private EquationDecomposition(EquationSystem eqsys)
{
/** This algorithm decomposes a given equation system into smaller equation blocks
* which can be solved subsequently. The algorithm is sometimes referred to as
* Tarjan's sorting or Dulmage-Mendelson decomposition.
*
* The algorithm consists of several steps. The description below will use the
* following notions:
* Var: The set of variables of the whole equation system
* Eqn: The set of equations of the whole equation system
* Var u Eqn: The set unification of Var and Eqn
*
* Step (1): Maximum matching
* Define an undirected graph G = (V, E) with V = Var u Eqn.
* Whenever an equation eqn \in Eqn references a variable var \in Var
* add the edge (eqn, var) to E.
* Compute a maximum matching M between variables and equations.
* M will now contain a set of equation/variable pairs.
*
* Step (2): Finding strongly connected components
* Define a directed graph Gd = (V, Ed) with V = Var u Eqn.
* Whenever an equation eqn \in Eqn references a variable var \in Var
* add the edge (eqn, var) to Ed.
* For each equation/variable pair (eqn, var) in M add furthermore
* the edge (var, eqn) to Ed.
* Note: Matched equation/variable pairs will result in bi-directional edges in Ed.
* Compute the strongly connected components SCC of Gd.
* SCC is a set of subsets of V such that each subset represents one component.
*
* Step (3): Building the tree of equation blocks
* Defined a directed graph Dt = (SCC, Et).
* For each equation eqn \in Eqn which is assigned to component c1 and
* references a variable var \in Var which is assigned to component c2
* add an edge (c1, c2) to Et of c1 and c2 are different.
*
* The resulting graph will have a tree/forest sutrcture. Each node
* represents an equation block. An edge b1 -> b2 represents a dependency in the
* that b2 must be solved prior to b1. A schedule can be found by topological
* sorting the tree.
* */
if (eqsys == null ||
eqsys.Equations == null ||
eqsys.Variables == null)
throw new ArgumentException("null reference in equation system argument");
if (eqsys.Equations.Count != eqsys.Variables.Count)
throw new ArgumentException("equation system is not well-constrained!");
/** Assign each equation and each variable an index which will be a unique node
* ID within the graphs. Indices are assigned as follows:
* - Equations get indices from 0...count(equations)-1
* - Variables get indices from count(equations)...count(equations)+count(variables)
* Note: Assuming a well-constrained equation system, count(equations) = count(variables)
**/
int curidx = 0;
foreach (Expression eq in eqsys.Equations)
{
eq.Cookie = curidx++;
}
Dictionary<object, int> idx = new Dictionary<object, int>();
foreach (Variable v in eqsys.Variables)
{
idx[v] = curidx++;
}
// Steps 1 and 2: Construct G and Gd
Graph g = new Graph(curidx);
Digraph dg = new Digraph(curidx);
for (int i = 0; i < eqsys.Equations.Count; i++)
{
Expression eq = eqsys.Equations[i];
LiteralReference[] lrs = eq.ExtractLiteralReferences();
foreach (LiteralReference lr in lrs)
{
int j;
if (idx.TryGetValue(lr.ReferencedObject, out j))
{
g.AddEdge(i, j);
dg.AddEdge(i, j);
}
}
}
// Step 1: Maximum matching
GraphAlgorithms.Matching m = g.GetMaximumMatching();
bool success = m.IsMaximumCardinality;
for (int i = 0; i < eqsys.Equations.Count; i++)
{
int k = m[i];
if (k >= 0)
dg.AddEdge(k, i);
}
// Step 2: Strongly connected components
StrongComponents sc = dg.GetStrongComponents();
int numc = sc.NumComponents;
EquationBlock[] blocks = new EquationBlock[numc];
for (int i = 0; i < numc; i++)
{
blocks[i] = new EquationBlock();
}
// Step 3: Construct the tree
Digraph dg2 = new Digraph(numc);
for (int i = 0; i < eqsys.Equations.Count; i++)
{
int c = sc[i];
blocks[c].EqSys.Equations.Add(eqsys.Equations[i]);
int vi = m[i];
Variable v = eqsys.Variables[vi - eqsys.Equations.Count];
blocks[c].EqSys.Variables.Add(v);
List<int> outset = dg.GetOutSet(i);
foreach (int ovi in outset)
{
int oc = sc[ovi];
if (c != oc)
dg2.AddEdge(c, oc);
}
}
List<int> rootSet = new List<int>();
List<int> sinkSet = new List<int>();
for (int c = 0; c < numc; c++)
{
if (dg2.GetOutDegree(c) == 0)
rootSet.Add(c);
if (dg2.GetInDegree(c) == 0)
sinkSet.Add(c);
List<int> outset = dg2.GetOutSet(c);
foreach (int oc in outset)
blocks[c].Predecessors.Add(blocks[oc]);
List<int> inset = dg2.GetInSet(c);
foreach (int ic in inset)
blocks[c].Successors.Add(blocks[ic]);
}
RootSet = (from int c in rootSet
select blocks[c]).ToArray();
SinkSet = (from int c in sinkSet
select blocks[c]).ToArray();
AllBlocks = blocks;
}
private EquationDecomposition(EquationSystem eqsys)
{
/** This algorithm decomposes a given equation system into smaller equation blocks
* which can be solved subsequently. The algorithm is sometimes referred to as
* Tarjan's sorting or Dulmage-Mendelson decomposition.
*
* The algorithm consists of several steps. The description below will use the
* following notions:
* Var: The set of variables of the whole equation system
* Eqn: The set of equations of the whole equation system
* Var u Eqn: The set unification of Var and Eqn
*
* Step (1): Maximum matching
* Define an undirected graph G = (V, E) with V = Var u Eqn.
* Whenever an equation eqn \in Eqn references a variable var \in Var
* add the edge (eqn, var) to E.
* Compute a maximum matching M between variables and equations.
* M will now contain a set of equation/variable pairs.
*
* Step (2): Finding strongly connected components
* Define a directed graph Gd = (V, Ed) with V = Var u Eqn.
* Whenever an equation eqn \in Eqn references a variable var \in Var
* add the edge (eqn, var) to Ed.
* For each equation/variable pair (eqn, var) in M add furthermore
* the edge (var, eqn) to Ed.
* Note: Matched equation/variable pairs will result in bi-directional edges in Ed.
* Compute the strongly connected components SCC of Gd.
* SCC is a set of subsets of V such that each subset represents one component.
*
* Step (3): Building the tree of equation blocks
* Defined a directed graph Dt = (SCC, Et).
* For each equation eqn \in Eqn which is assigned to component c1 and
* references a variable var \in Var which is assigned to component c2
* add an edge (c1, c2) to Et of c1 and c2 are different.
*
* The resulting graph will have a tree/forest sutrcture. Each node
* represents an equation block. An edge b1 -> b2 represents a dependency in the
* that b2 must be solved prior to b1. A schedule can be found by topological
* sorting the tree.
* */
if (eqsys == null ||
eqsys.Equations == null ||
eqsys.Variables == null)
{
throw new ArgumentException("null reference in equation system argument");
}
if (eqsys.Equations.Count != eqsys.Variables.Count)
{
throw new ArgumentException("equation system is not well-constrained!");
}
/** Assign each equation and each variable an index which will be a unique node
* ID within the graphs. Indices are assigned as follows:
* - Equations get indices from 0...count(equations)-1
* - Variables get indices from count(equations)...count(equations)+count(variables)
* Note: Assuming a well-constrained equation system, count(equations) = count(variables)
**/
int curidx = 0;
foreach (Expression eq in eqsys.Equations)
{
eq.Cookie = curidx++;
}
Dictionary <object, int> idx = new Dictionary <object, int>();
foreach (Variable v in eqsys.Variables)
{
idx[v] = curidx++;
}
// Steps 1 and 2: Construct G and Gd
Graph g = new Graph(curidx);
Digraph dg = new Digraph(curidx);
for (int i = 0; i < eqsys.Equations.Count; i++)
{
Expression eq = eqsys.Equations[i];
LiteralReference[] lrs = eq.ExtractLiteralReferences();
foreach (LiteralReference lr in lrs)
{
int j;
if (idx.TryGetValue(lr.ReferencedObject, out j))
{
g.AddEdge(i, j);
dg.AddEdge(i, j);
}
}
}
// Step 1: Maximum matching
GraphAlgorithms.Matching m = g.GetMaximumMatching();
bool success = m.IsMaximumCardinality;
for (int i = 0; i < eqsys.Equations.Count; i++)
{
int k = m[i];
if (k >= 0)
{
dg.AddEdge(k, i);
}
}
// Step 2: Strongly connected components
StrongComponents sc = dg.GetStrongComponents();
int numc = sc.NumComponents;
EquationBlock[] blocks = new EquationBlock[numc];
for (int i = 0; i < numc; i++)
{
blocks[i] = new EquationBlock();
}
// Step 3: Construct the tree
Digraph dg2 = new Digraph(numc);
for (int i = 0; i < eqsys.Equations.Count; i++)
{
int c = sc[i];
blocks[c].EqSys.Equations.Add(eqsys.Equations[i]);
int vi = m[i];
Variable v = eqsys.Variables[vi - eqsys.Equations.Count];
blocks[c].EqSys.Variables.Add(v);
List <int> outset = dg.GetOutSet(i);
foreach (int ovi in outset)
{
int oc = sc[ovi];
if (c != oc)
{
dg2.AddEdge(c, oc);
}
}
}
List <int> rootSet = new List <int>();
List <int> sinkSet = new List <int>();
for (int c = 0; c < numc; c++)
{
if (dg2.GetOutDegree(c) == 0)
{
rootSet.Add(c);
}
if (dg2.GetInDegree(c) == 0)
{
sinkSet.Add(c);
}
List <int> outset = dg2.GetOutSet(c);
foreach (int oc in outset)
{
blocks[c].Predecessors.Add(blocks[oc]);
}
List <int> inset = dg2.GetInSet(c);
foreach (int ic in inset)
{
blocks[c].Successors.Add(blocks[ic]);
}
}
RootSet = (from int c in rootSet
select blocks[c]).ToArray();
SinkSet = (from int c in sinkSet
select blocks[c]).ToArray();
AllBlocks = blocks;
}
