Algorithm Analysis. Chapter 3

Transcription

1 Data Structures Dr Ahmed Rafat Abas Computer Sciece Dept, Faculty of Computer ad Iformatio, Zagazig Uiversity

2 Algorithm Aalysis Chapter 3

3 3. Itroductio Algorithm aalysis is a method that measures the efficiecy of: a algorithm, or its implemetatio as a program, whe the iput size becomes large. This method aalyzes: the time required for a algorithm, or its implemetatio as a program, ad the storage space required for a data structure.

4 Estimatig a algorithm's performace ca be carried out through estimatig the umber of basic operatios required by the algorithm to process a iput of a certai size. The term size is the umber of iputs processed. example: whe comparig sortig algorithms, the size of the problem is measured by the umber of records to be sorted. A basic operatio requires a time to be completed, which does ot deped o the values of the operads of this basic operatio. example: addig two itegers is a basic operatio.

5 c Example: compute the ruig time of the followig code: Sum=0; For (i=; i<=; i++) For (j=; j<=; j++) Sum++; Assumig the time required for icremetig the sum variable is c, the ruig time is computed as: T( ) c c This equatio describes the growth rate of the ruig time of the previous code.

6 The cocept of growth rate is importat as it allows us to compare the ruig time of two algorithms without actually writig two programs ad ruig them o the same computer.

7 A graph illustratig the growth rates for five equatios. The horizotal axis represets iput size. The vertical axis ca represet time, space, or ay other measure of cost.

8 Slow T() Expoetial growth rate Quadratic growth rate 5log Logarithmic growth rate 0 Liear growth rate 0 Liear growth rate Fast Orderig of the ruig time correspodig to five growth rate equatios from slow to fast.

9 3. Best, Worst ad Average Cases The best case for a algorithm is the case correspodig to miimum ruig time for this algorithm. The worst case for a algorithm is the case correspodig to maximum ruig time for this algorithm. The average case for a algorithm is the case correspodig to ruig time equal to the average of the best ad the worst cases ruig time. I practice, it is preferred to deal with a worst case aalysis of a algorithm.

10 3.3 Asymptotic Aalysis 3.3. Upper Bouds The upper boud for the ruig time of a algorithm is the upper boud or the highest growth rate the algorithm ca have. The upper bouds are measured o the best case, average-case or worst case iputs. example: - if the growth rate i the worst case for a algorithm is T( ) c, - the this algorithm has a upper boud to its growth rate of i the worst case. -This phrase ca be rewritte usig a special otatio called the big-oh otatio as follows: this algorithm is i O(), i the worst case.

11 3.3. Lower Bouds The lower boud for the ruig time of a algorithm is the least amout of time required by this algorithm for some class of iput. This is a measure of the growth rate of this algorithm. The iput ca be the worst, average, or best-case of size. example: - if the lower boud of the ruig time of a algorithm i the worst case is T( ) c, - the this algorithm has a lower boud to its growth rate of i the worst case. - Alteratively, we ca use the big otatio as follows: this algorithm is i, i the worst case. ()

12 Notatio Whe the upper ad lower bouds are the same withi a costat factor, this case ca be idicated by the use of (big-theta) otatio. example: a algorithm is said to be i ( h( )) ad it is i ( h( )) ( h( )) if it is Note that the word "i" is dropped for otatio, sice there is a strict equality for two equatios with the same.

13 3.3.4 Simplifyig Rules I determiig,, from the ruig time equatio, the followig rules are cosidered:. if f () is i ( g( )) ad g () is i ( h( )), the f () is i ( h( )). This rule holds true for ad otatios.

14 . if f () is i ( Kg( )) for ay costat K 0, the f () is i ( g( )). This rule holds true for ad otatios.

15 3. if f ( ) is i ( g ( )) ad f ( ) is i ( g ( )), the f ) f ( ) is i g ( ), g ( ))) ( (max(. This rule holds true for ad otatios. 4. if f ( ) is i ( g ( )) ad f ( ) is i ( g ( )), the f ) f ( ) is i g ( ) g ( )) (. ( This rule holds true for ad otatios.

16 Example: If 4 3, the () T( ) ( 4 ). T is i

17 3.3.5 Summatios ad Closed Form Solutios Summatio = closed form solutio i f ( i) f () f ()... f ( ) 3 ( ) 3 i, i i i 6 log, i a i i a log,, for 0 a 0

18 i i, 0 a a a i i, for a i i i, 0 i i log log 0 i i Harmoic series i i, e e log log

19 3.4 Calculatig The Ruig Time for a Program Example: a=b; the T () is ().

20 Example: sum=0; for (i=; i<=; i++) sum += ; the T () is ( c c), which is simply ().

21 Example: Sum = 0; for (j=; j<=; j++) for (i=; i<=j; i++) sum ++; for (k=0; k<; k++) A[k] = k; T () The is ( c j cj c) ( ), which ca be simplified to ( c c c) that is ( ).

22 Example: Compare the asymptotic aalysis for the followig two code segmets: sum=0; sum=0; for (i=;i<=;i++) for (i=;i<=;i++) for (j=;j<=;j++) for (j=;j<=i;j++) sum++; sum++; the T ( ) is ( c c ), which is simply ( ). ( ) T is ( c ic), which is c ) i is simply ( ). ( ( ) c that

23 Therefore, both code segmets cost ( ), but the secod requires half the time of the first whe both of them are ru o the same computer.

24 Example: Compare the ruig time of the followig two code segmets: sum=0; sum=0; for (k=; k<=; k*=) for (k=; k<=; k*=) for (j=; j<=; j++) for (j=; j<=k; j++) sum++; sum++; the, T ( ) is ( c c ), which is ( c c ( log )) log i0 that is ( log ). T i ( ) is ( c c ), which is ( c c ( )) that is (). log i0

25 3.5 Space Bouds Storage space is also a importat factor for the program efficiecy. The aalysis techiques used to measure space requiremets are similar to those used to measure time requiremets.

26 Example: The space requiremets for a array of size itegers is ().

27 There are two importat priciples of algorithm desig: The space/time tradeoff this priciple says that oe ca ofte achieve a reductio i time if oe is willig to sacrifice space or vice versa. The disk-based space/time tradeoff this priciple states that the smaller you ca make your disk storage requiremets, the faster your program will ru.