O - en overskuelig matematisk model for vurdering af algoritmers effektivitet

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1 ODatalog C, Efterår 003 Opgørelse af effetvtet, esempel Verson med rettelser 1/ a=a+1 for(nt =1; <=n; ++) Td = 1 (af en slags) for(nt =1; <=n; ++) for(nt =1; <=n; ++) a=a+1; for(nt =1; <=n; ++) Td = n a=a+1; 3 Td = n Generelt: funton af nput (n, m) for(nt =1; <=n; ++) Mere omplceret ved: for(nt =1; <=m; ++) a=a+1; whle-løer Td = n*m f(betngelse) omrng ndre løe ndre løe afh. af ydre 1 Om algortmers effetvtet O - en oversuelg matemats model for vurderng af algortmers effetvtet En velunderbygget teor, som gver estmater der er uafhængge af atuelle computer! Vor ndfaldsfnel: Grundprncpperne og standard-esempler Matemat tl at aratersere effetvtet Illustraton af def. af O(-) Normalt e nteresseret esat tal, men asymptots opførsel (dvs. næ ) Hvs en funton T(n) står for esat tdsforbrug, bruges O(F(n)) tl at aratersere øvre mål for "opførsel" Foreløbg defnton (alternatv tl bogen): T(n) er O(F(n)) hvss T(n)/F(N) Æ onstant (vrer alle fornuftge tlfælde) når næ 3 nt; // td = 5 for(=1;<=n;++) // td = n*17 noget; for(=1;<=n;++) // td = n * 3 for(=1;<n=;++) noget_andet; Total tdsforbrug T(n) = 5 + n*17 + n *3 Påstand: T(n) er O(n ) Udtales også: "algortmen er af orden O(n )" "algortmen er vadrats" Bevs: T(n)/F(n) = T(n)/n = 5/n + n*17/n + n *3/n = 5/n + 17/n + 3 Æ 3 når n Æ Generelt: domnerende led bestemmer O onstant unteressant tl generel araterst størrelsesorden god tl estmater (esempel...) snyder ved små n 4 Esempel: estmat fra n= 1000 tl n=10000 Regneregler om O T(n) = 5 + n*17 + n *3 og F(n) = n T(10 n)/t(n) F(10 n)/f(n), dvs. T(10 n) 100 * T(n) Antag T(1000) = 10 se, dvs. T(1000) = t (hvor t er en passende brødel af et se.) Esat værd for T(10.000) = t Estmeret ved "*100": T(10.000) = 1000se T(10.000) omregnet fra "t" tl se: / * 10 se = 999,9 se Fel 0,1 Defnton: F(n) domnerer over G(n), F(n) > G(n) såfremt G(n)/F(n) Æ 0 når næ I så fald: O(F(n)+G(n)) = O(F(n)) hvor O(H(n)) = O(J(n)) betyder H(n)/J(n) Æ c >0 når næ Esempel: O(5 + n*17 + n *3) = O(n ) Esempler på regler: O(c*F(n)) = O(F(n)) O(n*F(n)) > O(F(n)) O(V(n)*F(n)) > O(F(n)) såfremt O(V(n)) > O(1) 5 6 1

2 Ofte foreommende størrelsesordner O(1) < O(log n) < O((log n) ) < O(n) < O(n log n) < O(n ) < O(n 3 ) <... < O( n ) Små drlse led: O(n + n / ) n T Hvem #"# an fnde den } llle tdrøver nden afleverngsprøven morgen formddag? 7 8 Små drlse led, med endnu mndre onstant: O(n + n / ) n T En orret defnton I tlfælde af en funton/algortme med sær opførsel: Defnton: f(n er lge tal) ; else for(nt =n; <=n; ++) Vl v gerne have tl at være af orden n... men grænseværd e veldef. T(n) er O(F(n)) hvss der fndes postve onstanter c og n 0, så 9 T(n) c F(n) for alle n n 0 10 Esempler på polynomelle alg. n 3, n, n Fnd masmal sum af delsevens Esempel: Esempler på polynomelle alg. n 3, n, n Fnd masmal sum af delsevens Esempel: 4 4 Den enle algortme: Generer samtlgt mulge summer og hold rede på den hdtl største. "Indlysende" ubs, dvs. n 3 1

3 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 14 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum)

4 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 19 0 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 1 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 3 4 4

5 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 5 6 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 7 8 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum)

6 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 31 3 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum)

7 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum)

8 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum)

9 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 51 5 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum)

10 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum)

11 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 61 6 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 65 66

12 Kubs algortme for max-sum-delsevens Kubs algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { for(=; <=; ++) sum += a[]; f(sum>maxsum) 1 + (1+) + (1++3) ( n) = n * (n+1) * (n+) / 6 dvs. O(n 3 ) Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens Udnyt at næste sum = gammel sum + næste led Dvs. Sum,+1 = Sum, + a[+1] for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum)

13 Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) 77 78

14 Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) 81 8 Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum)

15 Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) Kvadrats algortme for max-sum-delsevens Kvadrats algortme for max-sum-delsevens for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) for(=1; <=n; ++) { for(=; <=n; ++) { sum += a[]; f(sum>maxsum) Læs selv om lneær algortme bog: Esempel på at optmerede algortmer an være svært gennemsuelg! n = n * (n-1) / altså O(n )

16 Logartms, typs for del-og-hers algortmer Ldt om logartmefuntonen Esempel: Bnær søgnng for at fnde element sorteret sevens ~ a la telefonbog Prncppet: at fnde x sevens af lgd. 1: test "=x" a/ne ellers, hvs x <= mdt-element, så fnd x venstre halvdel ellers fnd x høre halvdel Tdsforbrug hvert srdt (" " ovenfor) onstant hvert srdt halverer længden af sevensen Total td for lgd. n: så mange gange n sal halveres for at blve "1" log n Defneret som omvendt tl esponentalfuntonen: hvs x =m så log m = x Esempler: n bnært log n Egensaber: Halvernger - Æ 1 Æ Æ 1 Æ 4 Æ Æ 1 Æ 8 Æ 4 Æ Æ 1 fordobles argumentet stger logartmen med 1 log n antal tegn n's bnære repræsentaton log n antal gange n sal halveres for at blve 1 Hvordan var det nu? 91 9 Funtonsurven for log n Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) log n n nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x low hgh low md hgh

17 Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x low hgh low md hgh Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x low hgh low md hgh Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x low hgh low md hgh

18 Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x low hgh md low hgh Bnær søgnng Java I: Reursv for "nt []" Bnær søgnng Java I: Reursv for "nt []" Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x Kaldes: bnarysearch(0,a.length-1,a,x) nt bnarysearch(nt low, nt hgh, nt [] a, nt x){ f(low>hgh) return -1; nt md = (low+hgh)/; f( a[md] < x ) bnarysearch(md+1,hgh,a,x) else f( a[md] > x ) bnarysearch(low,md-1,a,x); else return md; }} //a[md]==x hgh low 105 hgh low md 106 Bnær søgnng Java II: Reursv, geners pacage ava.lang; publc nterface Comparable {n compareto(obect other)}; publc statc nt bnarysearch(comparable [] a, Comparable x){ bnarysearch(0,a.length-1,a,x)} publc statc nt bnarysearch(nt low,nt hgh, Comparable [] a, Comparable x){ f(low>hgh) return -1; nt md = (low+hgh)/ f(a[md].compareto(x)<0) bnarysearch1(md+1,hgh,a,x) else f(a[md].compareto(x)>0) class Elephant mplements Comparable bnarysearch!(low,md-1,a,x); {...; publc nt compareto(...){...} } else return md; }} zoo = new Elephant []{...}; dumbo = new Elephant; nt where_s_dumbo = bnarysearch(zoo,dumbo); Bnær søgnng Java III: Iteratv, geners Lærebogens verson: nt bnarysearch(comparable[]a,comparable Iteratv vs. reursv: x){ nt low = 0; En smagssag, men reursve alg. nt hgh = a.length-1; ofte smplere for af-natur-re. nt md; problemer whle( low<=hgh){ md = (low+hgh)/; f(a[md].compareto(x)<0) Hvad er mest effetvt? low = md+1; Er reurson e meget dyrt? else f(a[md].compareto(x)>0) hgh = md-1; else return md; } return -1; } Det afhænger af compleren! Med en god Java compler, dentse øretder!

19 Afsluttende om O-notaton udvde med flere parametre, f.es. O(m* n ) O-notaton benyttes også for pladsforbrug ofte trade-off plads- vs. tdsomplestet Algortmer har bedste, værste og gennemsntlgt tdsforbrug f.es. qucsort, værst O(n ), gennemsnt O(n log n) "T(n) er O(F(n))" angver en overgrænse for T(n)'s asympttse opførsel; alternatve araterster: "T(n) er W(F(n))" angver undergrænse "T(n) er Q(F(n))" angver esat araterst "T(n) er o(f(n))" angver araterst som er lart for hø Opgaver tl øvelserne Opg. 1: Se på egensaber a la n = n(n 1)/ Opg. 5.7 bogen: Store-O for regnng med blyant og papr Opg. 5.8 m. tlføelse: Store-O for to måder at regne x n på Opg., afleverngsopgave: Implementér to måder at mplementere mængder af tal, med metoder publc vod ndsæt(nt x) publc boolean med_(nt x) 109 Afleverngsfrst 14. otober 0 19