Design and Analysis of Algorithms

Transcription

1 Desg ad Aalyss of Algorthms.. ก ก ก 54 ก ก ก กกก Algorthm Desg echques Greedy Algorthms Outle Geeral Idea Actvty-Selecto Problem wo Key Igredets More Problems : Kapsac, Huffma Codes Cocluso Caddate Set Algorthm emplate Why Greedy? Greedy( C ) Soluto Set S = empty whle (C empty) ad (! Soluto(S)) x = Select( C ) C = C - {x} f ( Feasble( S U {x} ) ) S = S U {x} f Soluto( S ) the retur S else No soluto Mae the choce that loos best at the momet Mae locally optmal choce Hope that ths choce wll lead to a globally optmal soluto Do ot always eld optmal solutos

2 s f Actvty-Selecto Problem Actvty Startg Fsh Brute Force ry all subsets of actvtes Choose the largest subset whch s feasble Ieffcet : "( ) choces Greedy Algorthm 5 Greedy Algorthm 5 Sort by fsh tmes Greedy Algorthm 5 Greedy Algorthm j = = = 4 j = 4 5 = 5 6 = 6

3 Greedy Algorthm Provg Optmalty Greedy_Actvty_Select( s[..], f[..] ) A = { } j = ; for = to f s[] >= f[j] A = A U { } j = retur A "( ) excludg sort Let f f Actvty # must be a optmal soluto 5 Provg Optmalty Provg Optmalty Let f f 5 Let f f Greedy choce produces a optmal soluto. 5 wo Key Igredets Optmal substructures a optmal soluto to the problem cotas wth t optmal solutos to subproblems Greedy-choce property a globally optmal soluto ca be arrved at by mag a locally optmal (greedy) choce. Greedy-Choce Property Mae whatever choce seems best at the momet Choce made caot deped o ay future choces Must prove that a greedy choce at each step elds a globally optmal soluto

4 Kapsac Problem tems ad a apsac tem -th s worth v ad weght w the apsac ca tae weght of at most W What tems should be tae to get the most valuable load? w Kapsac Problem v 6 W = 5 / Kapsac Fractoal Kapsac Kapsac : Optmal Substructure Kapsac : Optmal Substructure W W - w v + v + v 4 + v 7 (max.) v + v 4 + v 7 (max.) W W - w Fractoal Kapsac Fractoal Kapsac v = w = v = w = v = 66 w = v 4 = 4 w 4 = 4 v 5 = 6 w 5 = 5 v = w = v = w = v = 66 w = v 4 = 4 w 4 = 4 v 5 = 6 w 5 = $v = $v = 46

5 v = v = w = w = v /w..5 Fractoal Kapsac v = 66 w = v 4 = 4 w 4 = 4.. $v = 64 v 5 = 6 w 5 = Fractoal Kapsac : Greedy Alg. Greedy_Kapsac( v[..],w[..],w ) sort v, w by v/w (ocreasg) x[ ] =, wght =, = whle ( <= && wght < W ) dw = W - wght f dw < w[] the x[] = dw/w[] else x[] = wght += x[] * w[] ++ retur x Provg Optmalty Let v /w J v /w J... J v /w Let x be the soluto vector $ x v Let y be ay feasble soluto vector $ y v ( x y ) v J 4... Optmal Provg Optmalty... x... x < v v ( x w J ( x v ( x ) v v Г ( x y ) ( x y ) v v Г ( x J Г w ( x Г Provg Optmalty... x... x < / Kapsac : Greedy doest wor w v 6 v / w W = 5 v ( x w v v ( x ) v Г ( x y ) J ( x v v Г ( x Г w ( x Fract. / (optmal) 6 8 (optmal)

6 Huffma Codes Optmal Code & Full Bary ree a optmal prefx code prefx code : o codeword s also a prefx of some other codewords A : B : C : D : E : A : B : C : D : E : F : A C B F E D Optmal Code & Full Bary ree Coostructg a Huffma Code A:45 C: B: D:6 F:5 E:9 A : 45 K B : K C : K D : 6 K E : 9 K 4 F : 5 K 4 B ( ) f ( c) d ( c) cc 5 A:45 B: C: E:9 F:5 D: Costructg a Huffma Code Huffma( C, ) PQ = BuldHeap( C ) "( log ) for = to - z = Bode_Alloc( ) z.left = Extract_M( PQ ) z.rght = Extract_M( PQ ) z.freq = z.left.freq + z.rght.freq Isert( PQ, z ) retur Extract_M( PQ ) Provg Optmalty Greedy-choce property : buldg a optmal tree ca beg by mergg two lowest-frequecy characters Optmal-substructure property : Optmal Optmal

7 Greedy-Choce Property Let x ad y be two characters w/ lowest freq. Prove that there exsts a optmal-code tree where x ad y appear as sblg leaves of max. depth the tree. x y Greedy-Choce Property * ** x b c y b x c y b x y c cc f f ) ( c) d ( c) f ( c) d* ( c cc ( x) d ( x) Г f ( b) d ( b) f ( x) d *( x) f ( b) d *( b f ( x) d ( x) Г f ( b) d ( b) f ( x) d ( b) f ( b) d ( x) ( f ( b) f ( x))( d ( b) d ( x)) J ) Optmal-Substructure Property c x y * c f (c) = f (x) + f (y) B() = B(*) + ( f (x)d (x) + f (y)d (y) ) - f (c)d * (c) B() = B(*) + f (x) + f (y) f (x)d (x) + f (y)d (y) = ( f (x) + f (y) )( d * (c) + ) = f (c)d * (c) + f (x) + f (y) Prms algorthm Krusals algorthm Djstras algorthm More Greedy Algorthms Cocluso Greedy algorthm s smple easy to vet easy to mplemet effcet Do ot always eld optmal solutos requre greedy-choce ad optmal-substructure propertes for optmalty