptotic otatio - i.e., for ay two real umbers a ad b, we have either a < b, or a = b, or a > b - i.e., for two fuctios f() & g(), it may be the case that either f() = O(g()) or f() = (g()) holds - e.g., ad 1+si() caot be compared asymptotically CS 473 Lecture 9