/// <summary>
/// Compute coarse and then fine Dulmage-Mendelsohn decomposition. seed
/// optionally selects a randomized algorithm.
/// </summary>
/// <param name="matrix">column-compressed matrix</param>
/// <param name="seed">0: natural, -1: reverse, random order otherwise</param>
/// <returns>Dulmage-Mendelsohn analysis</returns>
public static DulmageMendelsohn Generate <T>(CompressedColumnStorage <T> matrix, int seed = 0)
where T : struct, IEquatable <T>, IFormattable
{
int i, j, k, cnz, nc, nb1, nb2;
int[] Cp, ps, rs;
bool ok;
// We are not interested in the actual matrix values.
var A = SymbolicColumnStorage.Create(matrix);
// Maximum matching
int m = A.RowCount;
int n = A.ColumnCount;
var result = new DulmageMendelsohn(m, n); // allocate result
int[] p = result.p;
int[] q = result.q;
int[] r = result.r;
int[] s = result.s;
int[] cc = result.cc;
int[] rr = result.rr;
int[] jimatch = MaximumMatching.Generate(A, seed); // max transversal
if (jimatch == null)
{
return(null);
}
// Coarse decomposition
for (j = 0; j < n; j++)
{
s[j] = -1; // unmark all cols for bfs
}
for (i = 0; i < m; i++)
{
r[i] = -1; // unmark all rows for bfs
}
result.BreadthFirstSearch(A, n, r, s, q, jimatch, m, 0, 1); // find C1, R1 from C0*/
ok = result.BreadthFirstSearch(A, m, s, r, p, jimatch, 0, m, 3); // find R3, C3 from R0*/
if (!ok)
{
return(null);
}
result.Unmatched(n, s, q, cc, 0); // unmatched set C0
result.Matched(n, s, jimatch, m, p, q, cc, rr, 1, 1); // set R1 and C1
result.Matched(n, s, jimatch, m, p, q, cc, rr, 2, -1); // set R2 and C2
result.Matched(n, s, jimatch, m, p, q, cc, rr, 3, 3); // set R3 and C3
result.Unmatched(m, r, p, rr, 3); // unmatched set R0
// Fine decomposition
int[] pinv = Permutation.Invert(p); // pinv=p'
var C = SymbolicColumnStorage.Create(matrix);
A.Permute(pinv, q, C); // C=A(p,q) (it will hold A(R2,C2))
Cp = C.ColumnPointers;
nc = cc[3] - cc[2]; // delete cols C0, C1, and C3 from C
if (cc[2] > 0)
{
for (j = cc[2]; j <= cc[3]; j++)
{
Cp[j - cc[2]] = Cp[j];
}
}
C.Reshape(-1, nc);
if (rr[2] - rr[1] < m) // delete rows R0, R1, and R3 from C
{
RowPrune(C, nc, rr);
cnz = Cp[nc];
int[] Ci = C.RowIndices;
if (rr[1] > 0)
{
for (k = 0; k < cnz; k++)
{
Ci[k] -= rr[1];
}
}
}
C.Reshape(nc, -1);
var scc = StronglyConnectedComponents.Generate(C, nc); // find strongly connected components of C*/
// Combine coarse and fine decompositions
ps = scc.p; // C(ps,ps) is the permuted matrix
rs = scc.r; // kth block is rs[k]..rs[k+1]-1
nb1 = scc.nb; // # of blocks of A(R2,C2)
for (k = 0; k < nc; k++)
{
s[k] = q[ps[k] + cc[2]];
}
for (k = 0; k < nc; k++)
{
q[k + cc[2]] = s[k];
}
for (k = 0; k < nc; k++)
{
r[k] = p[ps[k] + rr[1]];
}
for (k = 0; k < nc; k++)
{
p[k + rr[1]] = r[k];
}
nb2 = 0; // create the fine block partitions
r[0] = s[0] = 0;
if (cc[2] > 0)
{
nb2++; // leading coarse block A (R1, [C0 C1])
}
for (k = 0; k < nb1; k++) // coarse block A (R2,C2)
{
r[nb2] = rs[k] + rr[1]; // A (R2,C2) splits into nb1 fine blocks
s[nb2] = rs[k] + cc[2];
nb2++;
}
if (rr[2] < m)
{
r[nb2] = rr[2]; // trailing coarse block A ([R3 R0], C3)
s[nb2] = cc[3];
nb2++;
}
r[nb2] = m;
s[nb2] = n;
result.nb = nb2;
// Remove unused space
Array.Resize(ref result.r, nb2 + 1);
Array.Resize(ref result.s, nb2 + 1);
return(result);
}
/// <summary>
/// Finds the strongly connected components of a square matrix.
/// </summary>
/// <returns>strongly connected components, null on error</returns>
private static DulmageMendelsohn FindScc(SymbolicColumnStorage A, int n)
{
// matrix A temporarily modified, then restored
int i, k, b, nb = 0, top;
int[] xi, p, r, Ap, ATp;
var AT = A.Transpose(); // AT = A'
Ap = A.ColumnPointers;
ATp = AT.ColumnPointers;
xi = new int[2 * n + 1]; // get workspace
var D = new DulmageMendelsohn(n, 0); // allocate result
p = D.p;
r = D.r;
top = n;
for (i = 0; i < n; i++) // first dfs(A) to find finish times (xi)
{
if (!(Ap[i] < 0))
{
top = GraphHelper.DepthFirstSearch(i, A.ColumnPointers, A.RowIndices, top, xi, xi, n, null);
}
}
for (i = 0; i < n; i++)
{
//CS_MARK(Ap, i);
Ap[i] = -(Ap[i]) - 2; // restore A; unmark all nodes
}
top = n;
nb = n;
for (k = 0; k < n; k++) // dfs(A') to find strongly connnected comp
{
i = xi[k]; // get i in reverse order of finish times
if (ATp[i] < 0)
{
continue; // skip node i if already ordered
}
r[nb--] = top; // node i is the start of a component in p
top = GraphHelper.DepthFirstSearch(i, AT.ColumnPointers, AT.RowIndices, top, p, xi, n, null);
}
r[nb] = 0; // first block starts at zero; shift r up
for (k = nb; k <= n; k++)
{
r[k - nb] = r[k];
}
D.nb = nb = n - nb; // nb = # of strongly connected components
for (b = 0; b < nb; b++) // sort each block in natural order
{
for (k = r[b]; k < r[b + 1]; k++)
{
xi[p[k]] = b;
}
}
for (b = 0; b <= nb; b++)
{
xi[n + b] = r[b];
}
for (i = 0; i < n; i++)
{
p[xi[n + xi[i]]++] = i;
}
return(D);
}
