Algorithm Design Techniques

Transcription

1 Desg ad Aalyss of Algorthms.. ก ก ก 2542 ก ก ก กกก Algorthm Desg echques Greedy Algorthms Outle Geeral Idea Actvty-Selecto Problem wo Key Igredets More Problems : Kapsack, Huffma Codes Cocluso

2 Caddate Set Algorthm emplate Why Greedy? Greedy( C ) S = empty Soluto Set 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 Make the choce that looks best at the momet Make locally optmal choce Hope that ths choce wll lead to a globally optmal soluto Do ot always yeld optmal solutos Actvty-Selecto Problem Brute Force s f ry all subsets of actvtes Choose the largest subset whch s feasble Ieffcet : "( 2 ) choces Actvty Startg Fsh

3 Greedy Algorthm 5 Greedy Algorthm 5 Sort by fsh tmes Greedy Algorthm 5 Greedy Algorthm j = 2 = 2 3 = 3 4 j = 4 5 = 5 6 = 6

4 Greedy Algorthm Provg Optmalty Greedy_Actvty_Select( s[..], f[..] ) A = { } j = ; for = 2 to f s[] >= f[j] A = A U { } j = retur A "( ) excludg sort Let f Actvty # must be a optmal soluto 5 k Provg Optmalty Provg Optmalty Let f 5 k Let f Greedy choce produces a optmal soluto. 5

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 makg a locally optmal (greedy) choce. Greedy-Choce Property Make whatever choce seems best at the momet Choce made caot deped o ay future choces Must prove that a greedy choce at each step yelds a globally optmal soluto Kapsack Problem tems ad a kapsack tem -th s worth v ad weght w the kapsack ca take weght of at most W What tems should be take to get the most valuable load? w Kapsack Problem 2 3 v 6 2 W = / Kapsack Fractoal Kapsack

6 Kapsack : Optmal Substructure Kapsack : Optmal Substructure W W - w 3 v + v 3 + v 4 + v 7 (max.) v + v 4 + v 7 (max.) W W - w Fractoal Kapsack Fractoal Kapsack v = 2 w = v 2 = 3 w 2 = 2 v 3 = 66 w 3 = 3 v 4 = 4 w 4 = 4 v 5 = 6 w 5 = 5 v = 2 w = v 2 = 3 w 2 = 2 v 3 = 66 w 3 = 3 v 4 = 4 w 4 = 4 v 5 = 6 w 5 = $v = $v = 46

7 v = 2 w = v 2 = 3 w 2 = 2 v /w 2..5 Fractoal Kapsack v 3 = 66 w 3 = 3 v 4 = 4 w 4 = $v = 64 v 5 = 6 w 5 = Fractoal Kapsack : Greedy Alg. Greedy_Kapsack( 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 Provg Optmalty Let v /w J v 2 /w 2 J... J v /w 2 k... x k... x k < Let x be the soluto vector Let y be ay feasble soluto vector ( x y ) v J Optmal $ x v $ y v ( x k k v vk ( x y ) w w J ( x y ) w wk vk v y ) v k Г ( xk yk ) wk ( x k yk ) wk wk wk v vk Г ( x y ) w J k Г w ( x y ) w k Г wk

8 2 k... x k... Provg Optmalty x k < / Kapsack : Greedy doest work w 2 3 v 6 2 v / w W = 5 ( x k v ( x y ) w w vk y ) v Г ( xk yk ) wk J ( x wk v vk Г ( x y ) w k Г w wk y ) w ( x y ) w vk w k Fract. / (optmal) (optmal) 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

9 Optmal Code & Full Bary ree Coostructg a Huffma Code A:45 C:2 B:3 D:6 F:5 E:9 A : 45 K B : 3 K 3 C : 2 K 3 D : 6 K 3 E : 9 K 4 F : 5 K 4 B ( ) f ( c) d ( c) cc 25 A:45 B:3 C:2 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

10 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 y b y b c b c x c x y 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 Kruskals algorthm Djkstras algorthm More Greedy Algorthms

11 Cocluso Greedy algorthm s smple easy to vet easy to mplemet effcet Do ot always yeld optmal solutos requre greedy-choce ad optmal-substructure propertes for optmalty