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|
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 this program finds one or more hamiltonian circuits in a
c directed graph of n vertices and m arcs.
c input : n number of vertices, ar=ls, pr=lp-1
subroutine hamil(n,m,np1,pr,ar,s,nc,nb,
* pc,ac,vr,vc,p,subr,rbus,tor)
integer pr(np1),pc(np1),ar(m),ac(m),s(n),vr(n),
* vc(n),p(n),subr(n),rbus(n),tor(n)
c exact procedure to find a single hamiltonian circuit, if one exists.
do 1,i=1,n
s(i)=0
1 continue
nc = 1
nb = -1
call hproc(n,pr,ar,nout,nc,nb,s,n+1,pr(n+1),pc,ac,vr, vc,
*p,subr,rbus,tor)
c exact procedure to find all the hamiltonian circuits.
c nc = -1
c nb = -1
c call hproc(n,pr,ar,nout,nc,nb,s,n+1,pr(n+1),pc,ac,vr,vc,
c * p, subr, rbus, tor)
c heuristic procedure to find a single hamiltonian
c circuit, if one exists, without performing more than 2 backtrackings.
c nc = 1
c nb = 2
c call hproc(n,pr,ar,nout,nc,nb,s,n+1, pr(n+1),pc,ac,vr,vc,
c *p,subr,rbus,tor)
c heuristic procedure to find a single hamiltonian
c circuit, if one exists, without performing more than 4 backtrackings.
c nc = 1
c nb = 4
c call hproc(n,pr,ar,nout,nc,nb,s,n+1,pr(n+1),pc,ac,vr,vc,
c *p,subr,rbus,tor)
c heuristic procedure to find at most 2
c hamiltonian circuits, without performing more than 5 backtrackings.
c nc = 2
c nb = 5
c call hproc(n,pr,ar,-1,nc,nb,s,n+1,pr(n+1),pc,ac,vr,vc,
c *p,subr,rbus,tor)
c if (nc.eq.0) return
return
end
subroutine hproc(n,pr,ar,kw,nc,nb,s,np1,m,pc,ac,vr,vc,
*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 hproc 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 hproc is achieved by using the statement
c call hproc(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 hproc. hproc 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 hproc calls 5 subroutines: pathp, fupd, bupd, iupd, rarc.
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 hproc must be executed as an exact
c procedure.
c = upper bound on the number of backtrackings if
c hproc must be executed as a heuristic procedure.
c nb(output) = number of backtrackings performed. when hproc
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 hproc 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 pathp(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 pathp 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 pathp(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 pathp 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
c ww s(kj),kj=1,n)
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 pathp(i,j,subr,rbus,ar,pr,s,n,np,i1,i2,k,
* 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)
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)
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)
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,
* 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 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
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