C ALGORITHM 595, COLLECTED ALGORITHMS FROM ACM. C ALGORITHM APPEARED IN ACM-TRANS. MATH. SOFTWARE, VOL.9, NO. 1, C MAR., 1983, P. 131-138. C SAMPLE DRIVER PROGRAM FOR HC. MAN 10 C MAN 20 C THIS PROGRAM FINDS ONE OR MORE HAMILTONIAN CIRCUITS IN A MAN 30 C DIRECTED GRAPH OF N VERTICES AND M ARCS. MAN 40 C MAN 50 C COMPLETE DETAILS OF THE PARAMETERS MAY BE FOUND IN THE MAN 60 C DOCUMENTATION OF SUBROUTINE HC. MAN 70 C MAN 80 C THE INPUT UNIT NUMBER IS ASSUMED TO BE 5 . MAN 90 C THE OUTPUT UNIT NUMBER IS ASSUMED TO BE 6 . MAN 100 C THE ARRAYS ARE CURRENTLY DIMENSIONED TO ALLOW PROBLEMS FOR MAN 110 C WHICH N .LE. 250 AND M .LE. 2000 . MAN 120 C MAN 130 C THE PROGRAM MAY BE TESTED ON THE FOLLOWING DATA MAN 140 C MAN 150 C N = 6 MAN 160 C PR = 0 3 5 8 11 13 16 MAN 170 C AR = 3 5 6 6 3 6 1 4 5 1 MAN 180 C 2 2 3 4 2 5 MAN 190 C MAN 200 INTEGER PR(251), PC(251), AR(2000), AC(2000), S(250), VR(250), MAN 210 * VC(250), P(250), SUBR(250), RBUS(250), TOR(250) MAN 220 NIN = 5 MAN 230 NOUT = 6 MAN 240 C INPUT DATA MAN 250 READ (NIN,99999) N MAN 260 NP1 = N + 1 MAN 270 READ (NIN,99999) (PR(J),J=1,NP1) MAN 280 M = PR(NP1) MAN 290 READ (NIN,99999) (AR(J),J=1,M) MAN 300 WRITE (NOUT,99998) MAN 310 WRITE (NOUT,99997) N MAN 320 WRITE (NOUT,99996) (PR(J),J=1,NP1) MAN 330 WRITE (NOUT,99995) (AR(J),J=1,M) MAN 340 WRITE (NOUT,99994) MAN 350 C CALL HC AS EXACT PROCEDURE TO FIND (AND PRINT) A SINGLE MAN 360 C HAMILTONIAN CIRCUIT, IF ONE EXISTS. MAN 370 NC = 1 MAN 380 NB = -1 MAN 390 CALL HC(N, PR, AR, NOUT, NC, NB, S, N+1, PR(N+1), PC, AC, VR, VC, MAN 400 * P, SUBR, RBUS, TOR) MAN 410 WRITE (NOUT,99993) NC, NB MAN 420 C CALL HC AS EXACT PROCEDURE TO FIND (AND PRINT) ALL THE MAN 430 C HAMILTONIAN CIRCUITS. MAN 440 NC = -1 MAN 450 NB = -1 MAN 460 CALL HC(N, PR, AR, NOUT, NC, NB, S, N+1, PR(N+1), PC, AC, VR, VC, MAN 470 * P, SUBR, RBUS, TOR) MAN 480 WRITE (NOUT,99993) NC, NB MAN 490 C CALL HC AS HEURISTIC PROCEDURE TO FIND (AND PRINT) A MAN 500 C SINGLE HAMILTONIAN CIRCUIT, IF ONE EXISTS, WITHOUT MAN 510 C PERFORMING MORE THAN 2 BACKTRACKINGS. MAN 520 NC = 1 MAN 530 NB = 2 MAN 540 CALL HC(N, PR, AR, NOUT, NC, NB, S, N+1, PR(N+1), PC, AC, VR, VC, MAN 550 * P, SUBR, RBUS, TOR) MAN 560 WRITE (NOUT,99993) NC, NB MAN 570 C CALL HC AS HEURISTIC PROCEDURE TO FIND (AND PRINT) A MAN 580 C SINGLE HAMILTONIAN CIRCUIT, IF ONE EXISTS, WITHOUT MAN 590 C PERFORMING MORE THAN 4 BACKTRACKINGS. MAN 600 NC = 1 MAN 610 NB = 4 MAN 620 CALL HC(N, PR, AR, NOUT, NC, NB, S, N+1, PR(N+1), PC, AC, VR, VC, MAN 630 * P, SUBR, RBUS, TOR) MAN 640 WRITE (NOUT,99993) NC, NB MAN 650 C CALL HC AS HEURISTIC PROCEDURE TO FIND (WITHOUT PRINTING) MAN 660 C AT MOST 2 HAMILTONIAN CIRCUITS, WITHOUT PERFORMING MORE MAN 670 C THAN 5 BACKTRACKINGS. MAN 680 NC = 2 MAN 690 NB = 5 MAN 700 CALL HC(N, PR, AR, -1, NC, NB, S, N+1, PR(N+1), PC, AC, VR, VC, MAN 710 * P, SUBR, RBUS, TOR) MAN 720 WRITE (NOUT,99993) NC, NB MAN 730 IF (NC.EQ.0) STOP MAN 740 C PRINT THE LAST HAMILTONIAN CIRCUIT FOUND MAN 750 WRITE (NOUT,99992) (S(J),J=1,N) MAN 760 STOP MAN 770 99999 FORMAT (10I5) MAN 780 99998 FORMAT (1H1////////) MAN 790 99997 FORMAT (6H N = , I5) MAN 800 99996 FORMAT (6H PR = , 25I5) MAN 810 99995 FORMAT (6H AR = , 25I5) MAN 820 99994 FORMAT (//////) MAN 830 99993 FORMAT (/I5, 10H CIRCUITS , I5, 14H BACKTRACKINGS///) MAN 840 99992 FORMAT (4X, 4HS = , 25I5) MAN 850 END MAN 860 SUBROUTINE HC(N, PR, AR, KW, NC, NB, S, NP1, M, PC, AC, VR, VC, HC 10 * P, SUBR, RBUS, TOR) C C SUBROUTINE TO FIND ONE OR MORE HAMILTONIAN CIRCUITS IN A C DIRECTED GRAPH OF N VERTICES ( N .GT. 1 ) REPRESENTED C BY THE INTEGERS 1, 2, ..., N AND M ARCS. C C HC IS BASED ON AN ENUMERATIVE ALGORITHM AND CAN BE USED C EITHER AS AN EXACT PROCEDURE OR AS A HEURISTIC PROCEDURE C (BY LIMITING THE NUMBER OF BACKTRACKINGS ALLOWED). C C ENTRANCE TO HC IS ACHIEVED BY USING THE STATEMENT C CALL HC(N,PR,AR,KW,NC,NB,S,N+1,PR(N+1),PC,AC,VR,VC, C * P,SUBR,RBUS,TOR) C C THE VALUES OF THE FIRST SIX PARAMETERS MUST BE DEFINED C BY THE USER PRIOR TO CALLING HC. HC NEEDS 2 ARRAYS ( PR C AND PC ) OF LENGTH N + 1 , 2 ARRAYS ( AR AND AC ) C OF LENGTH M AND 7 ARRAYS ( S , VR , VC , SUBR , C RBUS AND TOR ) OF LENGTH N . THESE ARRAYS MUST BE C DIMENSIONED BY THE USER IN THE CALLING PROGRAM. C C HC CALLS 5 SUBROUTINES: PATH, FUPD, BUPD, IUPD, RARC. C THESE SUBROUTINES ARE COMPLETELY LOCAL, I.E. THE INFORMA- C TION THEY NEED IS PASSED THROUGH THE PARAMETER LIST. C THE WHOLE PACKAGE IS COMPLETELY SELF CONTAINED AND COMMU- C NICATION TO IT IS ACHIEVED SOLELY THROUGH THE PARAMETER C LIST OF HC. NO MACHINE DEPENDENT CONSTANTS ARE USED. C THE PACKAGE IS WRITTEN IN AMERICAN NATIONAL STANDARD C FORTRAN AND IS ACCEPTED BY THE FTN(EL=A) COMPILER OF THE C CDC CYBER 76 (OPTION EL=A CHECKS PROGRAM AND SUBROUTINES C FOR ADHERENCE TO ANSI) AS WELL AS BY THE PFORT VERIFIER. C THE PACKAGE HAS BEEN TESTED ON A CDC CYBER 76 , ON A CDC C 6600 , ON AN IBM 370/158 AND ON A DIGITAL VAX 11/780 . C C MEANING OF THE INPUT PARAMETERS: C N = NUMBER OF VERTICES. C PR(I) = SUM OF THE OUT-DEGREES OF VERTICES 1, ..., I-1 C ( PR(1) = 0 , PR(N+1) = M ). C AR = ADJACENCY LIST. THE ELEMENTS FROM AR(PR(I)+1) TO C AR(PR(I+1)) ARE A RECORD CONTAINING,IN ANY ORDER, C ALL THE VERTICES J SUCH THAT ARC (I,J) EXISTS. C THE GRAPH SHOULD NOT CONTAIN ARCS STARTING AND C ENDING AT THE SAME VERTEX. C KW = UNIT TAPE ON WHICH TO WRITE THE HAMILTONIAN CIR- C CUITS FOUND, ACCORDING TO FORMAT 20I5 . KW = - 1 C IF NO WRITING IS DESIRED. THE CIRCUITS ARE WRITTEN C AS ORDERED SEQUENCES OF N VERTICES. C C MEANING OF THE INPUT-OUTPUT PARAMETERS: C NC(INPUT) = UPPER BOUND ON THE NUMBER OF HAMILTONIAN C CIRCUITS TO BE FOUND ( NC = - 1 IF ALL THE C HAMILTONIAN CIRCUITS ARE TO BE FOUND). C NC(OUTPUT) = NUMBER OF HAMILTONIAN CIRCUITS FOUND. C NB(INPUT) = - 1 IF HC MUST BE EXECUTED AS AN EXACT C PROCEDURE. C = UPPER BOUND ON THE NUMBER OF BACKTRACKINGS IF C HC MUST BE EXECUTED AS A HEURISTIC PROCEDURE. C NB(OUTPUT) = NUMBER OF BACKTRACKINGS PERFORMED. WHEN HC C HAS BEEN EXECUTED AS A HEURISTIC PROCEDURE, C IF NB(OUTPUT) .LT. NB(INPUT) THEN THE C RESULT OBTAINED IS EXACT. C C MEANING OF THE OUTPUT PARAMETER: C S(I) = I-TH VERTEX IN THE LAST HAMILTONIAN CIRCUIT FOUND. C C ON RETURN OF HC N, PR AND KW ARE UNCHANGED, WHILE IN C AR THE ORDER OF THE ELEMENTS WITHIN EACH RECORD MAY BE C ALTERED. C C MEANING OF THE WORK ARRAYS: C PC(I) = SUM OF THE IN-DEGREES OF VERTICES 1, ..., I-1 C ( PC(1) = 0 ). C AC = ADJACENCY LIST (BACKWARD). THE ELEMENTS FROM C AC(PC(I)+1) TO AC(PC(I+1)) CONTAIN, IN ANY C ORDER, ALL THE VERTICES J SUCH THAT ARC (J,I) C EXISTS. C WHEN AN ARC IS REMOVED FROM THE GRAPH AT THE K-TH LEVEL C OF THE BRANCH-DECISION TREE, THE CORRESPONDING ELEMENTS C AR(Q) AND AC(T) ARE SET TO - (K*(N+1) + AR(Q)) AND C TO - (K*(N+1) + AC(T)) , RESPECTIVELY. C VR(I) = CURRENT OUT-DEGREE OF VERTEX I . C VC(I) = CURRENT IN-DEGREE OF VERTEX I . C SUBR(I) = - (K*(N+1) + J) IF ARC (I,J) WAS IMPLIED AT C THE K-TH LEVEL OF THE BRANCH-DECISION TREE. C = 0 OTHERWISE. C RBUS(I) = - J IF ARC (J,I) IS CURRENTLY IMPLIED. C = 0 OTHERWISE. C TOR(K) = Q*(M+1) + T IF THE ARC GOING FROM S(K) TO THE C ROOT, CORRESPONDING TO AR(Q) AND TO AC(T), C WAS REMOVED FROM THE GRAPH AT THE K-TH LEVEL C OF THE BRANCH-DECISION TREE. C = 0 OTHERWISE. C P(I) = POINTER FOR THE FORWARD STEP. THE NEXT ARC C STARTING FROM I TO BE CONSIDERED IN THE C BRANCH-DECISION TREE IS (I,AR(PR(I)+P(I)). C C MEANING OF THE MAIN WORK SIMPLE VARIABLES: C JR = ROOT. THE HAMILTONIAN CIRCUITS ARE DETERMINED AS C PATHS STARTING AND ENDING AT JR . C K = CURRENT LEVEL OF THE BRANCH-DECISION TREE. C M = NUMBER OF ARCS. C MP1 = M + 1 (USED FOR PACKING TOR ). C NP1 = N + 1 (USED FOR PACKING AR , AC AND SUBR ) C INTEGER PR(NP1), PC(NP1), AR(M), AC(M), S(N), VR(N), VC(N), P(N), * SUBR(N), RBUS(N), TOR(N) C C S T E P 0 (INITIALIZE). C NCO = NC NC = 0 NBO = NB NB = 0 DO 10 I=1,N VC(I) = 0 SUBR(I) = 0 RBUS(I) = 0 P(I) = 1 10 CONTINUE DO 30 I=1,N J1 = PR(I) + 1 J2 = PR(I+1) VR(I) = J2 - J1 + 1 IF (VR(I).EQ.0) RETURN DO 20 J=J1,J2 JA = AR(J) VC(JA) = VC(JA) + 1 20 CONTINUE 30 CONTINUE PC(1) = 0 DO 40 I=1,N IF (VC(I).EQ.0) RETURN PC(I+1) = PC(I) + VC(I) VC(I) = 0 40 CONTINUE DO 60 I=1,N J1 = PR(I) + 1 J2 = PR(I+1) DO 50 J=J1,J2 JJ = AR(J) VC(JJ) = VC(JJ) + 1 JA = PC(JJ) + VC(JJ) AC(JA) = I 50 CONTINUE 60 CONTINUE MP1 = M + 1 C SELECT AS ROOT JR THE VERTEX I WITH MAXIMUM VC(I) C (BREAK TIES BY CHOOSING I WITH MINIMUM VR(I) ). MAXE = VC(1) MINU = VR(1) JR = 1 DO 100 I=2,N IF (VC(I)-MAXE) 100, 70, 80 70 IF (VR(I).GE.MINU) GO TO 100 GO TO 90 80 MAXE = VC(I) 90 MINU = VR(I) JR = I 100 CONTINUE K1 = -NP1 K = 1 S(1) = JR C C S T E P 1 (SEARCH FOR IMPLIED ARCS). C 110 DO 120 J=1,N IF (VR(J).EQ.1) GO TO 130 IF (VC(J).EQ.1) GO TO 170 120 CONTINUE C NO FURTHER ARC CAN BE IMPLIED. GO TO 220 C ARC (J,JL) IS IMPLIED BECAUSE VR(J) = 1 . 130 L1 = PR(J) + 1 L2 = PR(J+1) DO 140 L=L1,L2 IF (AR(L).GT.0) GO TO 150 140 CONTINUE 150 JL = AR(L) C FIND THE STARTING VERTEX IT1 AND THE ENDING VERTEX IT2 C OF THE LARGEST PATH OF IMPLIED ARCS CONTAINING (J,JL) . CALL PATH(J, JL, SUBR, RBUS, AR, PR, S, N, NP, IT1, IT2, K, JR, * M, NP1) IF (NP.EQ.0) GO TO 160 IF (NP.EQ.(-1)) GO TO 340 C SUBROUTINE PATH FOUND A HAMILTONIAN CIRCUIT. K = K + 1 GO TO 320 160 SUBR(J) = K1 - JL RBUS(JL) = J C REMOVE FROM THE GRAPH ALL ARCS TERMINATING AT JL . CALL IUPD(J, JL, L, AC, AR, PC, PR, VC, VR, K1, N, M, NP1) IF (J.EQ.0) GO TO 340 GO TO 210 C ARC (JL,J) IS IMPLIED BECAUSE VC(J) = 1 . 170 L1 = PC(J) + 1 L2 = PC(J+1) DO 180 L=L1,L2 IF (AC(L).GT.0) GO TO 190 180 CONTINUE 190 JL = AC(L) C FIND THE STARTING VERTEX IT1 AND THE ENDING VERTEX IT2 C OF THE LARGEST PATH OF IMPLIED ARCS CONTAINING (JL,J) . CALL PATH(JL, J, SUBR, RBUS, AR, PR, S, N, NP, IT1, IT2, K, JR, * M, NP1) IF (NP.EQ.0) GO TO 200 IF (NP.EQ.(-1)) GO TO 340 C SUBROUTINE PATH FOUND A HAMILTONIAN CIRCUIT. I = S(K) K = K + 1 GO TO 320 200 SUBR(JL) = K1 - J RBUS(J) = JL C REMOVE FROM THE GRAPH ALL ARCS EMANATING FROM JL . CALL IUPD(J, JL, L, AR, AC, PR, PC, VR, VC, K1, N, M, NP1) IF (J.EQ.0) GO TO 340 C IF ARC (IT2,IT1) IS IN THE GRAPH, REMOVE IT. 210 CALL RARC(IT2, IT1, AR, AC, PR, PC, VR, VC, K1, JJ, LL, N, M, NP1) IF (JJ.EQ.(-1)) GO TO 340 GO TO 110 C C S T E P 2 (ADD IMPLIED ARCS TO S ). C 220 I = S(K) IF (SUBR(I).EQ.0) GO TO 230 JSUBR = -SUBR(I) + SUBR(I)/NP1*NP1 IF (JSUBR.EQ.JR) GO TO 340 K = K + 1 S(K) = JSUBR IF (K.NE.N) GO TO 220 IF (SUBR(JSUBR).LT.0) GO TO 320 GO TO 340 C C S T E P 3 (BRANCH). C 230 L1 = PR(I) + P(I) L2 = PR(I+1) IF (L1.GT.L2) GO TO 340 C FIND THE NEXT ARC (I,JL) TO BE ADDED TO S . DENS = N**3 J1 = 0 J2 = 0 DO 310 J=L1,L2 JL = AR(J) IF (JL.LT.0) GO TO 310 IF (VR(JL).GT.0) GO TO 260 IF (SUBR(JL).EQ.0) GO TO 310 IF (JL.EQ.JR) GO TO 310 IEND = JL 240 IEND = -SUBR(IEND) + SUBR(IEND)/NP1*NP1 IF (SUBR(IEND).NE.0) GO TO 240 IF (VC(JL).LT.VR(IEND)) GO TO 250 SCORE = VR(IEND)*N + VC(JL) GO TO 280 250 SCORE = VC(JL)*N + VR(IEND) GO TO 280 260 IF (VC(JL).LT.VR(JL)) GO TO 270 SCORE = VR(JL)*N + VC(JL) GO TO 280 270 SCORE = VC(JL)*N + VR(JL) 280 IF (DENS.LE.SCORE) GO TO 290 DENS = SCORE IPI = J 290 IF (J1.EQ.0) GO TO 300 IF (J2.EQ.0) J2 = J GO TO 310 300 J1 = J 310 CONTINUE IF (J1.EQ.0) GO TO 340 JL = AR(IPI) AR(IPI) = AR(J1) AR(J1) = JL IF (J2.EQ.0) J2 = PR(I+1) + 1 P(I) = J2 - PR(I) K = K + 1 S(K) = JL K1 = -K*NP1 C REMOVE FROM THE GRAPH ALL ARCS EMANATING FROM I . CALL FUPD(AR, AC, PR, PC, VR, VC, I, K1, N, M, NP1) C REMOVE FROM THE GRAPH ALL ARCS TERMINATING AT JL . CALL FUPD(AC, AR, PC, PR, VC, VR, JL, K1, N, M, NP1) TOR(K) = 0 C IF ARC (JL,JR) IS IN THE GRAPH, REMOVE IT. CALL RARC(JL, JR, AR, AC, PR, PC, VR, VC, K1, JJ, LL, N, M, NP1) IF (JJ.EQ.0) GO TO 110 IF (JJ.EQ.(-1)) GO TO 340 TOR(K) = JJ*MP1 + LL GO TO 110 C C S T E P 4 (HAMILTONIAN CIRCUIT FOUND). C 320 NC = NC + 1 IF (KW.EQ.(-1)) GO TO 330 WRITE (KW,99999) (S(KJ),KJ=1,N) 99999 FORMAT (20I5) 330 IF (NC.EQ.NCO) GO TO 430 K = K - 1 C C S T E P 5 (BACKTRACK). C 340 IF (K.LE.1) GO TO 430 JA = S(K) P(JA) = 1 JA = S(K-1) IF (SUBR(JA).EQ.0) GO TO 350 C BACKTRACKING FOR AN IMPLIED ARC. K = K - 1 GO TO 340 350 IF (NB.EQ.NBO) GO TO 430 NB = NB + 1 K1 = -K*NP1 K2 = -(K+1)*NP1 I = S(K-1) C BACKTRACKING FOR THE ARCS IMPLIED AT LEVEL K . IFF = 0 DO 360 J=1,N IF (SUBR(J).GT.K1) GO TO 360 IF (SUBR(J).LT.K2) GO TO 360 JA = K1 - SUBR(J) RBUS(JA) = 0 SUBR(J) = 0 IFF = 1 360 CONTINUE IF (IFF.EQ.1) GO TO 370 C NO ARC WAS IMPLIED AT LEVEL K . CALL BUPD(AR, AC, PR, PC, VR, VC, I, K1, K2, N, M, NP1) CALL BUPD(AC, AR, PC, PR, VC, VR, S(K), K1, K2, N, M, NP1) IF (TOR(K).EQ.0) GO TO 420 J1 = TOR(K)/MP1 J2 = TOR(K) - J1*MP1 AR(J1) = JR JA = S(K) VR(JA) = VR(JA) + 1 AC(J2) = S(K) VC(JR) = VC(JR) + 1 GO TO 420 C AT LEAST ONE ARC WAS IMPLIED AT LEVEL K . 370 DO 410 J=1,N L1 = PR(J) + 1 L2 = PR(J+1) DO 400 L=L1,L2 JL = AR(L) IF (JL.GT.K1) GO TO 400 IF (JL.LT.K2) GO TO 400 JL = K1 - JL AR(L) = JL VR(J) = VR(J) + 1 LL1 = PC(JL) + 1 LL2 = PC(JL+1) DO 380 LL=LL1,LL2 IF (K1-AC(LL).EQ.J) GO TO 390 380 CONTINUE 390 AC(LL) = J VC(JL) = VC(JL) + 1 400 CONTINUE 410 CONTINUE 420 K = K - 1 GO TO 230 C C RE-STORE THE ORIGINAL VECTOR AR . C 430 DO 440 J=1,M IF (AR(J).GT.0) GO TO 440 AR(J) = -AR(J) + AR(J)/NP1*NP1 440 CONTINUE RETURN END SUBROUTINE PATH(I, J, SUBR, RBUS, AR, PR, S, N, NP, I1, I2, K, PAT 10 * JR, M, NP1) C SUBROUTINE TO FIND THE STARTING VERTEX I1 AND THE ENDING C VERTEX I2 OF THE LARGEST PATH OF IMPLIED ARCS CONTAINING C ARC (I,J) . C MEANING OF THE OUTPUT PARAMETER NP : C NP = 0 IF THE PATH CONTAINS L .LT. N VERTICES. C = 1 IF THE PATH CONTAINS N VERTICES AND ARC (I2,I1) C EXISTS (THE HAMILTONIAN CIRCUIT IS STORED IN S ) C = -1 IF THE PATH CONTAINS N VERTICES BUT ARC (I2,I1) C DOES NOT EXIST. INTEGER SUBR(N), RBUS(N), AR(M), PR(NP1), S(N) NP = 0 L = 1 I1 = I 10 IF (RBUS(I1).EQ.0) GO TO 20 I1 = RBUS(I1) L = L + 1 GO TO 10 20 I2 = J L = L + 1 30 IF (SUBR(I2).EQ.0) GO TO 40 I2 = -SUBR(I2) + SUBR(I2)/NP1*NP1 L = L + 1 GO TO 30 40 CONTINUE IF (L.LT.N) RETURN C THE PATH CONTAINS N VERTICES. K1 = -K*NP1 L1 = PR(I2) + 1 L2 = PR(I2+1) DO 60 L=L1,L2 IF (AR(L).LT.0) GO TO 50 IF (AR(L).EQ.I1) GO TO 70 GO TO 60 50 IF (K1-AR(L).EQ.I1) GO TO 70 60 CONTINUE C NO HAMILTONIAN CIRCUIT CAN BE DETERMINED. NP = -1 RETURN C A HAMILTONIAN CIRCUIT EXISTS. STORE IT IN S . 70 NP = 1 RBUS(J) = I RBUS(I1) = I2 S(N) = RBUS(JR) L = N - 1 80 IF (L.EQ.K) GO TO 90 JA = S(L+1) S(L) = RBUS(JA) L = L - 1 GO TO 80 90 RBUS(I1) = 0 RBUS(J) = 0 RETURN END SUBROUTINE FUPD(A1, A2, P1, P2, V1, V2, I1, K1, N, M, NP1) FUP 10 C FORWARD STEP UPDATING INTEGER A1(M), A2(M), P1(NP1), P2(NP1), V1(N), V2(N) J1 = P1(I1) + 1 J2 = P1(I1+1) DO 30 J=J1,J2 IF (A1(J).LT.0) GO TO 30 IA = A1(J) L1 = P2(IA) + 1 L2 = P2(IA+1) DO 10 L=L1,L2 IF (A2(L).EQ.I1) GO TO 20 10 CONTINUE 20 V2(IA) = V2(IA) - 1 A2(L) = K1 - A2(L) A1(J) = K1 - IA 30 CONTINUE V1(I1) = 0 RETURN END SUBROUTINE BUPD(A1, A2, P1, P2, V1, V2, II, K1, K2, N, M, NP1) BUP 10 C BACKTRACKING STEP UPDATING INTEGER A1(M), A2(M), P1(NP1), P2(NP1), V1(N), V2(N) L1 = P1(II) + 1 L2 = P1(II+1) DO 30 L=L1,L2 IF (A1(L).GT.K1) GO TO 30 IF (A1(L).LT.K2) GO TO 30 IA = K1 - A1(L) A1(L) = IA V1(II) = V1(II) + 1 LL1 = P2(IA) + 1 LL2 = P2(IA+1) DO 10 LL=LL1,LL2 IF (K1-A2(LL).EQ.II) GO TO 20 10 CONTINUE 20 A2(LL) = II V2(IA) = V2(IA) + 1 30 CONTINUE RETURN END SUBROUTINE IUPD(IA, IB, L, A1, A2, P1, P2, V1, V2, K1, N, M, NP1) IUP 10 INTEGER A1(M), A2(M), P1(NP1), P2(NP1), V1(N), V2(N) C UPDATING FOR IMPLIED ARC M1 = P1(IB) + 1 M2 = P1(IB+1) DO 40 MM=M1,M2 IARC = A1(MM) IF (IARC.LT.0) GO TO 40 IF (V2(IARC).NE.1) GO TO 10 IF (IARC.NE.IA) GO TO 50 JJ = L GO TO 30 10 J1 = P2(IARC) + 1 J2 = P2(IARC+1) DO 20 JJ=J1,J2 IF (A2(JJ).EQ.IB) GO TO 30 20 CONTINUE 30 A2(JJ) = K1 - A2(JJ) V2(IARC) = V2(IARC) - 1 A1(MM) = K1 - IARC V1(IB) = V1(IB) - 1 40 CONTINUE RETURN 50 IA = 0 RETURN END SUBROUTINE RARC(IA, IB, AR, AC, PR, PC, VR, VC, K1, JJ, LL, N, M, RAR 10 * NP1) C SUBROUTINE TO REMOVE ARC (IA,IB) FROM THE GRAPH. C MEANING OF THE OUTPUT PARAMETERS JJ AND LL : C JJ = LOCATION OF THE ELEMENT OF AR CORRESPONDING TO THE C REMOVED ARC. C = 0 IF ARC (IA,IB) IS NOT IN THE GRAPH. C = -1 IF, AFTER THE REMOVAL OF ARC (IA,IB) , THE GRAPH C WOULD ADMIT NO HAMILTONIAN CIRCUIT. C LL = LOCATION OF THE ELEMENT OF AC CORRESPONDING TO THE C REMOVED ARC (DEFINED ONLY IF JJ .GT. 0 ). INTEGER AR(M), AC(M), PR(NP1), PC(NP1), VR(N), VC(N) J1 = PR(IA) + 1 J2 = PR(IA+1) DO 20 JJ=J1,J2 IF (AR(JJ).LT.0) GO TO 20 IF (AR(JJ).NE.IB) GO TO 20 L1 = PC(IB) + 1 L2 = PC(IB+1) DO 10 LL=L1,L2 IF (AC(LL).EQ.IA) GO TO 30 10 CONTINUE 20 CONTINUE C ARC (IA,IB) IS NOT IN THE GRAPH. JJ = 0 RETURN 30 IF (VR(IA).EQ.1) GO TO 40 IF (VC(IB).EQ.1) GO TO 40 AR(JJ) = K1 - IB VR(IA) = VR(IA) - 1 AC(LL) = K1 - IA VC(IB) = VC(IB) - 1 RETURN C ARC (IA,IB) CANNOT BE REMOVED FROM THE GRAPH. 40 JJ = -1 RETURN END 6 0 3 5 8 11 13 16 3 5 6 6 3 6 1 4 5 1 2 2 3 4 2 5