private void StrongConnect(int v, int vertexCount, uint[] edgeBits, ref TarjanOutput output)
{
Debug.Assert((vertexIndex[v] & OnStackBit) == 0);
vertexIndex[v] = index | OnStackBit;
vertexLowLink[v] = index;
index++;
stack[stackEnd++] = v; // Stack Push
// Consider successors of v
for (int w = 0; w < vertexCount; w++)
{
// NOTE: Considered using FFS-bit operation, but that only moves O(|V|) down to O(|E|) on a per-32-bit-block basis (by block will always be O(|V|)), probably not worth it -AR
// PERF: Validate that this is inlined by the JIT (I think it should be -AR)
if (!edgeBits.IsEdge(vertexCount, v, w))
{
continue;
}
if (vertexIndex[w] == 0) // "not visited"
{
// Successor w has not yet been visited; recurse on it
StrongConnect(w, vertexCount, edgeBits, ref output);
vertexLowLink[v] = System.Math.Min(vertexLowLink[v], vertexLowLink[w]);
}
else if ((vertexIndex[w] & OnStackBit) != 0)
{
// Successor w is in the stack and hence in the current strongly-connected-component
vertexLowLink[v] = System.Math.Min(vertexLowLink[v], vertexIndex[w] & ~OnStackBit);
}
}
// If v is a root node, pop the stack and generate a strongly-connected-component
if ((vertexIndex[v] & ~OnStackBit) == vertexLowLink[v])
{
int componentEnd = stackEnd; // <- rather than popping into a buffer, keep track of the top of the stack so we can return the range
// Repeatedly pop until we pop off v:
do
{
stackEnd--; // Stack Pop (stack[stackEnd] is what was popped)
Debug.Assert((vertexIndex[v] & OnStackBit) != 0);
vertexIndex[stack[stackEnd]] &= ~OnStackBit; // Clear stack bit
} while(stack[stackEnd] != v);
int count = componentEnd - stackEnd;
Debug.Assert(count >= 1);
// Write output:
if (count > 1)
{
output.sccList[output.sccCount++] = new SCC(output.vertexCount, count); // Add
Array.Copy(stack, stackEnd, output.vertexList, output.vertexCount, count); // AddRange
output.vertexCount += count;
}
else // Lone node (don't bother tracking these as SCCs, and don't bother with the bulk copy)
{
output.vertexList[output.vertexCount++] = stack[stackEnd]; // Add
}
}
}
public void FindStronglyConnectedComponents(int vertexCount, uint[] edgeBits, ref TarjanOutput output)
{
Debug.Assert(output.sccList.Length >= vertexCount / 3); // <- NOTE: we could check for "vertexCount/2" but having less space is an optimisation for when we don't have a bi-directional graph
Debug.Assert(output.vertexList.Length >= vertexCount);
// INITIALIZE:
{
index = 1; // <- 0 is "undefined" (so we can fast-clear our indicies)
stackEnd = 0;
if (stack == null || stack.Length < vertexCount)
{
// PERF: Convert this to use a single buffer, with sections reserved for each purpose, so there are no gaps
// Also interleave lowlink and index. And could use short instead of int. All probably overkill.
// NOTE: Lazy over-allocate here, to avoid reallocation
stack = new int[vertexCount * 2]; // <- the maximum size for the stack is vertexCount
vertexIndex = new uint[vertexCount * 2];
vertexLowLink = new uint[vertexCount * 2];
}
else // <- don't have to clear if we just got fresh memory from .NET
{
Array.Clear(vertexIndex, 0, vertexCount); // <- only have to clear as much as we'll use here...
// NOTE: don't need to inialize vertexLowLink or stack
}
output.sccCount = 0;
output.vertexCount = 0;
}
// RUN:
{
for (int v = 0; v < vertexCount; v++)
{
if (vertexIndex[v] == 0) // "not visited"
{
StrongConnect(v, vertexCount, edgeBits, ref output);
}
}
}
Debug.Assert(output.vertexCount == vertexCount);
}
SCC[] bestSCCs; // count is a local
/// <summary>NOTE: Returns a reference to an internal array which is cleared on subsequent sorts. May be longer than the passed in vertexCount - excess data is garbage.</summary>
public int[] Sort(int vertexCount, uint[] edgeBits, Action <int, int> removedEdge)
{
//
// INITIALIZE:
//
{
if (primaryVertices == null || primaryVertices.Length < vertexCount)
{
// NOTE: Lazy over-allocate here, to avoid reallocation
primaryVertices = new int[vertexCount * 2];
primarySCCStack = new SCC[(vertexCount * 2) / 3]; // worst-case capacity required is that every vertex triplet is an SCC
// Secondary stage (NOTE: worst-case is that the whole of primary is a SCC, so this must be the same size)
secondaryEdgeBits = EdgeBits.Create(vertexCount * 2); // NOTE: This allocates 4x as much memory as was passed in for edges
secondaryVertices = new int[vertexCount * 2];
secondarySCCs = new SCC[(vertexCount * 2) / 3];
bestVertices = new int[vertexCount * 2];
bestSCCs = new SCC[(vertexCount * 2) / 3];
tempOutput = new int[vertexCount * 2];
}
else
{
// NOTE: Nothing to clear (all items have counts and get re-initialized up to their count upon usage)
}
}
//
// RUN:
//
{
// Primary:
// --------
TarjanOutput primaryOutput = new TarjanOutput();
primaryOutput.sccList = primarySCCStack;
primaryOutput.vertexList = primaryVertices;
ta.FindStronglyConnectedComponents(vertexCount, edgeBits, ref primaryOutput);
// Secondary: (this is a greedy search)
// ----------
// Try to resolve any SCCs remaining in the output
// Note that changes to a SCC do not affect the ordering of the surrounding DAG
while (primaryOutput.sccCount > 0)
{
SCC workingSCC = primaryOutput.sccList[--primaryOutput.sccCount]; // Pop
// The secondary stage uses indirect indices (ie: vertex number in primary = scc[vertex number in secondary])
// This allows the secondary stage to use implicit vertex numbers (0 to N-1, rather than having to access workingSCC)
// This requires rebuilding edges to match this indexing scheme (secondaryEdge[v, w] = primaryEdge[scc[v], scc[w]])
// But this is desireable, because the access pattern into the primary edges would be cache-unfriendly
// Also secondary is generally much smaller than primary (because it works on individual SCCs, which are usually relatively small)
// Rebuild edge bits:
int v, w; // v is "from", w is "to"
for (v = 0; v < workingSCC.count; v++)
{
for (w = 0; w < workingSCC.count; w++)
{
secondaryEdgeBits.ChangeEdgeBit(workingSCC.count, v, w,
edgeBits.GetEdgeBit(vertexCount, primaryVertices[workingSCC.index + v], primaryVertices[workingSCC.index + w]));
}
}
int bestSCCCount = 0;
int bestSCCMaximumSize = workingSCC.count; // The size of the largest SCCs <- used to select "best" (reduces size of O(N^2) problem -- unverified -AR)
int bestSCCCumulativeSize = workingSCC.count; // Number of vertices in all SCCs <- used as a tie-break (reduces size of O(N) problem -- unverified -AR)
// These are for logging
int bestRemovedV = 0;
int bestRemovedW = 0;
// Try to find the best constraint (ie: edge) to remove:
for (v = 0; v < workingSCC.count; v++)
{
for (w = 0; w < workingSCC.count; w++)
{
if (secondaryEdgeBits.IsEdge(workingSCC.count, v, w)) // PERF: Consider only looking at "backwards" edges (requires primary input is spatially sorted, then compare primary indices)
{
TarjanOutput secondaryOutput = new TarjanOutput();
secondaryOutput.sccList = secondarySCCs;
secondaryOutput.vertexList = secondaryVertices;
secondaryEdgeBits.ClearEdge(workingSCC.count, v, w); // Temporarly remove an edge (see if it improves the result)
ta.FindStronglyConnectedComponents(workingSCC.count, secondaryEdgeBits, ref secondaryOutput);
int secondarySCCMaximumSize = 0;
int secondarySCCCumulativeSize = 0;
for (int i = 0; i < secondaryOutput.sccCount; i++)
{
int count = secondaryOutput.sccList[i].count;
if (count > secondarySCCMaximumSize)
{
secondarySCCMaximumSize = count;
}
secondarySCCCumulativeSize += count;
}
// NOTE: If we change to only looking at "backwards" edges, should probably add tie-break to select rearmost edge to remove
// (May require pre-sorting the SCC, to handle the early-out case properly)
if (secondaryOutput.sccCount == 0 ||
secondarySCCMaximumSize < bestSCCMaximumSize ||
(secondarySCCMaximumSize == bestSCCMaximumSize && secondarySCCCumulativeSize < bestSCCCumulativeSize))
{
bestRemovedV = v;
bestRemovedW = w;
bestSCCCount = secondaryOutput.sccCount;
bestSCCMaximumSize = secondarySCCMaximumSize;
bestSCCCumulativeSize = secondarySCCCumulativeSize;
Swap(ref bestSCCs, ref secondarySCCs);
Swap(ref bestVertices, ref secondaryVertices);
}
if (secondaryOutput.sccCount == 0)
{
goto earlyOut; // Found optimal solution
}
secondaryEdgeBits.SetEdge(workingSCC.count, v, w); // Restore Edge
}
}
}
if (bestSCCMaximumSize == workingSCC.count) // Failed to find any improvement for this SCC - give up!
{
// Just leave the primary output in its original order
// NOTE: If we change to sorted input (for "backwards" edges), consider re-sorting this SCC in output
continue;
}
earlyOut: // skips the above
// Notify any debug output that an edge was removed. NOTE: Before the SCC in primary gets re-ordered!
if (removedEdge != null)
{
removedEdge(primaryVertices[workingSCC.index + bestRemovedV], primaryVertices[workingSCC.index + bestRemovedW]);
}
// Apply the new ordering
for (int i = 0; i < workingSCC.count; i++)
{
tempOutput[i] = primaryVertices[workingSCC.index + bestVertices[i]];
}
Array.Copy(tempOutput, 0, primaryVertices, workingSCC.index, workingSCC.count);
// Transfer new SCCs to primary:
for (int i = 0; i < bestSCCCount; i++)
{
primaryOutput.sccList[primaryOutput.sccCount++] = new SCC(workingSCC.index + bestSCCs[i].index, bestSCCs[i].count);
}
}
// At this point, primarySCC stack is empty -- we're done
return(primaryVertices);
}
}
