static internal EdgesAndExpectations GetStaticStrategy(Graph graph,
int[] sources,
HSet targets,
int nStates,
int resetCost,
int [] deadStates)
{
foreach (int t in targets)
{
foreach (Edge l in graph.EdgesAtVertex(t))
{
l.weight = 0;
}
}
//fix the edges weights
foreach (Edge l in graph.MustEdges)
{
if (l.target >= nStates)
{
l.weight = 0;
}
}
foreach (Edge l in graph.OptionalEdges)
{
l.weight = resetCost;
}
if (graph.NumberOfVertices > 1000)//Value iteration becomes too slow
{
return(graph.GetStaticStrategyWithDistancesToAcceptingStates(sources, targets.ToArray(typeof(int)) as int[]));
}
HSet deadStatesSet = new HSet(deadStates);
//create reachableGraph
bool [] reachableVerts = new bool[graph.NumberOfVertices];
//we have to walk backwards from the targets avoiding dead states
graph.InitBackwardEdges();
foreach (int i in targets)
{
reachableVerts[i] = true;
}
System.Collections.Queue queue = new System.Collections.Queue(targets);
while (queue.Count > 0)
{
int i = (int)queue.Dequeue();
foreach (int v in graph.Pred(i))
{
if (!reachableVerts[v] && !deadStatesSet.Contains(v))
{
queue.Enqueue(v);
reachableVerts[v] = true;
}
}
}
int numberOfReachableVerts = 0;
foreach (bool b in reachableVerts)
{
if (b)
{
numberOfReachableVerts++;
}
}
Edge[] strategyEdges;
double [] expectations;
if (numberOfReachableVerts == graph.NumberOfVertices)
{
expectations = GetExpectations(graph, /* sources,*/ targets, nStates);
if (expectations == null)
{
return(new EdgesAndExpectations());
}
strategyEdges = new Edge[nStates];
for (int i = 0; i < nStates && i < graph.NumberOfVertices; i++)
{
if (targets.Contains(i) || deadStatesSet.Contains(i))
{
continue;
}
double min = Single.MaxValue;
Edge stEdge = null;
foreach (Edge l in graph.EdgesAtVertex(i))
{
int j = l.target;
if (expectations[j] < min)
{
min = expectations[j];
stEdge = l;
}
}
strategyEdges[i] = stEdge;
}
}
else
{ //numberOfReachableVerts<graph.NumberOfVertices)
int [] graphToRG = new int[graph.NumberOfVertices];
//reachable graph to graph
int [] rGToGraph = new int[numberOfReachableVerts];
int count = 0;
int rNStates = 0;
for (int i = 0; i < reachableVerts.Length; i++)
{
if (reachableVerts[i])
{
graphToRG[i] = count;
rGToGraph[count] = i;
count++;
if (i < nStates)
{
rNStates++;
}
}
}
System.Collections.ArrayList mustEdges = new System.Collections.ArrayList();
foreach (Edge l in graph.MustEdges)
{
if (reachableVerts[l.source] && reachableVerts[l.target])
{
Edge ml = new Edge(graphToRG[l.source], graphToRG[l.target], l.label, l.weight);
mustEdges.Add(ml);
}
}
System.Collections.ArrayList nondVerts = new System.Collections.ArrayList();
for (int i = nStates; i < graph.NumberOfVertices; i++)
{
if (reachableVerts[i])
{
nondVerts.Add(graphToRG[i]);
}
}
Graph rGraph = new Graph(0, mustEdges.ToArray(typeof(Edge)) as Edge[], new Edge[0],
nondVerts.ToArray(typeof(int)) as int[], true, WeakClosureEnum.DoNotClose);
int [] rSources = new int[sources.Length];
int c = 0;
foreach (int s in sources)
{
rSources[c++] = graphToRG[s];
}
HSet rTargets = new HSet();
foreach (int s in targets)
{
if (reachableVerts[s])
{
rTargets.Insert(graphToRG[s]);
}
}
double [] rExpectations = GetExpectations(rGraph, /*rSources,*/ rTargets, rNStates);
if (rExpectations == null)
{
return(new EdgesAndExpectations());
}
strategyEdges = new Edge[nStates];
for (int i = 0; i < nStates; i++)
{
if (!reachableVerts[i])
{
continue;
}
if (targets.Contains(i) || deadStatesSet.Contains(i))
{
continue;
}
double min = Single.MaxValue;
Edge stEdge = null;
foreach (Edge l in graph.EdgesAtVertex(i))
{
int j = l.target;
if (reachableVerts[j])
{
if (rExpectations[graphToRG[j]] < min)
{
min = rExpectations[graphToRG[j]];
stEdge = l;
}
}
}
strategyEdges[i] = stEdge;
}
expectations = new double[graph.NumberOfVertices];
if (expectations == null)
{
return(new EdgesAndExpectations());
}
for (int i = 0; i < expectations.Length; i++)
{
expectations[i] = Int32.MaxValue;
}
for (int i = 0; i < rExpectations.Length; i++)
{
expectations[rGToGraph[i]] = rExpectations[i];
}
}
graph.CleanTheStrategy(strategyEdges, sources);
return(new EdgesAndExpectations(strategyEdges, expectations));
}
internal static EdgesAndExpectations GetStaticStrategy(Graph graph,
int[] sources,
HSet targets,
int nStates,
int resetCost,
int []deadStates)
{
foreach(int t in targets){
foreach(Edge l in graph.EdgesAtVertex(t))
l.weight=0;
}
//fix the edges weights
foreach(Edge l in graph.MustEdges){
if(l.target>=nStates)
l.weight=0;
}
foreach(Edge l in graph.OptionalEdges)
l.weight=resetCost;
if(graph.NumberOfVertices>1000){//Value iteration becomes too slow
return graph.GetStaticStrategyWithDistancesToAcceptingStates(sources, targets.ToArray(typeof(int)) as int[]);
}
HSet deadStatesSet=new HSet(deadStates);
//create reachableGraph
bool []reachableVerts=new bool[graph.NumberOfVertices];
//we have to walk backwards from the targets avoiding dead states
graph.InitBackwardEdges();
foreach(int i in targets)
reachableVerts[i]=true;
System.Collections.Queue queue=new System.Collections.Queue(targets);
while(queue.Count>0)
{
int i=(int)queue.Dequeue();
foreach(int v in graph.Pred(i))
{
if(!reachableVerts[v] && !deadStatesSet.Contains(v))
{
queue.Enqueue(v);
reachableVerts[v]=true;
}
}
}
int numberOfReachableVerts=0;
foreach(bool b in reachableVerts)
if(b)
numberOfReachableVerts++;
Edge[] strategyEdges;
double [] expectations;
if(numberOfReachableVerts==graph.NumberOfVertices)
{
expectations=GetExpectations(graph,/* sources,*/targets,nStates);
if(expectations==null)
return new EdgesAndExpectations();
strategyEdges=new Edge[nStates];
for(int i=0;i<nStates&&i<graph.NumberOfVertices;i++){
if(targets.Contains(i)||deadStatesSet.Contains(i))
continue;
double min=Single.MaxValue;
Edge stEdge=null;
foreach(Edge l in graph.EdgesAtVertex(i)){
int j=l.target;
if(expectations[j]<min){
min=expectations[j];
stEdge=l;
}
}
strategyEdges[i]=stEdge;
}
}
else
{ //numberOfReachableVerts<graph.NumberOfVertices)
int [] graphToRG=new int[graph.NumberOfVertices];
//reachable graph to graph
int [] rGToGraph=new int[numberOfReachableVerts];
int count=0;
int rNStates=0;
for(int i=0;i<reachableVerts.Length;i++)
if(reachableVerts[i])
{
graphToRG[i]=count;
rGToGraph[count]=i;
count++;
if(i<nStates)
rNStates++;
}
System.Collections.ArrayList mustEdges=new System.Collections.ArrayList();
foreach(Edge l in graph.MustEdges)
{
if( reachableVerts[l.source]&& reachableVerts[l.target])
{
Edge ml=new Edge(graphToRG[l.source],graphToRG[l.target], l.label,l.weight);
mustEdges.Add(ml);
}
}
System.Collections.ArrayList nondVerts=new System.Collections.ArrayList();
for(int i=nStates;i<graph.NumberOfVertices;i++)
{
if(reachableVerts[i])
nondVerts.Add(graphToRG[i]);
}
Graph rGraph=new Graph(0,mustEdges.ToArray(typeof(Edge)) as Edge[],new Edge[0],
nondVerts.ToArray(typeof(int)) as int[],true,WeakClosureEnum.DoNotClose);
int []rSources=new int[sources.Length];
int c=0;
foreach(int s in sources)
{
rSources[c++]=graphToRG[s];
}
HSet rTargets=new HSet();
foreach(int s in targets)
{
if( reachableVerts[s])
{
rTargets.Insert(graphToRG[s]);
}
}
double []rExpectations=GetExpectations(rGraph,/*rSources,*/ rTargets,rNStates);
if(rExpectations==null)
return new EdgesAndExpectations();
strategyEdges=new Edge[nStates];
for(int i=0;i<nStates;i++){
if(!reachableVerts[i])
continue;
if(targets.Contains(i)||deadStatesSet.Contains(i))
continue;
double min=Single.MaxValue;
Edge stEdge=null;
foreach(Edge l in graph.EdgesAtVertex(i)){
int j=l.target;
if(reachableVerts[j])
if(rExpectations[graphToRG[j]]<min){
min=rExpectations[graphToRG[j]];
stEdge=l;
}
}
strategyEdges[i]=stEdge;
}
expectations=new double[graph.NumberOfVertices];
if(expectations==null)
return new EdgesAndExpectations();
for(int i=0;i<expectations.Length;i++)
expectations[i]=Int32.MaxValue;
for(int i=0;i<rExpectations.Length;i++)
expectations[rGToGraph[i]]=rExpectations[i];
}
graph.CleanTheStrategy(strategyEdges,sources);
return new EdgesAndExpectations(strategyEdges, expectations);
}
/*
* From "Play to test"
* Value iteration is the most widely used algorithm for solving discounted Markov decision
* problems (see e.g. [21]). Reachability games give rise to non-discounted Markov
* decision problems. Nevertheless the value iteration algorithm applies; this is a practical
* approach for computing strategies for transient test graphs. Test graphs, modified by inserting
* a zero-cost edge (0; 0), correspond to a subclass of negative stationary Markov
* decision processes (MDPs) with an infinite horizon, where rewards are negative and
* thus regarded as costs, strategies are stationary, i.e. time independent, and there is no
* finite upper bound on the number of steps in the process. The optimization criterion
* for our strategies corresponds to the expected total reward criterion, rather than the
* expected discounted reward criterion used in discounted Markov decision problems.
* Let G = (V;E; V a; V p; g; p; c) be a test graph modified by inserting a zero-cost
* edge (0; 0). The classical value iteration algorithm works as follows on G.
*
* Value iteration Let n = 0 and let M0 be the zero vector with coordinates V so that
* every M0[u] = 0. Given n and Mn, we compute Mn+1 (and then increment n):
* Mn+1[u] ={ min {c(u,v) +Mn[v]:(u,v) in E} if u is an active state}
* or sum {p(u,v)*(c(u,v) +Mn[v]); if u is a choice point
*
* Value iteration for negative MDPs with the expected total reward criterion, or negative
* Markov decision problems for short, does not in general converge to an optimal
* solution, even if one exists. However, if there exists a strategy for which the the expected
* cost is finite for all states [21, Assumption 7.3.1], then value iteration does converge for
* negative Markov decision problems [21, Theorem 7.3.10]. In light of lemmas 2 and 3,
* this implies that value iteration converges for transient test graphs. Let us make this
* more precise, as a corollary of Theorem 7.3.10 in [21].
*/
//nStates marks the end of active states, choice points start after that
static double[] ValueIteration(Graph graph, HSet targets, int nStates) //ValueIteration(Graph graph,int[] sources,HSet targets,int nStates)
{
graph.InitEdgeProbabilities();
double[] v0 = new double[graph.NumberOfVertices];
double[] v1 = new double[graph.NumberOfVertices];
double eps = 1.0E-6;
double delta;
double[] v = v0;
//double[] vnew=v1;
// CheckTransience(graph,targets);
int nOfIter = 0;
do
{
delta = 0;
for (int i = 0; i < nStates && i < graph.NumberOfVertices; i++)
{
if (targets.Contains(i))
{
continue;
}
double min = Double.MaxValue;
foreach (Edge l in graph.EdgesAtVertex(i))
{
double r = ((double)l.weight) + v[l.target];
if (r < min)
{
min = r;
}
}
if (min != Double.MaxValue)
{
v1[i] = min;
if (delta < min - v[i])
{
delta = min - v[i];
}
}
}
for (int i = nStates; i < graph.NumberOfVertices; i++)
{
if (targets.Contains(i))
{
continue;
}
double r = 0;
foreach (Edge l in graph.EdgesAtVertex(i))
{
r += graph.EdgeProbability(l) * (((double)l.weight) + v[l.target]);
}
v1[i] = r;
if (delta < r - v[i])
{
delta = r - v[i];
}
}
nOfIter++;
//swap v and v1
double[] vtmp = v;
v = v1;
v1 = vtmp;
}while(delta > eps && nOfIter < 10000);
if (delta > eps)
{
return(null); //the result is erroneous
}
return(v);
}
//ValueIteration(Graph graph,int[] sources,HSet targets,int nStates)
/*
From "Play to test"
Value iteration is the most widely used algorithm for solving discounted Markov decision
problems (see e.g. [21]). Reachability games give rise to non-discounted Markov
decision problems. Nevertheless the value iteration algorithm applies; this is a practical
approach for computing strategies for transient test graphs. Test graphs, modified by inserting
a zero-cost edge (0; 0), correspond to a subclass of negative stationary Markov
decision processes (MDPs) with an infinite horizon, where rewards are negative and
thus regarded as costs, strategies are stationary, i.e. time independent, and there is no
finite upper bound on the number of steps in the process. The optimization criterion
for our strategies corresponds to the expected total reward criterion, rather than the
expected discounted reward criterion used in discounted Markov decision problems.
Let G = (V;E; V a; V p; g; p; c) be a test graph modified by inserting a zero-cost
edge (0; 0). The classical value iteration algorithm works as follows on G.
Value iteration Let n = 0 and let M0 be the zero vector with coordinates V so that
every M0[u] = 0. Given n and Mn, we compute Mn+1 (and then increment n):
Mn+1[u] ={ min {c(u,v) +Mn[v]:(u,v) in E} if u is an active state}
or sum {p(u,v)*(c(u,v) +Mn[v]); if u is a choice point
Value iteration for negative MDPs with the expected total reward criterion, or negative
Markov decision problems for short, does not in general converge to an optimal
solution, even if one exists. However, if there exists a strategy for which the the expected
cost is finite for all states [21, Assumption 7.3.1], then value iteration does converge for
negative Markov decision problems [21, Theorem 7.3.10]. In light of lemmas 2 and 3,
this implies that value iteration converges for transient test graphs. Let us make this
more precise, as a corollary of Theorem 7.3.10 in [21].
*/
//nStates marks the end of active states, choice points start after that
static double[] ValueIteration(Graph graph, HSet targets, int nStates)
{
graph.InitEdgeProbabilities();
double[]v0=new double[graph.NumberOfVertices];
double[]v1=new double[graph.NumberOfVertices];
double eps=1.0E-6;
double delta;
double[] v=v0;
//double[] vnew=v1;
// CheckTransience(graph,targets);
int nOfIter=0;
do{
delta=0;
for(int i=0;i<nStates&&i<graph.NumberOfVertices;i++){
if(targets.Contains(i))
continue;
double min=Double.MaxValue;
foreach(Edge l in graph.EdgesAtVertex(i)){
double r=((double)l.weight)+v[l.target];
if(r<min)
min=r;
}
if(min!=Double.MaxValue){
v1[i]=min;
if(delta<min-v[i])
delta=min-v[i];
}
}
for(int i=nStates;i<graph.NumberOfVertices;i++){
if(targets.Contains(i))
continue;
double r=0;
foreach(Edge l in graph.EdgesAtVertex(i))
r+=graph.EdgeProbability(l)*(((double)l.weight)+v[l.target]);
v1[i]=r;
if(delta<r-v[i])
delta=r-v[i];
}
nOfIter++;
//swap v and v1
double[] vtmp=v;
v=v1;
v1=vtmp;
}
while(delta>eps && nOfIter<10000);
if(delta>eps){
return null; //the result is erroneous
}
return v;
}