/// <summary>
/// Solves a sparse instance of the linear assignment problem with integer costs.
/// </summary>
/// <param name="cost">The weights of the edges of the bipartite graph representing the problem.</param>
/// <param name="maximize">Whether or not to maximize total cost rather than minimize it.</param>
/// <param name="solver">The solver to use. If not given, this defaults to <see cref="ShortestPathSolver"/>.</param>
/// <param name="allowOverwrite">Allows the entries <paramref name="cost"/> to be changed; setting this to
/// <see langword="true" /> can give performance improvements in certain cases.</param>
/// <returns>An <see cref="Assignment"/> representing the solution.</returns>
public static Assignment Solve(SparseMatrixInt cost, bool maximize = false,
ISolver solver = null, bool allowOverwrite = false)
{
// Currently, the implemented solvers just translate to the floating point
// equivalents, so there's no reason to distinguish here.
return(Solve(new SparseMatrixDouble(cost), maximize, solver, true));
}
private Assignment Solve(int[,] costDense, SparseMatrixInt costSparse)
{
// The signature here is kind of nasty; it would be much nicer if we could split
// out the method into one for the dense case, and one for the sparse case. However,
// if we want to avoid code duplication, some sort of abstractions need to be in
// place for the (substantial) parts of the algorithm that they have in common, and
// including such abstractions is impossible without incurring a performance penalty
// in some of our hot loops, which is an even worse alternative.
if (costDense != null && costSparse != null)
{
throw new ArgumentException("Only one parameter may be non-null.");
}
var isSparse = costSparse != null;
List <int> ia = null;
List <int> a = null;
List <int> ca = null;
if (isSparse)
{
ia = costSparse.IA;
a = costSparse.A;
ca = costSparse.CA;
}
// TODO: Allow rectangular inputs
var nr = isSparse ? costSparse.NumRows : costDense.GetLength(0);
var nc = isSparse ? costSparse.NumColumns : costDense.GetLength(1);
if (nr != nc)
{
throw new NotImplementedException("Pseudoflow is only implemented for square matrices");
}
var n = nr;
// Initialize cost-scaling to be the configured value if given, and
// otherwise let it be the largest given cost.
double epsilon;
if (_initialEpsilon.HasValue)
{
epsilon = _initialEpsilon.Value;
}
else if (isSparse)
{
epsilon = costSparse.MaxValue;
}
else
{
epsilon = double.NegativeInfinity;
for (var i = 0; i < n; i++)
{
for (var j = 0; j < n; j++)
{
if (costDense[i, j] > epsilon && costDense[i, j] != int.MaxValue)
{
epsilon = costDense[i, j];
}
}
}
}
// Initialize dual variables and assignment variables keeping track of
// assignments as we move along: col maps a given row to the column
// it's assigned to, and conversely row maps a given column to its
// assigned row.
var v = new double[n];
var col = new int[n];
var row = new int[n];
while (epsilon >= 1d / n)
{
epsilon /= _alpha;
// A value in -1 in row and col corresponds to "unassigned"
for (var i = 0; i < n; i++)
{
col[i] = -1;
}
for (var j = 0; j < n; j++)
{
row[j] = -1;
}
// We also maintain a stack of rows that have not been assigned. We
// could get this information from the variable col, but being able to
// just pop the stack to get new unassigned rows is much faster. We put
// lower numbers at the top of the stack simply because that feels a bit
// more natural; we could get rid of the Reverse if we wanted to.
var unassigned = new Stack <int>(Enumerable.Range(1, n - 1).Reverse());
var k = 0;
// At this point, Burkard--Dell'Amico--Martello would update
// the dual variable u. However, as they note, and as is noted in
// Goldberg--Kennedy, the variable is only used to evaluate the reduced costs
// when determining the smallest and second smallest reduced costs in the first
// part of the double-push. However, we actually only care about the argmin and
// the arg-second-min, which end up being independent of the value of u. This is
// also apparent from the original implementation of CSA-Q, in which only the
// partial reduced costs are used. The bottom-line is that we never need to
// calculate u or any of the reduced costs; we can do with only v and partial
// reduced costs instead; doing so provides a large speed increase over the
// naive implementation -- and simpler code.
while (true)
{
// Perform double-push. The halting condition is that all rows
// have been assigned, which corresponds to the stack of unassigned
// rows having been emptied.
var smallest = double.PositiveInfinity;
var j = -1;
var secondSmallest = double.PositiveInfinity;
// Again, below we find some duplication in the gap calculation, but this
// is all in the name of performance.
if (isSparse)
{
for (var m = ia[k]; m < ia[k + 1]; m++)
{
var jp = ca[m];
var partialReducedCost = a[m] - v[jp];
if (partialReducedCost <= secondSmallest)
{
if (partialReducedCost <= smallest)
{
secondSmallest = smallest;
smallest = partialReducedCost;
j = jp;
}
else
{
secondSmallest = partialReducedCost;
}
}
}
}
else
{
for (var jp = 0; jp < n; jp++)
{
var partialReducedCost = costDense[k, jp] - v[jp];
if (partialReducedCost <= secondSmallest)
{
if (partialReducedCost <= smallest)
{
secondSmallest = smallest;
smallest = partialReducedCost;
j = jp;
}
else
{
secondSmallest = partialReducedCost;
}
}
}
}
col[k] = j;
// TODO: Detect infeasibility by investigating dual updates.
if (row[j] != -1)
{
var i = row[j];
row[j] = k;
// The Burkard--Dell'Amico--Martello updates v[j] to be cost[k, j] - u[k] - epsilon,
// but cost[k, j] - u[k] = v[j] + smallest - secondSmallest; using this instead avoids a
// few costly cost lookups.
// Note moreover that if k only has a single incident egde, then secondSmallest becomes
// double.NegativeInfinity. This in turn means that j will never be picked as the smallest
// or second smallest element again, which guarantees that j will be forever assigned to k,
// which is exactly what we want.
v[j] += smallest - secondSmallest - epsilon;
col[i] = -1;
k = i;
}
else
{
row[j] = k;
if (unassigned.Count == 0)
{
break;
}
k = unassigned.Pop();
}
}
}
return(new Assignment(col, row));
}
public Assignment Solve(SparseMatrixInt cost) => Solve(null, cost);
public SparseMatrixDouble(SparseMatrixInt sparse) :
this(sparse.A.ConvertAll(x => (double)x), new List <int>(sparse.IA), new List <int>(sparse.CA), sparse.NumColumns)
{
_max = sparse.MaxValue;
}
public Assignment Solve(SparseMatrixInt cost) => Solve(new SparseMatrixDouble(cost));
public SparseMatrixInt(SparseMatrixInt matrix) :
this(new List <int>(matrix.A), new List <int>(matrix.IA), new List <int>(matrix.CA), matrix.NumColumns)
{
}
