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Θεωρία Γράφων Θεμελιώσεις-Αλγόριθμοι-Εφαρμογές Κεφάλαιο 2: Μονοπάτια και Κύκλοι ( Hamilton ). Sir William Rowan Hamilton. O Ιρλανδός μαθηματικός Hamilton το 1856 κατασκεύασε το παιχνίδι «Γύρος του κόσμου» - PowerPoint PPT Presentation

### Transcript of Θεωρία Γράφων...

• -- 2: (Hamilton)Data Engineering Lab*

Data Engineering Lab

• Sir William Rowan HamiltonO Hamilton 1856 : ?Hamiltonian Data Engineering Lab*

Data Engineering Lab

• The Icosian GameData Engineering Lab*

Data Engineering Lab

• ( !)Data Engineering Lab*

Data Engineering Lab

• Hamiltonian Data Engineering Lab*

Data Engineering Lab

• Hamilton Hamilton 2-, Hamilton 2. 2- Hamilton.

Hamilton C=(v1,v2,v3,v4,v8,v7,v6,v5,v1)2- ( Hamilton) : (v1,v2,v6,v5,v1) (v3,v4,v8,v7,v3)Data Engineering Lab*

Data Engineering Lab

• HamiltonE-traceable ( Hamilton)O-homogenously ( ) -Hamilton Hamilton G-v Hamilton v G Hamilton ( Hamilton) Hamilton 3, Hamilton. .Data Engineering Lab*

Data Engineering Lab

• Hamiltonian : , Hamilton? (NP-complete): Hamilton: n (n ) (n-1)/2 Hamilton (Dirac 1952): n3 d(G)n/2 Hamiltonian (Ore 1960): n3 d(x)+d(y)n x,y HamiltonianData Engineering Lab*

Data Engineering Lab

• Hamiltonian : n3 d(x)+d(y)n x,y Hamiltonian, G+(x,y) Hamiltonian-closure c(G) x y, d(x)+d(y)n. (Bondy-Chvatal 1976): Hamiltonian, Hamiltonian. (Fraudee-Dould-Jacobsen-Schelp 1989): 2- x,y d(x)+d(y)(2n-1)/3 Hamiltonian.Data Engineering Lab*

Data Engineering Lab

• c(G)GData Engineering Lab* c1(G), c2(G) G, , c1(G)=c2(G)

Data Engineering Lab

• ACBDA 16 Hamiltonian K4 ? ABCDA 17ABDCA 17 Hamiltonian Data Engineering Lab*

Data Engineering Lab

• M= *=2;M2= 1+2++k i j 1,2,,k.Data Engineering Lab*

001100100010010100010011101001111000

020111102101110021312011122200111210

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• HamiltonM1MM2M4;Data Engineering Lab*

0B00000C00000DE0000EAB0D0

000ABCEDABCDEBCDEA00000CDEAB00000DEABC00000EABCD0

0B00000C00000CDCE0000DEEAEB0ED0

Data Engineering Lab

• Hamilton k- , Hamiltonian. (backtracking), (dynamic programming) (branch and bound).Data Engineering Lab*

Data Engineering Lab

• : ; : , . Hamilton .2135424433376Data Engineering Lab*

Data Engineering Lab

• Hamilton (n1)!/2 o O(nn), . , : O(n22n)., , (ant) Data Engineering Lab*

Data Engineering Lab

• - 1
• i1. v0 Pi=v0 i=n, C=Pn Hamilton, e Pi , Pi. Pi+1=(Pie). ii+1. 2.

=(|ln n|+1)/2 n ;Data Engineering Lab*

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• () 3-6 25-3-6 2+35=374-5-3-6 37+21=584-5-3-6-1 58+51=1094-5-3-6-1-2 109+13=1224-5-3-6-1-2-4 122+70=192Data Engineering Lab*

Data Engineering Lab

• i1. v0 Ci=v0 i=n, C=Cn Hamilton, vi Ci {wi,wi+1} Ci. Ci+1 vi wi wi+1. ii+1. 2.

2 dist(wi,vi)+dist(vi,wi+1)dist(wi, wi+1). .

2, O(n3).Data Engineering Lab*

Data Engineering Lab

• () (3)(3,6,3)(3,6,5,3)(3,6,5,4,3)(3,6,1,5,4,3)(3,6,2,1,5,4,3) 19212435165136056213626857615135706870Data Engineering Lab*

Data Engineering Lab

• G. . [ v0 v1, v1 v0. , .] T 2, Hamilton Data Engineering Lab*

Data Engineering Lab

• Prim 13 dfs 3(3,1,2,5,4,6,3) 212 dfs 1(1,2,3,5,4,6,1) 237 (m+nlogn)Data Engineering Lab*

Data Engineering Lab

• Hamiltonian C=(v1,v2,,vn,v1) i,j, 1
• (3,4,5,6,1,2,3) 237(3,6,5,4,1,2,3) 210(3,6,5,4,2,1,3) 193(3,6,1,2,4,5,3) 192 .12435165136056213626857615135706870Data Engineering Lab*

Data Engineering Lab

• TSP: G-v w(T) v A v=5, w(T)=122, 122+21+35=178= Data Engineering Lab*

Data Engineering Lab

• TSP100,000 = 105 5,500,000,000 = 5.5 109 1,000,000,000,000,000,000,000 = 1021 1010 years = 3 1017 seconds Data Engineering Lab*

# of cities npossible solutions (n-1)! = # of cyclic permutations10 181,00020 10,000,000,000,000,000 = 101650 100,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000 = 1062

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• 1962 Proctor & Gamble 33 1962. TSP Data Engineering Lab*

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• TSP Data Engineering Lab*

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• Data Engineering Lab TSP (2) *

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• 1954 TSP 49 George Dantzig, Ray Fulkerson, Selmer Johnson TSP Data Engineering Lab*

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• 21134 km. TSP 100 Data Engineering Lab*

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• (179 ) TSP 120 6942 kmData Engineering Lab*

Data Engineering Lab

• 1998 TSP 13,509 500 http://www.tsp.gatech.edu TSP Data Engineering Lab*

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• TSP http://www.tsp.gatech.edu TSP Data Engineering Lab*

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• Martin Grtschel3 http://www.math.princeton.edu/ tsp/d15sol/dhistory.htmlData Engineering Lab

Martin Grtschel

• Martin Grtschel 24.978 2004Data Engineering Lab*

Data Engineering Lab

• N = 85,900 (2006) Concorde 2719.5 cpu-daysData Engineering Lab*

Data Engineering Lab

• 11,849 programmed logic arrayhttp://www.tsp.gatech.edu TSP PLAData Engineering Lab*

Data Engineering Lab

• TSP http://www.tsp.gatech.edu TSP PLA ()Data Engineering Lab*

Data Engineering Lab

• TSPThe Traveling Salesman Problem, Lawler, Lenstra, Rinnooy Kan, and Shmoys (Editors), Wiley (1985).The Traveling Salesman Problem and Its Variations, Gutin and Punnen (Editors), Kluwer (2002). The Traveling Salesman Problem: A Computational Study, Applegate, Bixby, Chvatal, and Cook, Princeton University Press (2006). In Pursuit of the Traveling Salesman, Cook, Princeton University Press (2012). Data Engineering Lab*

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• , 1 Euler Hamilton, Euler (two-way) . Hamiltonian 2-. (one-way) Euler/Hamilton (space filling curve)Data Engineering Lab*

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• Data Engineering Lab*

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• , Data Engineering Lab*

16321351011896712415141

23122019221691445111315218121710187252463

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• ( ): 1,2,... , [1..9] 9 [3..17] Bachet ( ) Data Engineering Lab*

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• Bachet 123457131925101520981462412181123171622211262520244510162221Data Engineering Lab*

Data Engineering Lab

• - 3,34,4 Data Engineering Lab*

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• : 2 Hamilton, . . ( , , )Data Engineering Lab*

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• 8x8, ? /Data Engineering Lab*

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• 1759: 4000 .

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• /Data Engineering Lab* DeMoivre -

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• 4 4 (, , ). ( ) 4 . . 2424 243=41472Data Engineering Lab*

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• 4 ( ) 12341234Data Engineering Lab*

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• 4 . 2 Hamiltonian . Data Engineering Lab*

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