Design and Analysis of ALGORITHM (Topic 2)

Transcription

1 DR. Gatot F. Hertoo, MSc. Desig ad Aalysis of ALGORITHM (Topic 2) Algorithms + Data Structures = Programs Lessos Leared 1

2 Our Machie Model: Assumptios Geeric Radom Access Machie (RAM) Executes operatios sequetially Set of primitive operatios: Arithmetic. Logical, Comparisos, Fuctio calls Simplifyig assumptio: all ops cost 1 uit Elimiates depedece o the speed of our computer, otherwise impossible to verify ad to compare Examples of Basic Operatios Algorithm List Searchig List Sortig Matrix Product Prime Factorisatio Polyomial Evaluatio Tree Traversal Iput Types List with elemets List with elemets x matrices digit umbers degree polyomial Tree with odes Basic Operatios Comparatio Comparatio Scalar Products Scalar Divisio Scalar Products Visitig a ode Notes: The ruig time of a algorithm is determied by its iput size 2

3 Ruig time The ruig time depeds o the iput: a already sorted sequece is easier to sort. Major Simplifyig Covetio: Parameterize the ruig time by the size of the iput, sice short sequeces are easier to sort tha log oes. T A () = time of A o legth iputs Geerally, we seek upper bouds o the ruig time, to have a guaratee of performace. Time Complexity The complexity of a algorithm is determied by the umber of basic operatios ad how may time the algorithm computes those basic operatios. Notes: The complexity aalysis is machie idepedet. Time complexity of a algorithm will determie the ruig time depeds o its iput size, i.e. the time complexity is a fuctio of iput size. Time Complexity maps iput size to time T() executed. 3

4 Purpose To estimate how log a program will ru. To estimate the largest iput that ca reasoably be give to the program. To compare the efficiecy of differet algorithms. To help focus o the parts of code that are executed the largest umber of times. To choose a algorithm for a applicatio. Time Complexity: a example 4

5 Best, Worst ad Average Case Sometimes, give two differet iputs with a same size, a algorithm ca have differet ruig time. Example: Suppose a sortig algorithm has some iputs with a same size but differet order: I ascedig order -Iput 1: 10, 5, 23, 45, 1, 100 Average case -Iput 2: 1,5,10, 23,45, 100 -Iput 3: 100, 45, 23, 10, 5, 1 Best case Worst case Do those iputs give the same ruig time? Best, Worst ad Average Case (cot.) 5

6 Best, Worst ad Average Case (cot.) Best, Worst ad Average Cases (cot.) Let I deote a set of all iput with size of a algorithm ad τ(i) deote the umber of primitive operatios of the correspodig algorithm whe give iput i. Mathematically, we ca defie: Best-case Complexity: is a fuctio B() B() = mi{ τ(i) i I } Worst-case Complexity: is a fuctio W() W() = max{ τ(i) i I } Average-case Complexity: is a fuctio A() A() = τ ( i ). p( i) where p(i) is the probability of i occurs as a iput of a algorithm. i I 6

7 Isertio sort pseudocode INSERTION-SORT (A, ) A[1.. ] for j 2 to do key A[ j] i j 1 while i > 0 ad A[i] > key do A[i+1] A[i] i i 1 A[i+1] = key A: 1 i j key sorted Example of isertio sort 7

8 Example of isertio sort Example of isertio sort 8

9 Example of isertio sort Example of isertio sort 9

10 Example of isertio sort Example of isertio sort 10

11 Example of isertio sort Example of isertio sort

12 Example of isertio sort Example of isertio sort doe 12

13 Isertio Sort: a example 1 Isertio_Sort(A) for j 2 to legth(a) Cost c1 Times 2 key A(j) c i j-1 while i > 0 ad A(i) > key A(i+1) A(i) i i 1 c3 c4 c5 c6 j = t 2 j = t j 2 j = t j 2 j A(i+1) key c7-1 = legth(a) t j = umber of while loop executio for a certai value j Isertio Sort: a aalysis T( ) = c + c ( 1) + c ( 1) + c c 6 1 ( t 2 1) + c ( 1) j= 2 j t + c j= 2 j 5 j= 2 Best case: i a ordered list (i.e t j = 1, for j = 2,, ) T ( ) = ( c1 + c2 + c3 + c4 + c7 ) ( c2 + c3 + c4 + c7 ) Worst case: i a reverse ordered list (i.e t j = j, for j = 2,, ) c4 c5 c6 T ( ) = ( + + ) ( c + c + c + c ) ( c 1 + c 2 + c 3 ( t j 1) + c4 c5 c6 + c7 )

14 Time Complexity: a compariso Machie-idepedet time What is isertio sort s worst-case time? BIG IDEAS: Igore machie depedet costats, otherwise impossible to verify ad to compare algorithms Look at growth of T() as. Asymptotic Aalysis 14

15 Aalysis Simplificatios Igore actual ad abstract statemet costs Order of growth is the iterestig measure: Highest-order term is what couts Remember, we are doig asymptotic aalysis As the iput size grows larger it is the high order term that domiates Assigmet 1 I order to show that a algorithm is ot uique, desig two differet algorithms of a specific problem. Desig a algorithm ad show its time complexity to compute a product of two x matrices (how is the time complexity i the best ad worst cases?) 15