What are we going to learn? CSC Data Structures Analysis of Algorithms. Overview. Algorithm, and Inputs

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Allyson Allison
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1 What are we goig to lear? CSC Data Structures Aalysis of Algorithms Computer Sciece North Carolia State Uiversity Need to say that some algorithms are better tha others Criteria for evaluatio Structure of programs (simplicity, elegace, etc.) Ruig time Memory Space Overview Algorithm, ad Iputs Ruig Time Pseudo-Code Aalysis of algorithms Asymptotic otatios Asymptotic aalysis The ruig time of algorithms typically depeds o the iput set, ad its size (). We describe it usig fuctios of.

Average Case vs. Worst Case Measurig the ruig time The average case behavior is harder to aalyze because you eed to kow a probability distributio of the iput.

2 Average Case vs. Worst Case Measurig the ruig time The average case behavior is harder to aalyze because you eed to kow a probability distributio of the iput. I certai apps (air traffic cotrol, weapo systems,etc.), kowig the worst case time is importat. How should we measure the ruig time of a algorithm? Experimetal Studies Geeral Methodology Ru the program with various data sets i a computer, ad measure the wall clock time. Idepedet of implemetatio, hardware ad software eviromets Actual elapsed time depeds o hardware, software (os), compiler. Use high-level descriptio of the algorithm istead of oe of its implemetatios

3 Geeral Methodology Pseudo-Code Worry about order of magitude cout steps (do t worry about amout of time each step takes) igore multiplicative costats Take ito accout all possible iputs. Worst cast aalysis Pseudo-code is a descriptio of a algorithm for huma-eyes oly (mix of Eglish ad programmig laguages). Example: fidig the maximum elemet of a array. How to cout steps How to cout steps iteratio Commets, declarative statemets (0) expressios ad assigmets (1) except for fuctio calls cost for f eeds to be couted separately ad added to the cost for the callig statemet. Iteratio statemets -- for, while expressio + cout the umber of times that the body is executed, ad the multiply by the cost of body while <expr> do <body of while>

4 How to cout steps switch or if else Example Ruig time of worst case + expressio. switch <expr> case cod1: <statemet1> case cod2: <statemet2>.. Example of aalysis Aother Example Result <-0; m <-1; for i<-1 to m <- m*2; for j<- 1 to m do result <-result + i*m*j

5 Estimatig Ruig Time Growth Rate of Ruig Time Algorithm arraymax executes 3 1 primitive operatios i the worst case Defie a Time take by the fastest primitive operatio b Time take by the slowest primitive operatio Let T() be the actual worst-case ruig time of arraymax. We have a (3 1) T() b(3 1) Hece, the ruig time T() is bouded by two liear fuctios Chagig the hardware/ software eviromet Affects T() by a costat factor, but Does ot alter the growth rate of T() The liear growth rate of the ruig time T() is a itrisic property of algorithm arraymax Growth Rates Costat Factors Growth rates of fuctios: Liear Quadratic 2 Cubic 3 I a log-log chart, the slope of the lie correspods to the growth rate of the fuctio T ( ) 1E+30 1E+28 Cubic 1E+26 1E+24 Quadratic 1E+22 1E+20 Liear 1E+18 1E+16 1E+14 1E+12 1E+10 1E+8 1E+6 1E+4 1E+2 1E+0 1E+0 1E+2 1E+4 1E+6 1E+8 1E+10 The growth rate is ot affected by costat factors or lower-order terms Examples 1E E+10 is a liear 1E+8 fuctio 1E is a 1E+4 quadratic fuctio 1E+2 1E+0 T ( ) 1E+26 1E+24 1E+22 1E+20 1E+18 1E+16 1E+14 Quadratic Quadratic Liear Liear 1E+0 1E+2 1E+4 1E+6 1E+8 1E+10

6 Big-Oh Notatio Big-Oh Notatio (cot.) 10,000 Give fuctios f() ad g(), we say that f() is O(g()) if there are positive costats c ad 0 such that f() cg() for 0 Example: is O() c (c 2) 10 10/(c 2) Pick c = 3 ad 0 = 10 Or Pick c=4 ad 0 = 5 Just prove the existece of c ad 0 1, ,000 Example: the fuctio 2 is ot O() 2 c c The above iequality caot be satisfied sice c must be a costat 1,000, ,000 10,000 1, ^ ,000 Big-Oh ad Growth Rate Classes of Fuctios The big-oh otatio gives a upper boud o the growth rate of a fuctio The statemet f() is O(g()) meas that the growth rate of f() is o more tha the growth rate of g() We ca use the big-oh otatio to rak fuctios accordig to their growth rate Let {g()} deote the class (set) of fuctios that are O(g()) We have {} { 2 } { 3 } { 4 } { 5 } where the cotaimet is strict g() grows more f() grows more Same growth f() is O(g()) No g() is O(f()) No { 3 } { 2 } {}

7 Big-Oh Rules (of Thumb) Asymptotic Algorithm Aalysis If is f() a polyomial of degree d, the f() is O( d ), i.e., 1.Drop lower-order terms 2.Drop costat factors Use the smallest possible class of fuctios Say 2 is O() istead of 2 is O( 2 ) But it is true that 2 is O( 2 ) Use the simplest expressio of the class Say is O() istead of is O(3) But it is true that is O(3) The asymptotic aalysis of a algorithm determies the ruig time i big-oh otatio To perform the asymptotic aalysis We fid the worst-case umber of primitive operatios executed as a fuctio of the iput size We express this fuctio with big-oh otatio Example: We determie that algorithm arraymax executes at most 3 1 primitive operatios We say that algorithm arraymax rus i O() time Sice costat factors ad lower-order terms are evetually dropped ayhow i big-oh, we ca disregard them whe coutig primitive operatios Cautio! Relatives to Big-Oh s (Left for Exercise) Beware of very large costat factors. A algorithm ruig i time 1,000,000 is still O(), but might be less efficiet o your everyday data set tha oe ruig i time 2 2, which is O( 2 ). Big Omega(Ω(f())), Big-Theta (Θ (f()))

Outline and Reading. Analysis of Algorithms. Running Time. Experimental Studies. Limitations of Experiments. Theoretical Analysis

Outline and Reading. Analysis of Algorithms. Running Time. Experimental Studies. Limitations of Experiments. Theoretical Analysis Outlie ad Readig Aalysis of Algorithms Iput Algorithm Output Ruig time ( 3.) Pseudo-code ( 3.2) Coutig primitive operatios ( 3.3-3.) Asymptotic otatio ( 3.6) Asymptotic aalysis ( 3.7) Case study Aalysis