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package edu.stanford.nlp.fsm;
import java.text.NumberFormat;
import java.util.ArrayList;
import java.util.Collections;
import java.util.HashSet;
import java.util.Iterator;
import java.util.List;
import java.util.Map;
import java.util.Random;
import java.util.Set;
import edu.stanford.nlp.stats.ClassicCounter;
import edu.stanford.nlp.trees.TreebankLanguagePack;
import edu.stanford.nlp.util.Generics;
import edu.stanford.nlp.util.Maps;
import edu.stanford.nlp.util.Pair;
import edu.stanford.nlp.util.StringUtils;
/**
* TransducerGraph represents a deterministic finite state automaton
* without epsilon transitions.
*
* @author Teg Grenager
* @version 11/02/03
*/
// TODO: needs some work to make type-safe.
// (In several places, it takes an Object and does instanceof to see what
// it is, or assumes one of the alphabets is a Double, etc....)
public class TransducerGraph implements Cloneable {
public static final String EPSILON_INPUT = "EPSILON";
private static final String DEFAULT_START_NODE = "START";
private static final Random r = new Random();
// internal data structures
private final Set arcs;
private final Map> arcsBySource;
private final Map> arcsByTarget;
private final Map> arcsByInput;
private Map, Arc> arcsBySourceAndInput;
private Map> arcsByTargetAndInput;
private Object startNode;
private Set endNodes;
private boolean checkDeterminism = false;
public void setDeterminism(boolean checkDeterminism) {
this.checkDeterminism = checkDeterminism;
}
public TransducerGraph() {
arcs = Generics.newHashSet();
arcsBySource = Generics.newHashMap();
arcsByTarget = Generics.newHashMap();
arcsByInput = Generics.newHashMap();
arcsBySourceAndInput = Generics.newHashMap();
arcsByTargetAndInput = Generics.newHashMap();
endNodes = Generics.newHashSet();
setStartNode(DEFAULT_START_NODE);
}
public TransducerGraph(TransducerGraph other) {
this(other, (ArcProcessor) null);
}
public TransducerGraph(TransducerGraph other, ArcProcessor arcProcessor) {
this(other.getArcs(), other.getStartNode(), other.getEndNodes(), arcProcessor, null);
}
public TransducerGraph(TransducerGraph other, NodeProcessor nodeProcessor) {
this(other.getArcs(), other.getStartNode(), other.getEndNodes(), null, nodeProcessor);
}
public TransducerGraph(Set newArcs, Object startNode, Set endNodes, ArcProcessor arcProcessor, NodeProcessor nodeProcessor) {
this();
ArcProcessor arcProcessor2 = null;
if (nodeProcessor != null) {
arcProcessor2 = new NodeProcessorWrappingArcProcessor(nodeProcessor);
}
for (Arc a : newArcs) {
a = new Arc(a); // make a copy
if (arcProcessor != null) {
a = arcProcessor.processArc(a);
}
if (arcProcessor2 != null) {
a = arcProcessor2.processArc(a);
}
addArc(a);
}
if (nodeProcessor != null) {
this.startNode = nodeProcessor.processNode(startNode);
} else {
this.startNode = startNode;
}
if (nodeProcessor != null) {
if (endNodes != null) {
for (Object o : endNodes) {
this.endNodes.add(nodeProcessor.processNode(o));
}
}
} else {
if (endNodes != null) {
this.endNodes.addAll(endNodes);
}
}
}
/**
* Uses the Arcs newArcs.
*/
public TransducerGraph(Set newArcs) {
this(newArcs, null, null, null, null);
}
@Override
public TransducerGraph clone() throws CloneNotSupportedException {
super.clone();
TransducerGraph result = new TransducerGraph(this, (ArcProcessor) null);
return result;
}
public Set getArcs() {
return arcs;
}
/**
* Just does union of keysets of maps.
*/
public Set getNodes() {
Set result = Generics.newHashSet();
result.addAll(arcsBySource.keySet());
result.addAll(arcsByTarget.keySet());
return result;
}
public Set getInputs() {
return arcsByInput.keySet();
}
public void setStartNode(Object o) {
startNode = o;
}
public void setEndNode(Object o) {
//System.out.println(this + " setting endNode to " + o);
endNodes.add(o);
}
public Object getStartNode() {
return startNode;
}
public Set getEndNodes() {
//System.out.println(this + " getting endNode " + endNode);
return endNodes;
}
/**
* Returns a Set of type TransducerGraph.Arc.
*/
public Set getArcsByInput(Object node) {
return ensure(arcsByInput.get(node));
}
/**
* Returns a Set of type TransducerGraph.Arc.
*/
public Set getArcsBySource(Object node) {
return ensure(arcsBySource.get(node));
}
private static Set ensure(Set s) {
if (s == null) {
return Collections.emptySet();
}
return s;
}
/**
* Returns a Set of type TransducerGraph.Arc.
*/
public Set getArcsByTarget(Object node) {
return ensure(arcsByTarget.get(node));
}
/**
* Can only be one because automaton is deterministic.
*/
public Arc getArcBySourceAndInput(Object node, Object input) {
return arcsBySourceAndInput.get(Generics.newPair(node, input));
}
/**
* Returns a Set of type TransducerGraph.Arc.
*/
public Set getArcsByTargetAndInput(Object node, Object input) {
return ensure(arcsByTargetAndInput.get(Generics.newPair(node, input)));
}
/**
* Slow implementation.
*/
public Arc getArc(Object source, Object target) {
Set arcsFromSource = arcsBySource.get(source);
Set arcsToTarget = arcsByTarget.get(target);
Set result = Generics.newHashSet();
result.addAll(arcsFromSource);
result.retainAll(arcsToTarget); // intersection
if (result.size() < 1) {
return null;
}
if (result.size() > 1) {
throw new RuntimeException("Problem in TransducerGraph data structures.");
}
// get the only member
Iterator iterator = result.iterator();
return (Arc) iterator.next();
}
/**
* @return true if and only if it created a new Arc and added it to the graph.
*/
public boolean addArc(Object source, Object target, Object input, Object output) {
Arc a = new Arc(source, target, input, output);
return addArc(a);
}
/**
* @return true if and only if it added Arc a to the graph.
* determinism.
*/
protected boolean addArc(Arc a) {
Object source = a.getSourceNode();
Object target = a.getTargetNode();
Object input = a.getInput();
if (source == null || target == null || input == null) {
return false;
}
// add to data structures
if (arcs.contains(a)) {
return false;
}
// it's new, so add to the rest of the data structures
// add to source and input map
Pair p = Generics.newPair(source, input);
if (arcsBySourceAndInput.containsKey(p) && checkDeterminism) {
throw new RuntimeException("Creating nondeterminism while inserting arc " + a + " because it already has arc " + arcsBySourceAndInput.get(p) + checkDeterminism);
}
arcsBySourceAndInput.put(p, a);
Maps.putIntoValueHashSet(arcsBySource, source, a);
p = Generics.newPair(target, input);
Maps.putIntoValueHashSet(arcsByTargetAndInput, p, a);
Maps.putIntoValueHashSet(arcsByTarget, target, a);
Maps.putIntoValueHashSet(arcsByInput, input, a);
// add to arcs
arcs.add(a);
return true;
}
public boolean removeArc(Arc a) {
Object source = a.getSourceNode();
Object target = a.getTargetNode();
Object input = a.getInput();
// remove from arcs
if (!arcs.remove(a)) {
return false;
}
// remove from arcsBySourceAndInput
Pair p = Generics.newPair(source, input);
if (!arcsBySourceAndInput.containsKey(p)) {
return false;
}
arcsBySourceAndInput.remove(p);
// remove from arcsBySource
Set s = arcsBySource.get(source);
if (s == null) {
return false;
}
if (!s.remove(a)) {
return false;
}
// remove from arcsByTargetAndInput
p = Generics.newPair(target, input);
s = arcsByTargetAndInput.get(p);
if (s == null) {
return false;
}
if (!s.remove(a)) {
return false;
}
// remove from arcsByTarget
s = arcsByTarget.get(target);
if (s == null) {
return false;
}
s = arcsByInput.get(input);
if (s == null) {
return false;
}
if (!s.remove(a)) {
return false;
}
return true;
}
public boolean canAddArc(Object source, Object target, Object input, Object output) {
Arc a = new Arc(source, target, input, output);
if (arcs.contains(a)) // inexpensive check
{
return false;
}
Pair p = Generics.newPair(source, input);
return !arcsBySourceAndInput.containsKey(p); // expensive check
}
/** An arc in a finite state transducer.
*
* @param The type of the nodes that an Arc links
* @param The type of the input language of the transducer
* @param The type of the output language of the transducer
*/
public static class Arc {
private NODE sourceNode;
private NODE targetNode;
private IN input;
private OUT output;
public NODE getSourceNode() {
return sourceNode;
}
public NODE getTargetNode() {
return targetNode;
}
public IN getInput() {
return input;
}
public OUT getOutput() {
return output;
}
public void setSourceNode(NODE o) {
sourceNode = o;
}
public void setTargetNode(NODE o) {
targetNode = o;
}
public void setInput(IN o) {
input = o;
}
public void setOutput(OUT o) {
output = o;
}
@Override
public int hashCode() {
return sourceNode.hashCode() ^ (targetNode.hashCode() << 16) ^ (input.hashCode() << 16);
}
@Override
public boolean equals(Object o) {
if (o == this) {
return true;
}
if (!(o instanceof Arc)) {
return false;
}
Arc a = (Arc) o;
return ((sourceNode == null ? a.sourceNode == null : sourceNode.equals(a.sourceNode)) && (targetNode == null ? a.targetNode == null : targetNode.equals(a.targetNode)) && (input == null ? a.input == null : input.equals(a.input)));
}
// makes a copy of Arc a
protected Arc(Arc a) {
this(a.getSourceNode(), a.getTargetNode(), a.getInput(), a.getOutput());
}
protected Arc(NODE sourceNode, NODE targetNode) {
this(sourceNode, targetNode, null, null);
}
protected Arc(NODE sourceNode, NODE targetNode, IN input) {
this(sourceNode, targetNode, input, null);
}
protected Arc(NODE sourceNode, NODE targetNode, IN input, OUT output) {
this.sourceNode = sourceNode;
this.targetNode = targetNode;
this.input = input;
this.output = output;
}
@Override
public String toString() {
return sourceNode + " --> " + targetNode + " (" + input + " : " + output + ")";
}
} // end static class Arc
public static interface ArcProcessor {
/**
* Modifies Arc a.
*/
public Arc processArc(Arc a);
}
public static class OutputCombiningProcessor implements ArcProcessor {
@Override
public Arc processArc(Arc a) {
a = new Arc(a);
a.setInput(Generics.newPair(a.getInput(), a.getOutput()));
a.setOutput(null);
return a;
}
}
public static class InputSplittingProcessor implements ArcProcessor {
@Override
public Arc processArc(Arc a) {
a = new Arc(a);
Pair p = (Pair) a.getInput();
a.setInput(p.first);
a.setOutput(p.second);
return a;
}
}
public static class NodeProcessorWrappingArcProcessor implements ArcProcessor {
private final NodeProcessor nodeProcessor;
public NodeProcessorWrappingArcProcessor(NodeProcessor nodeProcessor) {
this.nodeProcessor = nodeProcessor;
}
@Override
public Arc processArc(Arc a) {
a = new Arc(a);
a.setSourceNode(nodeProcessor.processNode(a.getSourceNode()));
a.setTargetNode(nodeProcessor.processNode(a.getTargetNode()));
return a;
}
}
public static interface NodeProcessor {
public Object processNode(Object node);
}
public static class SetToStringNodeProcessor implements NodeProcessor {
private TreebankLanguagePack tlp;
public SetToStringNodeProcessor(TreebankLanguagePack tlp) {
this.tlp = tlp;
}
@Override
public Object processNode(Object node) {
Set s = null;
if (node instanceof Set) {
s = (Set) node;
} else {
if (node instanceof Block) {
Block b = (Block) node;
s = b.getMembers();
} else {
throw new RuntimeException("Unexpected node class");
}
}
Object sampleNode = s.iterator().next();
if (s.size() == 1) {
if (sampleNode instanceof Block) {
return processNode(sampleNode);
} else {
return sampleNode;
}
}
// nope there's a set of things
if (sampleNode instanceof String) {
String str = (String) sampleNode;
if (str.charAt(0) != '@') {
// passive category...
return tlp.basicCategory(str) + "-" + s.hashCode(); // TODO remove b/c there could be collisions
// return tlp.basicCategory(str) + "-" + System.identityHashCode(s);
}
}
return "@NodeSet-" + s.hashCode(); // TODO remove b/c there could be collisions
// return sampleNode.toString();
}
}
public static class ObjectToSetNodeProcessor implements NodeProcessor {
@Override
public Object processNode(Object node) {
return Collections.singleton(node);
}
}
public static interface GraphProcessor {
public TransducerGraph processGraph(TransducerGraph g);
}
public static class NormalizingGraphProcessor implements GraphProcessor {
boolean forward = true;
public NormalizingGraphProcessor(boolean forwardNormalization) {
this.forward = forwardNormalization;
}
public TransducerGraph processGraph(TransducerGraph g) {
g = new TransducerGraph(g);
Set nodes = g.getNodes();
for (Object node : nodes) {
Set myArcs = null;
if (forward) {
myArcs = g.getArcsBySource(node);
} else {
myArcs = g.getArcsByTarget(node);
}
// compute a total
double total = 0.0;
for (Arc a : myArcs) {
total += ((Double) a.getOutput()).doubleValue();
}
// divide each by total
for (Arc a : myArcs) {
a.setOutput(new Double(Math.log(((Double) a.getOutput()).doubleValue() / total)));
}
}
return g;
}
}
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
depthFirstSearch(true, sb);
return sb.toString();
}
private boolean dotWeightInverted = false;
private void setDotWeightingInverted(boolean inverted) {
dotWeightInverted = true;
}
public String asDOTString() {
NumberFormat nf = NumberFormat.getNumberInstance();
nf.setMaximumFractionDigits(3);
nf.setMinimumFractionDigits(1);
StringBuilder result = new StringBuilder();
Set nodes = getNodes();
result.append("digraph G {\n");
// result.append("page = \"8.5,11\";\n");
// result.append("margin = \"0.25\";\n");
// Heuristic number of pages
int sz = arcs.size();
int ht = 105;
int mag = 250;
while (sz > mag) {
ht += 105;
mag *= 2;
}
int wd = 8;
mag = 500;
while (sz > mag) {
wd += 8;
mag *= 4;
}
double htd = ht / 10.0;
result.append("size = \"" + wd + "," + htd + "\";\n");
result.append("graph [rankdir = \"LR\"];\n");
result.append("graph [ranksep = \"0.2\"];\n");
for (Object node : nodes) {
String cleanString = StringUtils.fileNameClean(node.toString());
result.append(cleanString);
result.append(" [ ");
// if (getEndNodes().contains(node)) {
// result.append("label=\"" + node.toString() + "\", style=filled, ");
// } else
result.append("label=\"" + node.toString() + "\"");
result.append("height=\"0.3\", width=\"0.3\"");
result.append(" ];\n");
for (Arc arc : getArcsBySource(node)) {
result.append(StringUtils.fileNameClean(arc.getSourceNode().toString()));
result.append(" -> ");
result.append(StringUtils.fileNameClean(arc.getTargetNode().toString()));
result.append(" [ ");
result.append("label=\"");
result.append(arc.getInput());
result.append(" : ");
// result.append(arc.getOutput());
Object output = arc.getOutput();
String wt = "";
if (output instanceof Number) {
double dd = ((Number) output).doubleValue();
if (dd == -0.0d) {
result.append(nf.format(0.0d));
} else {
result.append(nf.format(output));
}
int weight;
if (dotWeightInverted) {
weight = (int) (20.0 - dd);
} else {
weight = (int) dd;
}
if (weight > 0) {
wt = ", weight = \"" + weight + "\"";
}
if (dotWeightInverted && dd <= 2.0 || (!dotWeightInverted) && dd >= 20.0) {
wt += ", style=bold";
}
} else {
result.append(output);
}
result.append("\"");
result.append(wt);
// result.append("fontsize = 14 ");
if (arc.getInput().toString().equals("EPSILON")) {
result.append(", style = \"dashed\" ");
} else {
result.append(", style = \"solid\" ");
}
// result.append(", weight = \"" + arc.getOutput() + "\" ");
result.append("];\n");
}
}
result.append("}\n");
return result.toString();
}
public double inFlow(Object node) {
Set arcs = getArcsByTarget(node);
return sumOutputs(arcs);
}
public double outFlow(Object node) {
Set arcs = getArcsBySource(node);
return sumOutputs(arcs);
}
private static double sumOutputs(Set arcs) {
double sum = 0.0;
for (Arc arc : arcs) {
sum += ((Double) arc.getOutput()).doubleValue();
}
return sum;
}
private double getSourceTotal(Object node) {
double result = 0.0;
Set arcs = getArcsBySource(node);
if (arcs.isEmpty()) {
System.err.println("No outbound arcs from node.");
return result;
}
for (Arc arc : arcs) {
result += ((Double) arc.getOutput()).doubleValue();
}
return result;
}
/**
* For testing only. Doubles combined by addition.
*/
public double getOutputOfPathInGraph(List path) {
double score = 0.0;
Object node = getStartNode();
for (Object input : path) {
Arc arc = getArcBySourceAndInput(node, input); // next input in path
if (arc == null) {
System.out.println(" NOT ACCEPTED :" + path);
return Double.NEGATIVE_INFINITY;
}
score += ((Double) arc.getOutput()).doubleValue();
node = arc.getTargetNode();
}
return score;
}
/**
* for testing only. doubles combined by addition.
*/
public List sampleUniformPathFromGraph() {
List list = new ArrayList();
Object node = this.getStartNode();
Set endNodes = this.getEndNodes();
while (!endNodes.contains(node)) {
List arcs = new ArrayList<>(this.getArcsBySource(node));
TransducerGraph.Arc arc = arcs.get(r.nextInt(arcs.size()));
list.add(arc.getInput());
node = arc.getTargetNode();
}
return list;
}
private Map samplePathsFromGraph(int numPaths) {
Map result = Generics.newHashMap();
for (int i = 0; i < numPaths; i++) {
List l = sampleUniformPathFromGraph();
result.put(l, new Double(getOutputOfPathInGraph(l)));
}
return result;
}
/**
* For testing only.
*/
private static void printPathOutputs(List pathList, TransducerGraph graph, boolean printPaths) {
int i = 0;
for (List path : pathList) {
if (printPaths) {
for (Object aPath : path) {
System.out.print(aPath + " ");
}
} else {
System.out.print(i++ + " ");
}
System.out.print("output: " + graph.getOutputOfPathInGraph(path));
System.out.println();
}
}
/**
* For testing only.
*/
public List getPathOutputs(List pathList) {
List outputList = new ArrayList<>();
for (List path : pathList) {
outputList.add(new Double(getOutputOfPathInGraph(path)));
}
return outputList;
}
public static boolean testGraphPaths(TransducerGraph sourceGraph, TransducerGraph testGraph, int numPaths) {
for (int i = 0; i < numPaths; i++) {
List path = sourceGraph.sampleUniformPathFromGraph();
double score = sourceGraph.getOutputOfPathInGraph(path);
double newScore = testGraph.getOutputOfPathInGraph(path);
if ((score - newScore) / (score + newScore) > 1e-10) {
System.out.println("Problem: " + score + " vs. " + newScore + " on " + path);
return false;
}
}
return true;
}
/**
* For testing only. Doubles combined by multiplication.
*/
private boolean canAddPath(List path) {
Object node = this.getStartNode();
for (int j = 0; j < path.size() - 1; j++) {
Object input = path.get(j);
Arc arc = this.getArcBySourceAndInput(node, input); // next input in path
if (arc == null) {
return true;
}
node = arc.getTargetNode();
}
Object input = path.get(path.size() - 1); // last element
Arc arc = this.getArcBySourceAndInput(node, input); // next input in path
if (arc == null) {
return true;
} else {
return getEndNodes().contains(arc.getTargetNode());
}
}
/**
* If markovOrder is zero, we always transition back to the start state
* If markovOrder is negative, we assume that it is infinite
*/
public static TransducerGraph createGraphFromPaths(List paths, int markovOrder) {
ClassicCounter pathCounter = new ClassicCounter();
for (Object o : paths) {
pathCounter.incrementCount(o);
}
return createGraphFromPaths(pathCounter, markovOrder);
}
public static TransducerGraph createGraphFromPaths(ClassicCounter> pathCounter, int markovOrder) {
TransducerGraph graph = new TransducerGraph(); // empty
for (List path : pathCounter.keySet()) {
double count = pathCounter.getCount(path);
addOnePathToGraph(path, count, markovOrder, graph);
}
return graph;
}
// assumes that the path already has EPSILON as the last element.
public static void addOnePathToGraph(List path, double count, int markovOrder, TransducerGraph graph) {
Object source = graph.getStartNode();
for (int j = 0; j < path.size(); j++) {
Object input = path.get(j);
Arc a = graph.getArcBySourceAndInput(source, input);
if (a != null) {
// increment the arc weight
a.output = new Double(((Double) a.output).doubleValue() + count);
} else {
Object target;
if (input.equals(TransducerGraph.EPSILON_INPUT)) {
target = "END"; // to ensure they all share the same end node
} else if (markovOrder == 0) {
// we all transition back to the same state
target = source;
} else if (markovOrder > 0) {
// the state is described by the partial history
target = path.subList((j < markovOrder ? 0 : j - markovOrder + 1), j + 1);
} else {
// the state is described by the full history
target = path.subList(0, j + 1);
}
Double output = new Double(count);
a = new Arc(source, target, input, output);
graph.addArc(a);
}
source = a.getTargetNode();
}
graph.setEndNode(source);
}
/**
* For testing only. All paths will be added to pathList as Lists.
* // generate a bunch of paths through the graph with the input alphabet
* // and create new nodes for each one.
*/
public static TransducerGraph createRandomGraph(int numPaths, int pathLengthMean, double pathLengthVariance, int numInputs, List pathList) {
// compute the path length. Draw from a normal distribution
int pathLength = (int) (r.nextGaussian() * pathLengthVariance + pathLengthMean);
for (int i = 0; i < numPaths; i++) {
// make a path
List path = new ArrayList();
for (int j = 0; j < pathLength; j++) {
String input = Integer.toString(r.nextInt(numInputs));
path.add(input);
}
// TODO: createRandomPaths had the following difference:
// we're done, add one more arc to get to the endNode.
//input = TransducerGraph.EPSILON_INPUT;
//path.add(input);
pathList.add(path);
}
return createGraphFromPaths(pathList, -1);
}
public static List createRandomPaths(int numPaths, int pathLengthMean, double pathLengthVariance, int numInputs) {
List pathList = new ArrayList();
// make a bunch of paths, randomly
// compute the path length. Draw from a normal distribution
int pathLength = (int) (r.nextGaussian() * pathLengthVariance + pathLengthMean);
for (int i = 0; i < numPaths; i++) {
// make a path
List path = new ArrayList<>();
String input;
for (int j = 0; j < pathLength; j++) {
input = Integer.toString(r.nextInt(numInputs));
path.add(input);
}
// we're done, add one more arc to get to the endNode.
input = TransducerGraph.EPSILON_INPUT;
path.add(input);
pathList.add(path);
}
return pathList;
}
public void depthFirstSearch(boolean forward, StringBuilder b) {
if (forward) {
depthFirstSearchHelper(getStartNode(), new HashSet(), 0, true, b);
} else {
for (Object o : getEndNodes()) {
depthFirstSearchHelper(o, new HashSet(), 0, false, b);
}
}
}
/**
* For testing only.
*/
private void depthFirstSearchHelper(Object node, Set marked, int level, boolean forward, StringBuilder b) {
if (marked.contains(node)) {
return;
}
marked.add(node);
Set arcs;
if (forward) {
arcs = this.getArcsBySource(node);
} else {
arcs = this.getArcsByTarget(node);
}
if (arcs == null) {
return;
}
for (Arc newArc : arcs) {
// print it out
for (int i = 0; i < level; i++) {
b.append(" ");
}
if (getEndNodes().contains(newArc.getTargetNode())) {
b.append(newArc + " END\n");
} else {
b.append(newArc + "\n");
}
if (forward) {
depthFirstSearchHelper(newArc.getTargetNode(), marked, level + 1, forward, b);
} else {
depthFirstSearchHelper(newArc.getSourceNode(), marked, level + 1, forward, b);
}
}
}
/**
* For testing only.
*/
public static void main(String[] args) {
List pathList = new ArrayList();
TransducerGraph graph = createRandomGraph(1000, 10, 0.0, 10, pathList);
System.out.println("Done creating random graph");
printPathOutputs(pathList, graph, true);
System.out.println("Depth first search from start node");
StringBuilder b = new StringBuilder();
graph.depthFirstSearch(true, b);
System.out.println(b.toString());
b = new StringBuilder();
System.out.println("Depth first search back from end node");
graph.depthFirstSearch(false, b);
System.out.println(b.toString());
}
}
