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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

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  • -- 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*

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

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  • 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

    Data Engineering Lab

  • HamiltonM1MM2M4;Data Engineering Lab*

    0B00000C00000DE0000EAB0D0

    00ABC00000BCDBCECEACEB0CEDCDEDEADEB0000EABEBC00

    000ABCEDABCDEBCDEA00000CDEAB00000DEABC00000EABCD0

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

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

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  • 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*

    Data Engineering Lab

  • () 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*

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  • () (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

    Data Engineering Lab

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

    Data Engineering Lab

  • TSP Data Engineering Lab*

    Data Engineering Lab

  • Data Engineering Lab TSP (2) *

    Data Engineering Lab

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

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  • 21134 km. TSP 100 Data Engineering Lab*

    Data Engineering Lab

  • (179 ) TSP 120 6942 kmData Engineering Lab*

    Data Engineering Lab

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

    Data Engineering Lab

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

    Data Engineering Lab

  • 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*

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  • 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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  • Bachet 123457131925101520981462412181123171622211262520244510162221Data Engineering Lab*

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