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/* ffalg.c (Ford-Fulkerson algorithm) */
/***********************************************************************
* This code is part of GLPK (GNU Linear Programming Kit).
*
* Copyright (C) 2009-2013 Andrew Makhorin, Department for Applied
* Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
* reserved. E-mail: <mao@gnu.org>.
*
* GLPK is free software: you can redistribute it and/or modify it
* under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* GLPK is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
* License for more details.
*
* You should have received a copy of the GNU General Public License
* along with GLPK. If not, see <http://www.gnu.org/licenses/>.
***********************************************************************/
#include "env.h"
#include "ffalg.h"
/***********************************************************************
* NAME
*
* ffalg - Ford-Fulkerson algorithm
*
* SYNOPSIS
*
* #include "ffalg.h"
* void ffalg(int nv, int na, const int tail[], const int head[],
* int s, int t, const int cap[], int x[], char cut[]);
*
* DESCRIPTION
*
* The routine ffalg implements the Ford-Fulkerson algorithm to find a
* maximal flow in the specified flow network.
*
* INPUT PARAMETERS
*
* nv is the number of nodes, nv >= 2.
*
* na is the number of arcs, na >= 0.
*
* tail[a], a = 1,...,na, is the index of tail node of arc a.
*
* head[a], a = 1,...,na, is the index of head node of arc a.
*
* s is the source node index, 1 <= s <= nv.
*
* t is the sink node index, 1 <= t <= nv, t != s.
*
* cap[a], a = 1,...,na, is the capacity of arc a, cap[a] >= 0.
*
* NOTE: Multiple arcs are allowed, but self-loops are not allowed.
*
* OUTPUT PARAMETERS
*
* x[a], a = 1,...,na, is optimal value of the flow through arc a.
*
* cut[i], i = 1,...,nv, is 1 if node i is labelled, and 0 otherwise.
* The set of arcs, whose one endpoint is labelled and other is not,
* defines the minimal cut corresponding to the maximal flow found.
* If the parameter cut is NULL, the cut information are not stored.
*
* REFERENCES
*
* L.R.Ford, Jr., and D.R.Fulkerson, "Flows in Networks," The RAND
* Corp., Report R-375-PR (August 1962), Chap. I "Static Maximal Flow,"
* pp.30-33. */
void ffalg(int nv, int na, const int tail[], const int head[],
int s, int t, const int cap[], int x[], char cut[])
{ int a, delta, i, j, k, pos1, pos2, temp,
*ptr, *arc, *link, *list;
/* sanity checks */
xassert(nv >= 2);
xassert(na >= 0);
xassert(1 <= s && s <= nv);
xassert(1 <= t && t <= nv);
xassert(s != t);
for (a = 1; a <= na; a++)
{ i = tail[a], j = head[a];
xassert(1 <= i && i <= nv);
xassert(1 <= j && j <= nv);
xassert(i != j);
xassert(cap[a] >= 0);
}
/* allocate working arrays */
ptr = xcalloc(1+nv+1, sizeof(int));
arc = xcalloc(1+na+na, sizeof(int));
link = xcalloc(1+nv, sizeof(int));
list = xcalloc(1+nv, sizeof(int));
/* ptr[i] := (degree of node i) */
for (i = 1; i <= nv; i++)
ptr[i] = 0;
for (a = 1; a <= na; a++)
{ ptr[tail[a]]++;
ptr[head[a]]++;
}
/* initialize arc pointers */
ptr[1]++;
for (i = 1; i < nv; i++)
ptr[i+1] += ptr[i];
ptr[nv+1] = ptr[nv];
/* build arc lists */
for (a = 1; a <= na; a++)
{ arc[--ptr[tail[a]]] = a;
arc[--ptr[head[a]]] = a;
}
xassert(ptr[1] == 1);
xassert(ptr[nv+1] == na+na+1);
/* now the indices of arcs incident to node i are stored in
* locations arc[ptr[i]], arc[ptr[i]+1], ..., arc[ptr[i+1]-1] */
/* initialize arc flows */
for (a = 1; a <= na; a++)
x[a] = 0;
loop: /* main loop starts here */
/* build augmenting tree rooted at s */
/* link[i] = 0 means that node i is not labelled yet;
* link[i] = a means that arc a immediately precedes node i */
/* initially node s is labelled as the root */
for (i = 1; i <= nv; i++)
link[i] = 0;
link[s] = -1, list[1] = s, pos1 = pos2 = 1;
/* breadth first search */
while (pos1 <= pos2)
{ /* dequeue node i */
i = list[pos1++];
/* consider all arcs incident to node i */
for (k = ptr[i]; k < ptr[i+1]; k++)
{ a = arc[k];
if (tail[a] == i)
{ /* a = i->j is a forward arc from s to t */
j = head[a];
/* if node j has been labelled, skip the arc */
if (link[j] != 0) continue;
/* if the arc does not allow increasing the flow through
* it, skip the arc */
if (x[a] == cap[a]) continue;
}
else if (head[a] == i)
{ /* a = i<-j is a backward arc from s to t */
j = tail[a];
/* if node j has been labelled, skip the arc */
if (link[j] != 0) continue;
/* if the arc does not allow decreasing the flow through
* it, skip the arc */
if (x[a] == 0) continue;
}
else
xassert(a != a);
/* label node j and enqueue it */
link[j] = a, list[++pos2] = j;
/* check for breakthrough */
if (j == t) goto brkt;
}
}
/* NONBREAKTHROUGH */
/* no augmenting path exists; current flow is maximal */
/* store minimal cut information, if necessary */
if (cut != NULL)
{ for (i = 1; i <= nv; i++)
cut[i] = (char)(link[i] != 0);
}
goto done;
brkt: /* BREAKTHROUGH */
/* walk through arcs of the augmenting path (s, ..., t) found in
* the reverse order and determine maximal change of the flow */
delta = 0;
for (j = t; j != s; j = i)
{ /* arc a immediately precedes node j in the path */
a = link[j];
if (head[a] == j)
{ /* a = i->j is a forward arc of the cycle */
i = tail[a];
/* x[a] may be increased until its upper bound */
temp = cap[a] - x[a];
}
else if (tail[a] == j)
{ /* a = i<-j is a backward arc of the cycle */
i = head[a];
/* x[a] may be decreased until its lower bound */
temp = x[a];
}
else
xassert(a != a);
if (delta == 0 || delta > temp) delta = temp;
}
xassert(delta > 0);
/* increase the flow along the path */
for (j = t; j != s; j = i)
{ /* arc a immediately precedes node j in the path */
a = link[j];
if (head[a] == j)
{ /* a = i->j is a forward arc of the cycle */
i = tail[a];
x[a] += delta;
}
else if (tail[a] == j)
{ /* a = i<-j is a backward arc of the cycle */
i = head[a];
x[a] -= delta;
}
else
xassert(a != a);
}
goto loop;
done: /* free working arrays */
xfree(ptr);
xfree(arc);
xfree(link);
xfree(list);
return;
}
/* eof */
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