diff --git a/macros/math/SimpleGraph.pl b/macros/math/SimpleGraph.pl new file mode 100644 index 000000000..a471568bc --- /dev/null +++ b/macros/math/SimpleGraph.pl @@ -0,0 +1,2670 @@ +BEGIN { strict->import; } + +loadMacros('PGauxiliaryFunctions.pl', 'PGbasicmacros.pl', 'MathObjects.pl', 'plots.pl'); + +sub _SimpleGraph_init { + my $context = $main::context{EdgeSet} = Parser::Context->getCopy('Numeric'); + $context->{name} = 'EdgeSet'; + $context->{value}{EdgeSet} = 'GraphTheory::SimpleGraph::Value::EdgeSet'; + $context->{value}{Edge} = 'GraphTheory::SimpleGraph::Value::Edge'; + $context->lists->set( + # The "List" type list is set to use the GraphTheory::SimpleGraph::Parser::List class so that implicit lists can + # be turned into edges or edge sets, and empty lists can be interpreted as edge sets for a graph with no edges. + # All explicit lists that open with something other than a brace are untouched. + List => { %{ $context->lists->get('List') }, class => 'GraphTheory::SimpleGraph::Parser::List' }, + EdgeSet => + { class => 'GraphTheory::SimpleGraph::Parser::List', open => '{', close => '}', separator => ', ' }, + Edge => { class => 'GraphTheory::SimpleGraph::Parser::List', open => '{', close => '}', separator => ', ' }, + ); + $context->parens->set( + '{' => { close => '}', type => 'Edge', formList => 1, formMatrix => 0, removable => 0, emptyOK => 1 }); +} + +sub SimpleGraph { GraphTheory::SimpleGraph->new(@_) } +sub EdgeSet { GraphTheory::SimpleGraph::Value::EdgeSet->new(@_) } +sub Edge { GraphTheory::SimpleGraph::Value::Edge->new(@_) } + +sub randomSimpleGraph { + my ($size, %options) = @_; + + my $graph; + my $edgeCount = 0; + my $edgeProbability = $options{edgeProbability} // 0.5; + + if (ref $size eq 'ARRAY') { + $graph = + GraphTheory::SimpleGraph->new($size->[0] * $size->[1], %options, gridLayout => [ $size->[0], $size->[1] ]); + + for my $i (0 .. $size->[0] - 1) { + for my $j (0 .. $size->[1] - 1) { + my $location = $j * $size->[0] + $i; + + if ($j < $size->[1] - 1 && main::random(0, 100) <= 100 * $edgeProbability) { + $graph->addEdge($location, $location + $size->[0]); + ++$edgeCount; + } + if ($i < $size->[0] - 1 && main::random(0, 100) <= 100 * $edgeProbability) { + $graph->addEdge($location, $location + 1); + ++$edgeCount; + } + if ($i < $size->[0] - 1 && $j < $size->[1] - 1 && main::random(0, 100) <= 100 * $edgeProbability) { + $graph->addEdge($location, $location + $size->[0] + 1); + ++$edgeCount; + } + if ($i >= 1 && $j < $size->[1] - 1 && main::random(0, 100) <= 100 * $edgeProbability) { + $graph->addEdge($location, $location + $size->[0] - 1); + ++$edgeCount; + } + } + } + } else { + $graph = GraphTheory::SimpleGraph->new($size, %options); + + for my $i (0 .. $size - 1) { + for my $j (0 .. $i - 1) { + if (main::random(0, 100) <= 100 * $edgeProbability) { + $graph->addEdge($i, $j); + ++$edgeCount; + } + } + } + } + + return $graph->setRandomWeights(%options, edgeCount => $edgeCount); +} + +sub randomGraphWithEulerCircuit { + my ($size, %options) = @_; + + die 'A graph with an Euler circuit must have at least 5 vertices.' if $size < 5; + + # Remove these from the options so that setting weights is deferred until the return. + my ($startEdgeWeight, $edgeWeightIncrement, $edgeWeightRange) = + delete @options{qw(startEdgeWeight edgeWeightIncrement edgeWeightRange)}; + + my $graph; + + do { + $graph = simpleGraphWithDegreeSequence([ map { main::random(2, $size - 1, 2) } 0 .. $size - 1 ], %options); + } while !defined $graph || $graph->numComponents > 1; + + return $graph->setRandomWeights( + %options, + startEdgeWeight => $startEdgeWeight, + edgeWeightIncrement => $edgeWeightIncrement, + edgeWeightRange => $edgeWeightRange + )->shuffle; +} + +sub randomGraphWithEulerTrail { + my ($size, %options) = @_; + + die 'A graph with an Euler trail must have at least 5 vertices.' if $size < 5; + + # Remove these from the options so that setting weights is deferred until the return. + my ($startEdgeWeight, $edgeWeightIncrement, $edgeWeightRange) = + delete @options{qw(startEdgeWeight edgeWeightIncrement edgeWeightRange)}; + + my $graph = randomGraphWithEulerCircuit($size, %options); + + my ($vertex1, $vertex2); + + do { + ($vertex1, $vertex2) = main::random_subset(2, 0 .. $size - 1); + } until $graph->hasEdge($vertex1, $vertex2); + + $graph->removeEdge($vertex1, $vertex2); + + return $graph->setRandomWeights( + %options, + startEdgeWeight => $startEdgeWeight, + edgeWeightIncrement => $edgeWeightIncrement, + edgeWeightRange => $edgeWeightRange + ); +} + +sub randomGraphWithoutEulerTrail { + my ($size, %options) = @_; + + die 'A graph without an Euler trail must have at least 5 vertices.' if $size < 5; + + # Remove these from the options so that setting weights is deferred until the return. + my ($startEdgeWeight, $edgeWeightIncrement, $edgeWeightRange) = + delete @options{qw(startEdgeWeight edgeWeightIncrement edgeWeightRange)}; + + my $graph; + + do { + $graph = simpleGraphWithDegreeSequence([ map { main::random(2, $size - 1, 1) } 0 .. $size - 1 ], %options); + } while !defined $graph || $graph->hasEulerTrail; + + return $graph->setRandomWeights( + %options, + startEdgeWeight => $startEdgeWeight, + edgeWeightIncrement => $edgeWeightIncrement, + edgeWeightRange => $edgeWeightRange + )->shuffle; +} + +sub randomBipartiteGraph { + my ($size, %options) = @_; + + my ($s1, $s2); + + if (ref $size eq 'ARRAY' && @$size == 2) { + ($s1, $s2) = @$size; + die 'A bipartite graph must have at least 1 vertex in each partition.' unless $s1 > 0 && $s2 > 0; + } else { + die 'A bipartite graph must have at least 2 vertices.' if $size < 2; + $s1 = main::random(1, $size - 1); + $s2 = $size - $s1; + } + + my $graph = GraphTheory::SimpleGraph->new($s1 + $s2, %options); + + my $edgeProbability = $options{edgeProbability} // 0.5; + my $edgeCount = 0; + + for my $i (0 .. $s1 - 1) { + for my $j ($s1 .. $s1 + $s2 - 1) { + next unless main::random(0, 100) <= 100 * $edgeProbability; + $graph->addEdge($i, $j); + ++$edgeCount; + } + } + + return $graph->setRandomWeights(%options, edgeCount => $edgeCount)->shuffle; +} + +sub randomTreeGraph { + my ($size, %options) = @_; + + die 'A tree graph must have at least 2 vertices.' if $size < 2; + + my $graph = GraphTheory::SimpleGraph->new($size, %options); + + my @available = main::random_subset($size, 0 .. $size - 1); + + my @used; + push @used, pop @available; + do { + my $j = pop @available; + my $i = main::random(0, $#used); + $graph->addEdge($used[$i], $j); + push @used, $j; + } while @available > 0; + + return $graph->setRandomWeights(%options, edgeCount => @used - 1); +} + +sub randomForestGraph { + my ($size, %options) = @_; + + die 'A forest graph must have at least 2 vertices.' if $size < 2; + + # Remove these from the options so that setting weights is deferred until the return. + my ($startEdgeWeight, $edgeWeightIncrement, $edgeWeightRange) = + delete @options{qw(startEdgeWeight edgeWeightIncrement edgeWeightRange)}; + + my $graph = randomTreeGraph($size, %options); + + my ($vertex1, $vertex2); + do { + ($vertex1, $vertex2) = main::random_subset(2, 0 .. $size - 1); + } until $graph->hasEdge($vertex1, $vertex2); + $graph->removeEdge($vertex1, $vertex2); + + return $graph->setRandomWeights( + %options, + startEdgeWeight => $startEdgeWeight, + edgeWeightIncrement => $edgeWeightIncrement, + edgeWeightRange => $edgeWeightRange + ); +} + +# Returns a Hamiltonian graph of the given $size. Note that $size must be 5 or greater. +sub randomHamiltonianGraph { + my ($size, %options) = @_; + + die 'A Hamiltonian graph must have at least 5 vertices.' if $size < 5; + + my $graph = GraphTheory::SimpleGraph->new($size, %options); + + my $comp = $size * ($size - 1) / 2; + + my ($low, $high) = $size <= 5 ? ($size + 1, $comp - 1) : (int($comp / 3) + 1, int($comp / 2) + 1); + + for my $i (0 .. $size - 1) { + $graph->addEdge($i, ($i + 1) % $size); + } + + my $edges = int main::random($low, $high); + + my $edgeCount = $size; + while ($edgeCount < $edges) { + my ($i, $j) = main::random_subset(2, 0 .. $size - 1); + unless ($graph->hasEdge($i, $j)) { + $graph->addEdge($i, $j); + ++$edgeCount; + } + } + + return $graph->setRandomWeights(%options, edgeCount => $edgeCount)->shuffle; +} + +sub randomNonHamiltonianGraph { + my ($size, $type, %options) = @_; + + my $graph = GraphTheory::SimpleGraph->new($size); + my $edgeCount = 0; + + if ($type % 2 == 0) { + die 'A non Hamiltonian graph with a degree 1 vertex must have at least 5 vertices.' if $size < 5; + + my $numEdges = main::random($size + 1, ($size - 1) * ($size - 2) / 3 + 1); + + for my $i (0 .. $size - 2) { + $graph->addEdge($i, $i + 1); + } + $graph->addEdge($size - 2, 0); + + $edgeCount = $size; + do { + my ($i, $j) = main::random_subset(2, 0 .. $size - 2); + unless ($graph->hasEdge($i, $j)) { + $graph->addEdge($i, $j); + ++$edgeCount; + } + } while $edgeCount < $numEdges; + } else { + die 'A non Hamiltonian graph with two cycles must have at least 6 vertices.' if $size < 6; + + my $split = int($size / 2); + + for my $i (0 .. $split - 1) { + $graph->addEdge($i, ($i + 1) % $split); + } + for my $i ($split .. $size - 1) { + $graph->addEdge($i, ($i + 1) % $size); + } + $graph->addEdge($size - 1, $split); + + my $numEdges = 2 * $size - 5; + $edgeCount = $size + 1; + + my $maxtry = 4; # Protection against a possibly infinite loop. + while ($edgeCount < $numEdges && $maxtry > 0) { + --$maxtry; + my $v1 = main::random(0, $split - 1); + my $v2 = ($v1 + 2) % $split; + unless ($graph->hasEdge($v1, $v2)) { + $graph->addEdge($v1, $v2); + ++$edgeCount; + } + unless ($graph->hasEdge($v1 + $split, $v2 + $split)) { + $graph->addEdge($v1 + $split, $v2 + $split); + ++$edgeCount; + } + } + } + + return $graph->setRandomWeights(%options, edgeCount => $edgeCount)->shuffle; +} + +sub simpleGraphWithDegreeSequence { + my ($degrees, %options) = @_; + + my @degrees = reverse num_sort(@$degrees); + + return if $degrees[0] >= @degrees; + + my $graph = GraphTheory::SimpleGraph->new(scalar @degrees, %options); + + my $value = 0; + my $vertex = 0; + my $edgeCount = 0; + + while ($vertex < @degrees && $value == 0) { + $value = $degrees[$vertex] - $graph->vertexDegree($vertex); + my $otherVertex = $vertex + 1; + while ($value > 0 && $otherVertex < @degrees) { + if ($graph->vertexDegree($otherVertex) < $degrees[$otherVertex]) { + $graph->addEdge($vertex, $otherVertex); + ++$edgeCount; + --$value; + } + ++$otherVertex; + } + ++$vertex; + } + + return if $value != 0; + + return $graph->setRandomWeights(%options, edgeCount => $edgeCount); +} + +sub cycleGraph { + my ($size, %options) = @_; + + my $graph = GraphTheory::SimpleGraph->new($size, %options); + + for (0 .. $graph->lastVertexIndex) { + $graph->addEdge($_, ($_ + 1) % $graph->numVertices); + } + + return $graph->setRandomWeights(%options, edgeCount => $graph->numVertices); +} + +sub completeGraph { + my ($size, %options) = @_; + + my $graph = GraphTheory::SimpleGraph->new($size, %options); + + for my $i (0 .. $size - 1) { + for my $j ($i + 1 .. $size - 1) { + $graph->addEdge($i, $j); + } + } + + return $graph->setRandomWeights(%options, edgeCount => $graph->numVertices * ($graph->numVertices - 1) / 2); +} + +sub wheelGraph { + my ($size, %options) = @_; + + my $graph = GraphTheory::SimpleGraph->new($size, %options, wheelLayout => 0); + + for my $i (1 .. $size - 2) { + $graph->addEdge($i, $i + 1); + $graph->addEdge(0, $i); + } + $graph->addEdge($size - 1, 1); + $graph->addEdge(0, $size - 1); + + return $graph->setRandomWeights(%options, edgeCount => ($graph->numVertices - 1) * 2); +} + +sub completeBipartiteGraph { + my ($m, $n, %options) = @_; + + my $graph = + GraphTheory::SimpleGraph->new($m + $n, %options, bipartiteLayout => [ [ 0 .. $m - 1 ], [ $m .. $m + $n - 1 ] ]); + + for my $i (0 .. $m - 1) { + for my $j ($m .. $m + $n - 1) { + $graph->addEdge($i, $j); + } + } + + return $graph->setRandomWeights(%options, edgeCount => $m * $n); +} + +package GraphTheory::SimpleGraph; + +sub new { + my ($class, $definition, %options) = @_; + my $self = bless {}, ref($class) || $class; + + die 'A graph definition in the form of a numeric size, adjacency matrix, ' + . 'edge set, or another simple graph object is required.' + unless defined $definition; + + if (ref $definition eq 'GraphTheory::SimpleGraph') { + $self->{adjacencyMatrix} = [ map { [@$_] } @{ $definition->adjacencyMatrix } ]; + $self->{labels} = [ @{ $definition->{labels} } ]; + $options{gridLayout} //= $definition->{gridLayout}; + $options{bipartiteLayout} //= $definition->{bipartiteLayout}; + $options{wheelLayout} //= $definition->{wheelLayout}; + } elsif (Value::classMatch($definition, 'Matrix') + || Value::classMatch($definition, 'EdgeSet') + || ref $definition eq 'ARRAY') + { + $definition = [ map { [ $_->value ] } @{ $definition->data } ] if Value::classMatch($definition, 'Matrix'); + $definition = $definition->data if Value::classMatch($definition, 'EdgeSet'); + + my $haveLabels = ref $options{labels} eq 'ARRAY' && @{ $options{labels} }; + die 'Graphs with no vertices are not supported.' unless @$definition || $haveLabels; + + my @edgeSet; + for (@$definition) { + if (ref $_ ne 'GraphTheory::SimpleGraph::Value::Edge') { @edgeSet = (); last; } + push(@edgeSet, $_->data); + } + if (!@edgeSet) { + for (@$definition) { + if ( + ref $_ ne 'ARRAY' + || @$_ < 2 + || @$_ > 3 + || (!Value::classMatch($_->[0], 'String') + && (Value::isReal($_->[0]) || ($_->[0] ^ $_->[0]) eq '0')) + || (!Value::classMatch($_->[1], 'String') + && (Value::isReal($_->[1]) || ($_->[1] ^ $_->[1]) eq '0')) + ) + { + @edgeSet = (); + last; + } + push(@edgeSet, $_); + } + } + + if (@edgeSet || (!@$definition && $haveLabels)) { + die 'Labels must be provided when using an edgeset definition.' unless $haveLabels; + + $definition = [ map { $_->{data} } @$definition ] + if ref $definition->[0] eq 'GraphTheory::SimpleGraph::Value::Edge'; + + $self->{labels} = [ @{ $options{labels} } ]; + $self->{adjacencyMatrix} = [ map { [ (0) x @{ $self->{labels} } ] } @{ $self->{labels} } ]; + + my %labelIndices = map { $self->{labels}[$_] => $_ } 0 .. $#{ $self->{labels} }; + + for my $i (0 .. $#$definition) { + die 'Invalid edge set format.' unless ref $definition->[$i] eq 'ARRAY'; + my @edge = @{ $definition->[$i] }; + die 'Invalid edge format.' unless @edge >= 2; + die "Invalid vertex $edge[0] in edge set." unless defined $labelIndices{ $edge[0] }; + die "Invalid vertex $edge[1] in edge set." unless defined $labelIndices{ $edge[1] }; + $self->edgeWeight($labelIndices{ $edge[0] }, $labelIndices{ $edge[1] }, $edge[2] // 1); + } + } else { + $self->{adjacencyMatrix} = []; + for my $i (0 .. $#$definition) { + die 'Invalid adjacency matrix format.' unless ref $definition->[$i] eq 'ARRAY'; + die 'The adjacency matrix for a graph must be a square matrix.' + unless @{ $definition->[$i] } == @$definition; + die 'The diagonal entries of the adjacency matrix must be zero.' if $definition->[$i][$i]; + for my $j ($i + 1 .. $#{ $definition->[$i] }) { + die 'The adjacency matrix for a graph must be symmetric.' + unless $definition->[$i][$j] == $definition->[$j][$i]; + } + push(@{ $self->{adjacencyMatrix} }, [ @{ $definition->[$i] } ]); + } + } + } else { + die 'Graphs with no vertices are not supported.' unless $definition > 0; + $self->{adjacencyMatrix} = [ map { [ (0) x $definition ] } 0 .. ($definition - 1) ]; + } + + if (ref $options{labels} eq 'ARRAY') { + die 'Not enough vertex labels provided.' if @{ $options{labels} } < $self->numVertices; + for (0 .. $self->lastVertexIndex) { + die 'Labels cannot be undefined.' unless defined $options{labels}[$_]; + } + $self->{labels} = [ @{ $options{labels} }[ 0 .. $self->lastVertexIndex ] ]; + } + + unless (defined $self->{labels}) { + my $alphaOffset = main::random(0, 25 - $self->lastVertexIndex); + $self->{labels} = [ ('A' .. 'Z')[ $alphaOffset .. $alphaOffset + $self->lastVertexIndex ] ]; + } + + $self->{gridLayout} = [ @{ $options{gridLayout} } ] + if ref $options{gridLayout} eq 'ARRAY' && @{ $options{gridLayout} } == 2; + + if (ref $options{bipartiteLayout} eq 'ARRAY' + && @{ $options{bipartiteLayout} } == 2 + && !grep { ref $_ ne 'ARRAY' } @{ $options{bipartiteLayout} }) + { + $self->{bipartiteLayout} = [ map { [@$_] } @{ $options{bipartiteLayout} } ]; + } elsif ($options{bipartiteLayout}) { + $self->{bipartiteLayout} = 1; + } + + $self->{wheelLayout} = $options{wheelLayout} if defined $options{wheelLayout}; + + return $self; +} + +sub adjacencyMatrix { + my $self = shift; + return $self->{adjacencyMatrix}; +} + +sub edgeSet { + my ($self, %options) = @_; + + my $context = Value::isContext($options{context}) ? $options{context} : Parser::Context->getCopy('EdgeSet'); + $self->addVerticesToContext($options{caseSensitive} // 1, $context); + + my @edgeSet; + for my $i (0 .. $self->lastVertexIndex) { + for my $j ($i + 1 .. $self->lastVertexIndex) { + next unless $self->hasEdge($i, $j); + push( + @edgeSet, + GraphTheory::SimpleGraph::Value::Edge->new( + $context, $self->vertexLabel($i), $self->vertexLabel($j) + ) + ); + $edgeSet[-1]->{open} = '{'; + $edgeSet[-1]->{close} = '}'; + } + } + + my $edgeSet = GraphTheory::SimpleGraph::Value::EdgeSet->new($context, @edgeSet); + $edgeSet->{open} = '{'; + $edgeSet->{close} = '}'; + return $edgeSet; +} + +sub addVerticesToContext { + my ($self, $caseSensitive, $context) = @_; + $context = Value::isContext($context) ? $context : main::Context(); + $context->strings->add( + map { $_ => { caseSensitive => $caseSensitive // 1 } } + grep { !defined $context->strings->all->{$_} } @{ $self->labels } + ); + $context->strings->all->{$_}{isVertex} = 1 for @{ $self->labels }; + return; +} + +sub numVertices { + my $self = shift; + return scalar @{ $self->{adjacencyMatrix} }; +} + +sub lastVertexIndex { + my $self = shift; + return $#{ $self->{adjacencyMatrix} }; +} + +sub numEdges { + my $self = shift; + my $edgeCount = 0; + for my $i (0 .. $self->lastVertexIndex) { + for my $j (0 .. $i - 1) { + next unless $self->hasEdge($i, $j); + ++$edgeCount; + } + } + return $edgeCount; +} + +sub labels { + my ($self, $labels) = @_; + if (ref $labels eq 'ARRAY') { + die 'Not enough vertex labels provided.' if @$labels < $self->numVertices; + $self->{labels} = [ @$labels[ 0 .. $self->numVertices ] ]; + } + return $self->{labels}; +} + +sub labelsString { + my $self = shift; + return join(', ', @{ $self->{labels} }); +} + +sub vertexLabel { + my ($self, $vertexIndex) = @_; + return $self->{labels}[$vertexIndex]; +} + +sub vertexIndex { + my ($self, $vertexLabel) = @_; + for (0 .. $#{ $self->{labels} }) { + return $_ if $vertexLabel eq $self->{labels}[$_]; + } + return -1; +} + +sub vertexDegree { + my ($self, $vertex) = @_; + my $degree = 0; + for my $j (0 .. $self->lastVertexIndex) { + ++$degree if $self->hasEdge($vertex, $j); + } + return $degree; +} + +sub degrees { + my $self = shift; + return map { $self->vertexDegree($_) } 0 .. $self->lastVertexIndex; +} + +sub numComponents { + my $self = shift; + + my @adjacencyMatrix = map { [@$_] } @{ $self->{adjacencyMatrix} }; + + my $result = @adjacencyMatrix; + for my $i (0 .. $#adjacencyMatrix) { + my $connected = 0; + for my $j ($i + 1 .. $#adjacencyMatrix) { + if ($adjacencyMatrix[$i][$j] != 0) { + ++$connected; + for my $k (0 .. $#adjacencyMatrix) { + $adjacencyMatrix[$j][$k] += $adjacencyMatrix[$i][$k]; + $adjacencyMatrix[$k][$j] += $adjacencyMatrix[$k][$i]; + } + } + } + --$result if $connected > 0; + } + return $result; +} + +sub edgeWeight { + my ($self, $i, $j, $weight) = @_; + if (defined $weight) { + $self->{adjacencyMatrix}[$i][$j] = $weight; + $self->{adjacencyMatrix}[$j][$i] = $weight; + } + return $self->{adjacencyMatrix}[$i][$j]; +} + +sub addEdge { + my ($self, $i, $j, $weight) = @_; + $self->edgeWeight($i, $j, $weight || 1); + return; +} + +sub removeEdge { + my ($self, $i, $j) = @_; + $self->edgeWeight($i, $j, 0); + return; +} + +sub hasEdge { + my ($self, $i, $j) = @_; + return $self->edgeWeight($i, $j) != 0; +} + +sub setRandomWeights { + my ($self, %options) = @_; + + my $incrementalRandom = + defined $options{startEdgeWeight} + && $options{startEdgeWeight} > 0 + && ($options{edgeWeightIncrement} // 1) > 0; + + return $self + unless $incrementalRandom + || (ref $options{edgeWeightRange} eq 'ARRAY' && @{ $options{edgeWeightRange} } >= 2); + + my $edgeCount = $options{edgeCount} // $self->numEdges; + + my @weights = + $incrementalRandom + ? main::random_subset($edgeCount, + map { $options{startEdgeWeight} + $_ * ($options{edgeWeightIncrement} // 1) } 0 .. $edgeCount - 1) + : map { main::random(@{ $options{edgeWeightRange} }) } 1 .. $edgeCount; + + for my $i (0 .. $self->lastVertexIndex) { + for my $j ($i + 1 .. $self->lastVertexIndex) { + $self->edgeWeight($i, $j, shift(@weights) // 1) if $self->hasEdge($i, $j); + } + } + + return $self; +} + +sub isEqual { + my ($self, $other) = @_; + return 0 unless ref $other eq 'GraphTheory::SimpleGraph'; + return 0 if @{ $self->{adjacencyMatrix} } != @{ $other->{adjacencyMatrix} }; + for my $i (0 .. $#{ $self->{adjacencyMatrix} }) { + return 0 if @{ $self->{adjacencyMatrix}[$i] } != @{ $other->{adjacencyMatrix}[$i] }; + for my $j (0 .. $i - 1) { + return 0 if $self->{adjacencyMatrix}[$i][$j] != $other->{adjacencyMatrix}[$i][$j]; + } + } + return 1; +} + +sub isIsomorphic { + my ($self, $other) = @_; + + return 0 unless ref $other eq 'GraphTheory::SimpleGraph' && $self->numVertices == $other->numVertices; + + my @degrees = main::num_sort($self->degrees); + my @otherDegrees = main::num_sort($other->degrees); + for (0 .. $#degrees) { + return 0 unless $degrees[$_] == $otherDegrees[$_]; + } + + my $permutations = [ [0] ]; + + for my $i (1 .. $self->lastVertexIndex) { + my @newPermutations; + for my $permutation (@$permutations) { + for my $j (0 .. @$permutation) { + my @new = @$permutation; + splice(@new, $j, 0, $i); + push(@newPermutations, \@new); + } + } + $permutations = \@newPermutations; + } + + # The last permutation is the original vertex order, so remove it. + pop @$permutations; + + for my $permutation (@$permutations) { + my @shuffledGraph; + for my $i (0 .. $other->lastVertexIndex) { + for my $j (0 .. $other->lastVertexIndex) { + $shuffledGraph[ $permutation->[$i] ][ $permutation->[$j] ] = $other->edgeWeight($i, $j); + } + } + return 1 if $self->isEqual($self->new(\@shuffledGraph)); + } + + return 0; +} + +sub image { + my ($self, %options) = @_; + + return $self->gridLayoutImage(%options) if $self->{gridLayout}; + if ($self->{bipartiteLayout}) { + return $self->bipartiteLayoutImage(%options) if ref $self->{bipartiteLayout} eq 'ARRAY'; + # Attempt to partition the graph into two sets in which no edge connects vertices in the same set. + # If not found, then fall through and use the default circle layout. + my ($top, $bottom) = $self->bipartitePartition; + if (ref $top eq 'ARRAY' && ref $bottom eq 'ARRAY' && @$top && @$bottom) { + $self->{bipartiteLayout} = [ $top, $bottom ]; + return $self->bipartiteLayoutImage(%options); + } + } + return $self->wheelLayoutImage(%options) if defined $self->{wheelLayout}; + + $options{width} //= 250; + $options{height} //= $options{width}; + $options{showLabels} //= 1; + + my $plot = main::Plot( + xmin => -1.5, + xmax => 1.5, + ymin => -1.5, + ymax => 1.5, + width => $options{width}, + height => $options{height}, + xlabel => '', + ylabel => '', + xvisible => 0, + yvisible => 0, + show_grid => 0 + ); + + my $gap = 2 * $main::PI / ($self->numVertices || 1); + + for my $i (0 .. $self->lastVertexIndex) { + my $iVertex = [ cos($i * $gap), sin($i * $gap) ]; + $plot->add_stamp(@$iVertex, color => 'blue'); + + $plot->add_label( + 1.25 * $iVertex->[0], 1.25 * $iVertex->[1], + label => "\\\\($self->{labels}[$i]\\\\)", + color => 'blue', + h_align => 'center', + v_align => 'middle' + ) if $options{showLabels}; + + my $u = 0.275; + my $v = 1 - $u; + + for my $j ($i + 1 .. $self->lastVertexIndex) { + if ($self->hasEdge($i, $j)) { + my $jVertex = [ cos($j * $gap), sin($j * $gap) ]; + $plot->add_dataset($iVertex, $jVertex, color => 'black'); + + if ($options{showWeights}) { + my @vector = ($jVertex->[0] - $iVertex->[0], $jVertex->[1] - $iVertex->[1]); + my $norm = sqrt($vector[0]**2 + $vector[1]**2); + my @perp = ($vector[1] / $norm, -$vector[0] / $norm); + $plot->add_label( + $u * $iVertex->[0] + $v * $jVertex->[0] + $perp[0] * 0.06, + $u * $iVertex->[1] + $v * $jVertex->[1] + $perp[1] * 0.06, + label => "\\\\($self->{adjacencyMatrix}->[$i][$j]\\\\)", + color => 'red', + rotate => ($perp[0] < 0 ? 1 : -1) * + atan2(sqrt(1 - $perp[1] * $perp[1]), $perp[1]) * 180 / + $main::PI - ($perp[1] < 0 ? 180 : 0) + ); + } + } + } + } + + return $plot; +} + +sub gridLayoutImage { + my ($self, %options) = @_; + + die 'Grid layout is not defined, or is but does not have a row and column dimension.' + unless ref $self->{gridLayout} eq 'ARRAY' && @{ $self->{gridLayout} } == 2; + + $options{showLabels} //= 1; + + my $gridGap = 20; + my $gridShift = $gridGap / 2; + my $labelShift = $gridGap / 15; + + my $plot = main::Plot( + xmin => -$gridShift, + xmax => $self->{gridLayout}[1] * $gridGap - $gridShift, + ymin => -$gridShift, + ymax => $self->{gridLayout}[0] * $gridGap - $gridShift, + width => 7 * ($self->{gridLayout}[1] - 1) * $gridGap, + height => 7 * ($self->{gridLayout}[0] - 1) * $gridGap, + xlabel => '', + ylabel => '', + xvisible => 0, + yvisible => 0, + show_grid => 0 + ); + + for my $i (0 .. $self->{gridLayout}[0] - 1) { + for my $j (0 .. $self->{gridLayout}[1] - 1) { + my $x = $gridGap * $j; + my $y = $gridGap * ($self->{gridLayout}[0] - $i - 1); + $plot->add_stamp($x, $y, color => 'blue'); + $plot->add_label( + $x - $labelShift, $y + 2 * $labelShift, + label => "\\\\($self->{labels}[$i + $self->{gridLayout}[0] * $j]\\\\)", + color => 'blue', + h_align => 'center', + v_align => 'middle' + ) if $options{showLabels}; + } + } + + my $u = 0.6666; + my $v = 1 - $u; + + for my $i (0 .. $self->lastVertexIndex) { + my $iVertex = [ + int($i / $self->{gridLayout}[0]) * $gridGap, + ($self->{gridLayout}[0] - ($i % $self->{gridLayout}[0]) - 1) * $gridGap + ]; + for my $j ($i + 1 .. $self->lastVertexIndex) { + if ($self->hasEdge($i, $j)) { + my $jVertex = [ + int($j / $self->{gridLayout}[0]) * $gridGap, + ($self->{gridLayout}[0] - ($j % $self->{gridLayout}[0]) - 1) * $gridGap + ]; + $plot->add_dataset($iVertex, $jVertex, color => 'black', width => 1); + my $vector = [ $jVertex->[0] - $iVertex->[0], $jVertex->[1] - $iVertex->[1] ]; + if ($options{showWeights}) { + my $norm = sqrt($vector->[0]**2 + $vector->[1]**2); + $plot->add_label( + $u * $iVertex->[0] + $v * $jVertex->[0] - $vector->[1] / $norm * 2, + $u * $iVertex->[1] + $v * $jVertex->[1] + $vector->[0] / $norm * 2, + label => "\\\\($self->{adjacencyMatrix}[$i][$j]\\\\)", + color => 'red' + ); + } + } + } + } + + return $plot; +} + +sub bipartiteLayoutImage { + my ($self, %options) = @_; + + my ($top, $bottom); + + if (ref $self->{bipartiteLayout} eq 'ARRAY' + && @{ $self->{bipartiteLayout} } == 2 + && !grep { ref $_ ne 'ARRAY' } @{ $self->{bipartiteLayout} }) + { + ($top, $bottom) = @{ $self->{bipartiteLayout} }; + } elsif ($self->{bipartiteLayout}) { + ($top, $bottom) = $self->bipartitePartition; + die 'Graph is not bipartite.' unless ref $top eq 'ARRAY' && ref $bottom eq 'ARRAY' && @$top && @$bottom; + } else { + die 'Bipartite layout is not defined.'; + } + + $options{width} //= 250; + $options{height} //= $options{width}; + $options{showLabels} //= 1; + + my ($low, $high, $width) = (0, 15, 20); + my @shift = (0, 0); + + my $diff = @$top - @$bottom; + + my $x_max; + if ($diff == 0) { + $x_max = @$top * $width - 10; + } elsif ($diff % 2 == 0 && $diff > 0) { + $x_max = @$top * $width - 10; + $shift[1] += $width * $diff / 2; + } elsif ($diff % 2 == 0) { + $x_max = @$bottom * $width - 10; + $shift[0] += -$width * $diff / 2; + } elsif ($diff > 0) { + $x_max = @$top * $width - 10; + $shift[1] += ($width / 2) * $diff; + } else { + $x_max = @$bottom * $width - 10; + $shift[0] += (-$width / 2) * $diff; + } + + my $plot = main::Plot( + xmin => -10, + xmax => $x_max, + ymin => -5, + ymax => 20, + width => $options{width}, + height => $options{height}, + xlabel => '', + ylabel => '', + xvisible => 0, + yvisible => 0, + show_grid => 0 + ); + + for my $i (0 .. $#$top) { + $plot->add_stamp($i * $width + $shift[0], $high, color => 'blue'); + $plot->add_label( + $i * $width + $shift[0], $high + 2 / 3, + label => "\\\\($self->{labels}[$top->[$i]]\\\\)", + color => 'blue', + h_align => 'center', + v_align => 'bottom' + ) if $options{showLabels}; + } + for my $j (0 .. $#$bottom) { + $plot->add_stamp($j * $width + $shift[1], $low, color => 'blue'); + $plot->add_label( + $j * $width + $shift[1], $low - 2 / 3, + label => "\\\\($self->{labels}[$bottom->[$j]]\\\\)", + color => 'blue', + h_align => 'center', + v_align => 'top' + ) if $options{showLabels}; + } + + my ($u, $v) = $diff >= 0 ? (2 / 3, 1 / 3) : (1 / 3, 2 / 3); + + for my $i (0 .. $#$top) { + for my $j (0 .. $#$bottom) { + next unless $self->hasEdge($top->[$i], $bottom->[$j]); + my $point1 = [ $i * $width + $shift[0], $high ]; + my $point2 = [ $j * $width + $shift[1], $low ]; + $plot->add_dataset($point1, $point2, color => 'black'); + if ($options{showWeights}) { + my $vector = [ $point2->[0] - $point1->[0], $point2->[1] - $point1->[1] ]; + my $norm = sqrt($vector->[0]**2 + $vector->[1]**2); + $plot->add_label( + $u * $point1->[0] + $v * $point2->[0] - $vector->[1] / $norm * 5 / 4, + $u * $point1->[1] + $v * $point2->[1] + $vector->[0] / $norm * 5 / 4, + label => "\\\\($self->{adjacencyMatrix}[ $top->[$i] ][ $bottom->[$j] ]\\\\)", + color => 'red' + ); + } + } + } + + return $plot; +} + +sub wheelLayoutImage { + my ($self, %options) = @_; + + die 'Wheel layout is not defined.' unless defined $self->{wheelLayout}; + + $options{width} //= 250; + $options{height} //= $options{width}; + $options{showLabels} //= 1; + + my $plot = main::Plot( + xmin => -1.5, + xmax => 1.5, + ymin => -1.5, + ymax => 1.5, + width => $options{width}, + height => $options{height}, + xlabel => '', + ylabel => '', + xvisible => 0, + yvisible => 0, + show_grid => 0 + ); + + my $gap = 2 * $main::PI / ($self->lastVertexIndex || 1); + + $plot->add_stamp(0, 0, color => 'blue'); + $plot->add_label( + 0.1, 0.2, + label => "\\\\($self->{labels}[ $self->{wheelLayout} ]\\\\)", + color => 'blue', + h_align => 'center', + v_align => 'middle' + ) if $options{showLabels}; + + for my $i (0 .. $self->lastVertexIndex) { + next if $i == $self->{wheelLayout}; + + my $iRel = $i > $self->{wheelLayout} ? $i - 1 : $i; + + my $iVertex = [ cos($iRel * $gap), sin($iRel * $gap) ]; + $plot->add_stamp(@$iVertex, color => 'blue'); + + $plot->add_label( + 1.25 * $iVertex->[0], 1.25 * $iVertex->[1], + label => "\\\\($self->{labels}[$i]\\\\)", + color => 'blue', + h_align => 'center', + v_align => 'middle' + ) if $options{showLabels}; + + if ($self->hasEdge($self->{wheelLayout}, $i)) { + $plot->add_dataset([ 0, 0 ], $iVertex, color => 'black'); + if ($options{showWeights}) { + my $norm = sqrt($iVertex->[0]**2 + $iVertex->[1]**2); + my @perp = ($iVertex->[1] / $norm, -$iVertex->[0] / $norm); + $plot->add_label( + 0.5 * $iVertex->[0] + $iVertex->[1] / $norm * 0.1, + 0.5 * $iVertex->[1] - $iVertex->[0] / $norm * 0.1, + label => "\\\\($self->{adjacencyMatrix}->[ $self->{wheelLayout} ][$i]\\\\)", + color => 'red', + rotate => ($perp[0] < 0 ? 1 : -1) * + atan2(sqrt(1 - $perp[1] * $perp[1]), $perp[1]) * 180 / + $main::PI - ($perp[1] < 0 ? 180 : 0) + ); + } + } + + for my $j ($i + 1 .. $self->lastVertexIndex) { + next if $j == $self->{wheelLayout}; + + my $jRel = $j > $self->{wheelLayout} ? $j - 1 : $j; + + if ($self->hasEdge($i, $j)) { + my $jVertex = [ cos($jRel * $gap), sin($jRel * $gap) ]; + $plot->add_dataset($iVertex, $jVertex, color => 'black'); + + if ($options{showWeights}) { + my @vector = ($jVertex->[0] - $iVertex->[0], $jVertex->[1] - $iVertex->[1]); + my $norm = sqrt($vector[0]**2 + $vector[1]**2); + my @perp = ($vector[1] / $norm, -$vector[0] / $norm); + $plot->add_label( + 0.5 * $iVertex->[0] + 0.5 * $jVertex->[0] + $vector[1] / $norm * 0.1, + 0.5 * $iVertex->[1] + 0.5 * $jVertex->[1] - $vector[0] / $norm * 0.1, + label => "\\\\($self->{adjacencyMatrix}->[$i][$j]\\\\)", + color => 'red', + rotate => ($perp[0] < 0 ? 1 : -1) * + atan2(sqrt(1 - $perp[1] * $perp[1]), $perp[1]) * 180 / + $main::PI - ($perp[1] < 0 ? 180 : 0) + ); + } + } + } + } + + return $plot; +} + +sub copy { + my $self = shift; + return $self->new($self); +} + +sub shuffle { + my ($self, $permuteLabels) = @_; + my @shuffledGraph; + my @vertexPermutation = main::random_subset($self->numVertices, 0 .. $self->lastVertexIndex); + my @inverseVertexPermutation; # Only needed if labels are also permuted. + @inverseVertexPermutation[@vertexPermutation] = 0 .. $#vertexPermutation; + for my $i (0 .. $self->lastVertexIndex) { + for my $j (0 .. $self->lastVertexIndex) { + $shuffledGraph[ $vertexPermutation[$i] ][ $vertexPermutation[$j] ] = $self->edgeWeight($i, $j); + } + } + return $self->new( + \@shuffledGraph, + labels => $permuteLabels ? [ @{ $self->{labels} }[@inverseVertexPermutation] ] : $self->{labels}, + ref $self->{bipartiteLayout} eq 'ARRAY' ? () : (bipartiteLayout => $self->{bipartiteLayout}), + defined $self->{wheelLayout} ? (wheelLayout => $vertexPermutation[ $self->{wheelLayout} ]) : () + ); +} + +sub nearestNeighborPath { + my ($self, $vertex) = @_; + + my @visited = (undef) x $self->numVertices; + $visited[$vertex] = 1; + + my @path = ($vertex); + my $weight = 0; + my $currentVertex = $vertex; + + while (@path < $self->numVertices) { + my $nearest; + my $min = 0; + for my $i (0 .. $self->lastVertexIndex) { + next if $i == $currentVertex || defined $visited[$i] || $self->hasEdge($currentVertex, $i); + if ($min == 0 || $self->edgeWeight($currentVertex, $i) < $min) { + $min = $self->edgeWeight($currentVertex, $i); + $nearest = $i; + } + } + last unless defined $nearest; + push @path, $nearest; + $visited[$nearest] = 1; + $weight += $self->edgeWeight($currentVertex, $nearest); + $currentVertex = $nearest; + } + + if ($self->hasEdge($currentVertex, $vertex)) { + push @path, $vertex; + $weight += $self->edgeWeight($currentVertex, $vertex); + } + + return (\@path, $weight); +} + +sub kruskalGraph { + my $self = shift; + + my $graph = $self->copy; + my $tree = GraphTheory::SimpleGraph->new($graph->numVertices, labels => $graph->labels); + my $numTreeComponents = $tree->numComponents; + + my $treeWeight = 0; + + my $weight = 0; + my @treeWeights; + + my @weights; + for my $i (0 .. $graph->lastVertexIndex) { + for my $j ($i + 1 .. $graph->lastVertexIndex) { + push(@weights, $graph->edgeWeight($i, $j)) if $graph->hasEdge($i, $j); + } + } + @weights = main::num_sort(@weights); + + while (@weights > 0) { + $weight = shift @weights; + for my $i (0 .. $graph->lastVertexIndex) { + for my $j ($i + 1 .. $graph->lastVertexIndex) { + if ($graph->edgeWeight($i, $j) == $weight) { + $graph->removeEdge($i, $j); + $tree->addEdge($i, $j, $weight); + my $currentTreeNumComponents = $tree->numComponents; + if ($currentTreeNumComponents < $numTreeComponents) { + $numTreeComponents = $currentTreeNumComponents; + $treeWeight += $weight; + push @treeWeights, $weight; + } else { + $tree->removeEdge($i, $j); + } + last; + } + } + } + } + + return ($tree, $treeWeight, \@treeWeights); +} + +sub hasEulerCircuit { + my $self = shift; + + return wantarray ? (0, $main::PG->maketext('This graph is not connected.')) : 0 if $self->numComponents != 1; + + for my $degree ($self->degrees) { + return wantarray ? (0, $main::PG->maketext('The degrees of the vertices in this graph are not all even.')) : 0 + if $degree % 2 != 0; + } + + return 1; +} + +sub hasEulerTrail { + my $self = shift; + + return wantarray ? (0, $main::PG->maketext('This graph is not connected.')) : 0 if $self->numComponents != 1; + + my ($odd, $even) = (0, 0); + + for my $degree ($self->degrees) { + if ($degree % 2 == 0) { ++$even } + else { ++$odd } + } + + return wantarray + ? ( + 0, + $main::PG->maketext( + 'This graph does not have two vertices of odd degree and all other vertices of even degree.') + ) + : 0 + if $even != $self->numVertices && $odd != 2; + return 1; +} + +sub pathIsEulerTrail { + my ($self, @path) = @_; + + my $graph = $self->copy; + + my $i = shift @path; + do { + my $j = shift @path; + return + wantarray ? (0, $main::PG->maketext('An edge traversed by this path does not exist in the graph.')) : 0 + unless $graph->hasEdge($i, $j); + $graph->removeEdge($i, $j); + $i = $j; + } while @path > 0; + + for my $i (0 .. $graph->lastVertexIndex) { + for my $j (0 .. $i - 1) { + return wantarray ? (0, $main::PG->maketext('This path does not traverse all edges.')) : 0 + if $graph->hasEdge($i, $j); + } + } + + return 1; +} + +sub pathIsEulerCircuit { + my ($self, @path) = @_; + my @isEulerTrail = $self->pathIsEulerTrail(@path); + return wantarray ? @isEulerTrail : 0 unless $isEulerTrail[0]; + return wantarray ? (0, $main::PG->maketext('This path is not a circuit.')) : 0 unless $path[0] == $path[-1]; + return 1; +} + +sub hasCircuit { + my $self = shift; + + my $graph = $self->copy; + + for (my $i = 0; $i < $graph->numVertices; ++$i) { + my $sum = 0; + for my $j (0 .. $graph->lastVertexIndex) { + $sum += $graph->hasEdge($i, $j) ? 1 : 0; + } + if ($sum == 1) { + for my $j (0 .. $graph->lastVertexIndex) { + $graph->removeEdge($i, $j); + } + $i = -1; + } + } + + for my $i (0 .. $graph->lastVertexIndex) { + for my $j ($i + 1 .. $graph->lastVertexIndex) { + return 1 if $graph->hasEdge($i, $j); + } + } + + return 0; +} + +sub isTree { + my $self = shift; + return wantarray ? (0, $main::PG->maketext('This graph has a circuit.')) : 0 if $self->hasCircuit; + return wantarray ? (0, $main::PG->maketext('This graph is not connected.')) : 0 if $self->numComponents > 1; + return 1; +} + +sub isForest { + my $self = shift; + return wantarray ? (0, $main::PG->maketext('This graph has a circuit.')) : 0 if $self->hasCircuit; + return 1; +} + +# Returns 1 if the given $graph is bipartite, and 0 otherwise. +sub isBipartite { + my $self = shift; + + my @vertexColors = (0) x $self->numVertices; + my @color = (1, 2); + + my $done; + do { + my $i = 0; + while ($vertexColors[$i] != 0) { ++$i; } + $vertexColors[$i] = $color[0]; + + my @verticesToClassify; + push @verticesToClassify, $i; + + do { + my $vertex = shift @verticesToClassify; + my $currentColor = $vertexColors[$vertex]; + for my $i (0 .. $self->lastVertexIndex) { + if ($self->hasEdge($vertex, $i)) { + return 0 if $currentColor == $vertexColors[$i]; + if ($vertexColors[$i] == 0) { + push @verticesToClassify, $i; + $vertexColors[$i] = $color[ 2 - $currentColor ]; + } + } + } + } while @verticesToClassify; + + $done = 1; + for my $i (0 .. $self->lastVertexIndex) { + $done *= $vertexColors[$i]; + } + } while $done == 0; + + return 1; +} + +sub bipartitePartition { + my $self = shift; + + my @partition = ([], []); + + my %partitionAssignments; + + my $done; + do { + my $i = 0; + ++$i while defined $partitionAssignments{$i}; + $partitionAssignments{$i} = 0; + push(@{ $partition[0] }, $i); + + my @vertices; + push @vertices, $i; + + do { + my $vertex = shift @vertices; + my $currentPartition = $partitionAssignments{$vertex}; + for my $i (0 .. $self->lastVertexIndex) { + if ($self->hasEdge($vertex, $i)) { + return if defined $partitionAssignments{$i} && $currentPartition == $partitionAssignments{$i}; + if (!defined $partitionAssignments{$i}) { + push @vertices, $i; + $partitionAssignments{$i} = $currentPartition ? 0 : 1; + push(@{ $partition[ $partitionAssignments{$i} ] }, $i); + } + } + } + } while @vertices; + + $done = 1; + for my $i (0 .. $self->lastVertexIndex) { + unless (defined $partitionAssignments{$i}) { + $done = 0; + last; + } + } + } while !$done; + + $partition[0] = [ main::num_sort(@{ $partition[0] }) ]; + $partition[1] = [ main::num_sort(@{ $partition[1] }) ]; + + return @partition; +} + +sub dijkstraPath { + my ($self, $start, $end) = @_; + + my ($ind, $new, @dist, @path, @prev, @used); + + for my $i (0 .. $self->lastVertexIndex) { + $dist[$i] = -1; + $prev[$i] = -1; + } + $dist[$start] = 0; + + my @available = (0 .. $self->lastVertexIndex); + + while (@available) { + my $min = 1000000000; # Infinity (for all practical purposes) + for my $i (0 .. $#available) { + my $loc = $available[$i]; + if ($dist[$loc] >= 0 && $dist[$loc] < $min) { + $new = $loc; + $ind = $i; + $min = $dist[$new]; + } + } + push @used, $new; + splice @available, $ind, 1; + @used = main::num_sort(@used); + + for my $i (0 .. $self->lastVertexIndex) { + my $weight = $self->edgeWeight($new, $i); + if ($weight != 0) { + if ($dist[$i] > $min + $weight || $dist[$i] < 0) { + $dist[$i] = $min + $weight; + $prev[$i] = $new; + } + } + } + } + + my $loc = $end; + while ($loc != $start) { + unshift @path, $loc; + $loc = $prev[$loc]; + } + unshift @path, $start; + + return ($dist[$end], @path); +} + +sub sortedEdgesPath { + my $self = shift; + + my @weights; + my $sortedGraph = GraphTheory::SimpleGraph->new($self->numVertices, labels => $self->labels); + + for my $i (0 .. $self->lastVertexIndex) { + for my $j ($i + 1 .. $self->lastVertexIndex) { + next unless $self->hasEdge($i, $j); + push @weights, $self->edgeWeight($i, $j); + } + } + + @weights = main::num_sort(@weights); + + # Returns 1 if an edge can be added to the sorted edges based graph and 0 otherwise. An edge can be added if it does + # not make a vertex have more than two edges connected to it, and it does not create a circuit in the graph (unless + # it is the last vertex in which case that is okay since it completes the circuit). + my $goodEdge = sub { + my $graph = shift; + + my $sum = 0; + + for my $i (0 .. $graph->lastVertexIndex) { + my $degree = $graph->vertexDegree($i); + return 0 if $degree > 2; + $sum += $degree; + } + + return $sum < 2 * $graph->numVertices && $graph->hasCircuit ? 0 : 1; + }; + + my @pathWeights; + + do { + my $weight = shift @weights; + for my $i (0 .. $sortedGraph->lastVertexIndex) { + for my $j ($i + 1 .. $sortedGraph->lastVertexIndex) { + if ($weight == $self->edgeWeight($i, $j)) { + $sortedGraph->addEdge($i, $j, $self->edgeWeight($i, $j)); + if ($goodEdge->($sortedGraph)) { + push @pathWeights, $weight; + } else { + $sortedGraph->removeEdge($i, $j); + } + } + } + } + } while @pathWeights < $sortedGraph->numVertices && @weights > 0; + + return ($sortedGraph, \@pathWeights); +} + +sub chromaticNumber { + my $self = shift; + return Chromatic::computeBestColoring(@{ $self->{adjacencyMatrix} }); +} + +sub kColoring { + my ($self, $k) = @_; + + my @colors = (-1) x $self->numVertices; + my @nextColor = (0) x $self->numVertices; + my $v = 0; + + while ($v >= 0) { + my $assigned = 0; + while ($nextColor[$v] < $k) { + my $c = ++$nextColor[$v]; + my $ok = 1; + for my $u (0 .. $self->lastVertexIndex) { + if ($self->hasEdge($v, $u) && $colors[$u] == $c) { + $ok = 0; + last; + } + } + if ($ok) { + $colors[$v] = $c; + ++$v; + return @colors if $v == $self->numVertices; + $nextColor[$v] = 0; + $assigned = 1; + last; + } + } + unless ($assigned) { + $colors[$v] = -1; + $nextColor[$v] = 0; + --$v; + } + } + + return; +} + +# The GraphTheory::SimpleGraph::Parser::List, GraphTheory::SimpleGraph::EdgeSet and GraphTheory::SimpleGraph::Edge +# packages are special lists for the edgeSet return value context. + +package GraphTheory::SimpleGraph::Parser::List; +our @ISA = qw(Parser::List::List); + +sub _check { + my $self = shift; + $self->SUPER::_check; + + # Only handle implicit lists or lists explicitly opened with a brace. + return if $self->{open} && $self->{open} ne '{'; + + my $entryType = $self->typeRef->{entryType}; + + # Since there can only be one brace paren, the distinction between an edge and an edge set needs to be made here. + # An empty list or a list that contains another list is an edge set. All other lists are edges. + my $isEdgeSet = $self->length ? 0 : 1; + for (@{ $self->{coords} }) { + if ($_->{type}{list} && $_->{type}{list} == 1) { $isEdgeSet = 1; last; } + } + + if ($isEdgeSet) { + $self->{type} = Value::Type('EdgeSet', scalar(@{ $self->{coords} }), $entryType, list => 1); + } elsif ($self->{type}{name} ne 'Edge') { + $self->{type} = Value::Type('Edge', scalar(@{ $self->{coords} }), $entryType, list => 1); + } + + if ($self->{type}{name} eq 'Edge') { + my $strings = $self->context->strings->all; + for (@{ $self->{coords} }) { + $self->Error('An edge may only contain vertices.') + unless ref $_ eq 'Parser::String' + && defined $strings->{ $_->{value} } + && $strings->{ $_->{value} }{isVertex}; + } + $self->Error('An edge must contain exactly two vertices.') if $self->length != 2; + } elsif ($self->{type}{name} eq 'EdgeSet') { + for (@{ $self->{coords} }) { + $self->Error('An edge set may only contain edges.') unless $_->{type}{name} eq 'Edge'; + } + } +} + +package GraphTheory::SimpleGraph::Value::EdgeSet; +our @ISA = qw(Value::List); + +sub cmp_defaults { + my ($self, %options) = @_; + return ( + $self->SUPER::cmp_defaults(%options), + entry_type => 'edge', + list_type => 'edge set', + removeParens => 0, + showParenHints => 1, + implicitList => 0 + ); +} + +sub compare { + my ($l, $r, $flag) = @_; + my $self = $l; + $r = $self->promote($r); + if ($flag) { my $tmp = $l; $l = $r; $r = $tmp } + my @l = main::num_sort($l->value); + my @r = main::num_sort($r->value); + while (@l && @r) { + my $cmp = shift(@l) <=> shift(@r); + return $cmp if $cmp; + } + return @l - @r; +} + +package GraphTheory::SimpleGraph::Value::Edge; +our @ISA = qw(Value::List); + +sub cmp_defaults { + my ($self, %options) = @_; + return ( + $self->SUPER::cmp_defaults(%options), + entry_type => 'vertex', + list_type => 'edge', + removeParens => 0, + showParenHints => 1, + implicitList => 0 + ); +} + +sub compare { + my ($l, $r, $flag) = @_; + my $self = $l; + $r = $self->promote($r); + if ($flag) { my $tmp = $l; $l = $r; $r = $tmp } + my @l = main::num_sort($l->value); + my @r = main::num_sort($r->value); + while (@l && @r) { + my $cmp = shift(@l) <=> shift(@r); + return $cmp if $cmp; + } + return @l - @r; +} + +1; + +=head1 NAME + +SimpleGraph.pl - Tools for displaying and manipulating simple graphs from graph +theory. + +=head1 DESCRIPTION + +The core of this macro is the C object which +represents simple graphs from graph theory via an adjacency matrix. + +=head1 FUNCTIONS + +The following functions can be used to construct a C +object. + +=head2 SimpleGraph + + $graph = SimpleGraph($definition, %options); + +This is an alias for the C constructor. + +The C<$definition> argument is required and can be the number of vertices in the +graph, another C object, a reference to an array of +arrays of numbers, a reference to a MathObject C, a reference to an +array of arrays containing two strings and possibly a number (an edge set with +optional edge weights), a reference to an array of +C objects, or a reference to a +C object. + +If C<$definition> is the number of vertices in the graph, then the graph that is +returned will have that number of vertices and no edges. Note that the number of +vertices must be greater than 0. + +If C<$definition> is another C object, then a copy of +the graph represented by that object will be returned. + +If C<$definition> is a one of the matrix forms (a reference to an array of +arrays of numbers or a reference to a MathObject C), then that matrix +will be used for the adjacency matrix of the graph. Note that it must be a +square symmetric matrix with zero entries along the diagonal and must have size +greater than or equal to 1. For example, + + [ + [ 0, 1, 0], + [ 1, 0, 2], + [ 0, 2, 0] + ] + +represents a graph that has three vertices, an edge between the first and +second vertices with weight one, and an edge between the second and third +vertices with weight 2. + +If C<$definition> is one of the edge set definition forms (a reference to an +array of arrays containing two strings and possibly a number, a reference to an +array of C objects, or a reference to a +C object), then the C option +is required. For the reference to an array of arrays form each edge in the edge +set must be an array containing two strings and possibly a third numeric +element. The two strings are the labels for the vertices connected by the edge. +The optional third element is the weight of the edge. If the optional third +element is not given, then the edge will have a weight of one. Note that all +labels provided in the C option will be used with this definition, and +will all be vertices in the graph. For example, if C<$definition> is given as + + [ + [ 'A', 'B' ], + [ 'A', 'C', 2 ], + [ 'B', 'C', 3 ], + ] + +and the C option is C<['A', 'B', 'C', 'D']>, then the return object will +represent a graph that has at four vertices labeled 'A', 'B', 'C', and 'D', +respectively, an edge between 'A' and 'B' with weight one, an edge between 'A' +and 'C' with weight 2, an edge between 'B' and 'C' with weight 3, and no edges +connecting to the vertex 'D'. Note that setting weights when a graph is +constructed via a reference to an array of C +objects or a reference to a C object is not +supported. However, weights can still be set after construction with the +L method or the L method. + +The arguments that may be passed in C<%options> are as follows. Note that all of +these options are undefined by default. + +=over + +=item labels + +The value of this option must be a reference to an array of strings. These are +the labels for the vertices of the graph. The array must contain at least as +many strings as there are vertices of the graph. Any extraneous strings will not +be used (except in the edge set definition case as mentioned above). If this +option is not given, then a random set of consecutive letters from the alphabet +will be used for the labels of the vertices. Note that if an edge set version of +the C<$definition> argument is used, then this option must be provided. + +=item gridLayout + +If this option is given, then it must be a reference to an array containing two +numbers, for example, C<[3, 4]>. In this case when the L method is +called to create the image of the graph, the vertices will be arranged into a +grid with the number of rows equal to the first number in the array, and the +number of columns equal to the second number in the array. Note that many graphs +will not work well in a grid layout. This is primarily intended for graphs +created via the L function where its first argument is used +as the value for this option. This option is generally intended for internal use +in this macro specifically for that function. However, there may be other graphs +that may also work well in a grid layout. + +=item bipartiteLayout + +If this option is given, then it must either be 1 or a reference to an array +containing two arrays of vertex indices. If this is 1, then a partition of the +vertices into two sets in which no two vertices in the same set are connected by +an edge will attempt to be obtained internally via the L +method. If the graph is not bipartite then no such partition will be found and +the L method will not use the bipartite layout. Instead the image will +be displayed in the default circle layout. If this is an array containing two +arrays of vertex indices, then the two arrays must be the partition of the +vertices of the graph in which no two vertices in the same set are connected by +an edge. In this case when the L method is called to create the image of +the graph, the vertices will be arranged into to rows with the vertices in the +first set in the top row, and the vertices in the second set in the bottom row. +Note that an exception will be thrown if this option is 1 and the graph is not +bipartite. + +=item wheelLayout + +If this option is given, then it must be an index of one of the vertices in the +graph (0, 1, 2, ...), and when the L method is called to create the image +of the graph, that vertex will be placed in the center of a circle, and the +other vertices will be evenly spaced around the perimeter of the circle. + +=back + +=head2 randomSimpleGraph + + $graph = randomSimpleGraph($size, %options); + +This function returns a C object in which the +existence of an edge between any two vertices is randomly determined. + +The C<$size> argument is required and must either be an integer that is greater +than zero, or a reference to an array containing two integers, both of which are +greater than zero. If this argument is an integer that is greater than zero, +then that will be the number of vertices of the graph. If this argument is a +reference to an array containing two integers that are greater than zero, then +the number of vertices in the graph will be the product of those two integers. +Furthermore, this array reference will be passed as the C option to +the C constructor, and the image of the graph will +have the vertices arranged into rows and columns. Furthermore, in the graph will +be generated such that the only possible edges will connect vertices that are +adjacent in that grid (including diagonal adjacency). + +The arguments that may be passed in C<%options> are as follows. + +=over + +=item labels + +The value of this option, if given, must be a reference to an array of strings. +These are the labels for the vertices of the graph. The array must contain at +least as many strings as there are vertices of the graph. Any extraneous strings +will not be used (except in the edge set definition case as mentioned above). If +this option is not given, then a random set of consecutive letters from the +alphabet will be used for the labels of the vertices. + +=item edgeProbability + +This is the probability that there will be an edge between any two vertices. By +default this is 0.5. Note that if C<$size> is a reference to an array of +integers, then this only applies to vertices that are adjacent in the grid, and +there will never be edges between vertices that are not adjacent in the grid. + +=item edgeWeightRange + +If this is given, then it must be a reference to an array of two or three +numbers. In this case the weights of the edges will be a random number from the +first number to the second number with increments of the optional third number +(the increment defaults to 1). In fact the elements of this array are passed +directly to the C function (see L). Note that the C option takes precedence over this +option, and if it is given and is an integer greater than one, then it will be +used instead of this option. Furthermore, if neither this option nor the +C option are given, then all edge weights will be 1. + +=item startEdgeWeight + +If this option is given, then it must be an integer that is greater than zero. +In this case this will be the first edge weight used in the generated graph, and +the other edge weights will be determined by adding increments of the +following C option. + +=item edgeWeightIncrement + +If this option is given, then it must be an integer that is greater than zero. +Note that this is only used if C is also given and is greater +than zero, and in that case increments of this number will be added to the +C and used for the weights of the edges. If the +C is given, is an integer greater than 0, and this option is +undefined, then an edge weight increment of 1 will be used. + +=back + +The examples below demonstrate the usage of this function and its options in +more details. + + $graph = randomSimpleGraph(random(3, 5), edgeProbability => 1); + +will return a complete graph with 3, 4, or 5 vertices and all edge weights equal +to 1. + + $graph = randomSimpleGraph(random(3, 5), edgeProbability => 0); + +will return a graph with 3, 4, or 5 vertices that has no edges. Note that +C<$graph = SimpleGraph(random(3, 5))> is equivalent and is slightly more +efficient. + + $graph = randomSimpleGraph( + 5, + edgeProbability => 0.3, + edgeWeightRange => [1, 10] + ); + +will return a graph with 5 vertices with probability 0.3 of an edge between any +two vertices, and random edge weights from 1 to 10 (in increments of 1). + + $graph = randomSimpleGraph( + 5, + startEdgeWeight => 2, + edgeWeightIncrement => 3 + ); + +will return a graph with 5 vertices with probability 0.5 of an edge between any +two vertices, and the edge weights will be 2, 5, 8, 11, ... (which will be +randomly assigned to the edges in the graph). + + $graph = randomSimpleGraph([3, 4], edgeProbability => 0.6); + +will return a graph with 12 vertices and all edge weights equal to 1. When the +image of this graph is displayed the vertices will be arranged into a grid with +3 rows and 4 columns. The probability of an edge between vertices that are +adjacent in the grid is 0.6. There will be no edges between vertices that are +not adjacent in the grid. + +=head2 randomGraphWithEulerCircuit + + $graph = randomGraphWithEulerCircuit($size, %options); + +This function returns a C object that represents a +random graph with C<$size> vertices that has an Euler circuit. This function +also accepts the C, C, C, and +C options accepted by the L function. + +=head2 randomGraphWithEulerTrail + + $graph = randomGraphWithEulerTrail($size, %options); + +This function returns a C object that represents a +random graph with C<$size> vertices that has an Euler trail, but does not have +an Euler circuit. This function also accepts the C, C, +C, and C options accepted by the +L function. + +=head2 randomGraphWithoutEulerTrail + + $graph = randomGraphWithoutEulerTrail($size, %options); + +This function returns a C object that represents a +random graph with C<$size> vertices that does not have an Euler trail (or +circuit). This function also accepts the C, C, +C, and C options accepted by the +L function. + +=head2 randomBipartiteGraph + + $graph = randomBipartiteGraph($size, %options); + +This function returns a C object that represents a +random bipartite graph. + +The C<$size> argument can either be a number or a reference to an array of two +integers that are greater than zero. If it is a number, then the graph will have +C<$size> vertices, and the number of vertices in the two parts of the bipartite +partition will be randomly determined. If it is a reference to an array of two +integers, then graph will have the number of vertices equal to the sum of those +two numbers, and the two sets in the bipartite partition will have those numbers +of elements. + +This function also accepts the C, C, +C, C, and C options +accepted by the L function. + +=head2 randomTreeGraph + + $graph = randomTreeGraph($size, %options); + +This function returns a C object that represents a +random graph with C<$size> vertices that is a tree. This function also accepts +the C, C, C, and +C options accepted by the L function. + +=head2 randomForestGraph + + $graph = randomForestGraph($size, %options); + +This function returns a C object that represents a +random graph with C<$size> vertices that is a forest, but is not a tree. This +function also accepts the C, C, C, +and C options accepted by the L function. + +=head2 randomHamiltonianGraph + + $graph = randomHamiltonianGraph($size, %options); + +This function returns a C object that represents a +random graph with C<$size> vertices that is Hamiltonian. Note that C<$size> must +be 5 or more, or an exception will be thrown. The C may be passed in +C<%options>, and is the same as for the above functions. + +=head2 randomNonHamiltonianGraph + + $graph = randomNonHamiltonianGraph($size, $type, %options); + +This function returns a C object that represents a +random graph with C<$size> vertices that is not Hamiltonian. + +If C<$type> is even, then the graph will have a vertex of degree one. If +C<$type> is odd, then the graph will consist of two cycles joined by a single +edge. Note that for the odd C<$type> graph, the C<$size> must be at least 6, or +an exception will be thrown. + +This function also accepts the C, C, +C, and C options accepted by the +L function. + +=head2 randomGraphWithDegreeSequence + + $graph = randomGraphWithDegreeSequence($degrees, %options); + +This function returns a C object that represents a +random graph that has the given C<$degree> sequence, if such a graph is +possible, and undefined otherwise. + +The C<$degrees> argument must be a reference to an array containing the desired +vertex degrees. + +This function also accepts the C, C, +C, and C options accepted by the +L function. + +=head2 cycleGraph + + $graph = cycleGraph($size, %options); + +This function returns a C object that represents a +graph with C<$size> vertices that is a cycle. This function also accepts the +C, C, C, and C +options accepted by the L function. + +=head2 completeGraph + + $graph = completeGraph($size, %options); + +This function returns a C object that represents a +graph with C<$size> vertices that is a complete graph. This function also +accepts the C, C, C, and +C options accepted by the L function. + +=head2 wheelGraph + + $graph = wheelGraph($size, %options); + +This function returns a C object that represents a +graph with C<$size> vertices that is a wheel. This function also accepts the +C, C, C, and C +options accepted by the L function. + +Note that the returned graph object will have the C graph option +set to the vertex at the hub of the wheel, and that vertex will be displayed in +the center in the graph image. + +=head2 completeBipartiteGraph + + $graph = completeBipartiteGraph($m, $n, %options); + +This function returns a C object that represents a +graph with C<$m * $n> vertices that is a complete bipartite graph with C<$m> +vertices in first set in the bipartite partition, and C<$n> vertices in the +other set in the bipartite partition. This function also accepts the C, +C, C, and C options +accepted by the L function. + +Note that the returned graph object will have the C graph +option set so that the graph image is nicely displayed with the vertices in the +first set in the bipartite partition in a row on the top, and the vertices in +the other set in a row on the bottom. + +=head2 EdgeSet + + $edgeSet = EdgeSet($edges); + $edgeSet = EdgeSet($edge1, $edge2, ...); + +This method is an alias for the C +MathObject constructor. It can be passed a reference to an array of edges or +just a list of edges. An edges can be a reference to an array containing two +vertices, or can be a C MathObject. As +with all MathObject constructors, the first argument can optionally be a +context. Note that vertices must be strings in the current context or the +context that is passed. + +Since this derives from a MathObject C, a C must be used for +a custom checker routine. + +Usually this would only be used in the C context. Otherwise the +returned value will not display correctly in the problem and will not work well +as an answer. However, the object can still be passed to the C +function to create a graph with that edge set. + +=head2 Edge + + $edge = Edge($vertex1, $vertex2); + $edge = EdgeSet([ $vertex1, $vertex2 ]); + +This method is an alias for the C +MathObject constructor. It can be passed a reference to an array containing two +vertices or a list of two vertices. As with all MathObject constructors, the +first argument can optionally be a context. Note that vertices must be strings +in the current context or the context that is passed. + +Since this derives from a MathObject C, a C must be used for +a custom checker routine. + +Usually this would only be used in the C context. Otherwise the +returned value will not display correctly in the problem and will not work well +as an answer. However, the object can still be passed in an array reference to +the C function to create a graph with that edge set. + +=head1 GraphTheory::SimpleGraph Methods + +The C package is the heart of this macro. All of the +functions described above return an object that is an instance of this package +class. See the L function for the constructor usage. In addition, +the following object methods are available. + +=head2 adjacencyMatrix + + $matrix = $graph->adjacencyMatrix; + +This method returns the adjacency matrix that defines the graph. That is a +reference to an array of array references each containing numbers. The matrix +will be a symmetric square matrix with zero entries along the diagonal. The +entry in the i-th row and j-th column will be zero if there is not an edge +connecting the i-th and j-th vertex, and will be nonzero if there is such an +edge. Furthermore, the entry represents the edge weight in the case that it is +not zero. + +Note that indexing of vertices is zero based. So the first vertex has index 0, +and the last vertex has index one less than the number of vertices in the graph. + +=head2 edgeSet + + $edgeSet = $graph->edgeSet(%options); + +This method returns the edge set for the graph as a MathObject C +containing MathObject Cs. For example, if a graph has adjacency matrix + + [ + [ 0, 1, 0, 0 ], + [ 1, 0, 1, 0 ], + [ 0, 1, 0, 1 ] + [ 0, 0, 1, 0 ] + ] + +and labels 'A', 'B', 'C', and 'D', then this method would return + + {{A, B}, {B, C}, {C, D}} + +where the outer set is a MathObject C, the elements of that set are +MathObject Cs, and the elements of those sets are MathObject Cs. +The returned C will be either be in a created C context with +the vertex labels of the graph defined as strings and marked as being vertices +in that context, or in a context provided in the C<%options> as described below. + +This return value can be used as an answer to a question, and will display in +output as shown above. + +Note, that edge weights are not represented in the return value of this method. + +The following options that may be passed in C<%options>. + +=over + +=item context (Default: C<< context => undef >>) + +The context to use for the returned MathObject C object. If this is not +provided, then a C context will be created, the vertex labels added as +strings to the context and marked as being vertices and the returned object will +be in the created context. + +=item caseSensitive (Default: C<< caseSensitive => 1 >>) + +Whether vertex labels that are added to the context are case sensitive or not. + +=back + +=head2 addVerticesToContext + + $graph->addVerticesToContext($caseSensitive, $context); + +This adds the vertex labels of the graph to the context and marks them as being +vertices (by adding the C flag). + +The C<$caseSensitive> and C<$context> arguments are optional. + +If C<$caseSensitive> is 1 or the argument is not provided, then the vertices +will be case sensitive. So entering "a" will not be accepted as correct for the +vertex labeled "A". If C<$caseSensitive> is 0, then vertices will not be case +sensitive. So "a" will be accepted as correct for the vertex labeled "A". + +If C<$context> is provided, then the strings will be added to that context. +Otherwise the strings will be added to the current context. + +=head2 numVertices + + $n = $graph->numVertices; + +This method returns the number of vertices in the graph. + +=head2 lastVertex + + $n = $graph->lastVertexIndex; + +This method returns the index of the last vertex in the graph (i.e., it is one +less then the number of vertices). + +=head2 numEdges + + $n = $graph->numEdges; + +This method returns the number of edges in the graph. + +=head2 labels + + $labels = $graph->labels; + $graph->labels($labels); + +Get or set the vertex labels for the graph. In both examples above, C<$labels> +is a reference to an array of strings, for example, C<[ 'M', 'N', 'O', 'P' ]>. + +=head2 labelsString + + $labelsString = $graph->labelsString; + +This returns the vertex labels joined with a comma. This is a convenience method +for displaying labels in a problem. For example, the labels string can be +inserted into a PGML block with C<< [`[$graph->labelsString]`] >>. + +=head2 vertexLabel + + $label = $graph->vertexLabel($index); + +This method returns the label of the vertex at index C<$index> in the graph. + +=head2 vertexIndex + + $index = $graph->vertexIndex($label); + +This method returns the index of the vertex in the graph that is labeled +C<$label> if the given C<$label> exists for a vertex, and -1 otherwise. + +=head2 vertexDegree + + $degree = $graph->vertexDegree($index); + +This method returns the degree of the vertex at index C<$index> in the graph. + +=head2 degrees + + @degrees = $graph->degrees; + +This method returns an array of the degrees of the vertices in the graph. + +=head2 numComponents + + $c = $graph->numComponents; + +This method returns the number of connected components in the graph. + +=head2 edgeWeight + + $c = $graph->edgeWeight($i, $j); + $graph->edgeWeight($i, $j, $weight); + +Get or set the weight of the edge between the vertices at indices C<$i> and +C<$i>. If the optional third C<$weight> argument is provided, then the weight +of the edge between the vertices at indices C<$i> and C<$j> will be set to +C<$weight>. In any case, the weight of the edge between the vertices at indices +C<$i> and C<$j> will be returned. Note that an edge weight of zero means that +there is no edge connecting the vertices. + +Setting the edge weight to zero is equivalent to removing the edge if it was +nonzero, and setting the edge weight to a nonzero number when it was zero adds +an edge. However, if your only intent is to add or remove an edge, then it is +recommended to use the C or C methods instead for clarity +in the code. + +=head2 addEdge + + $graph->addEdge($i, $j, $weight); + +Add and edge that connects the vertices at indices C<$i> and C<$j>. The +C<$weight> argument is optional, and if not provided the weight of the edge will +be set to 1. + +=head2 removeEdge + + $graph->removeEdge($i, $j); + +Remove the edge that connects the vertices at indices C<$i> and C<$j>. This is +accomplished by setting the weight of the edge between the two vertices to 0. + +=head2 hasEdge + + $graph->hasEdge($i, $j); + +This method returns a true value if the graph has an edge that connects the +vertices at indices C<$i> and C<$j>, and a false value otherwise. + +=head2 setRandomWeights + + $graph->setRandomWeights(%options); + +Set random weights for the edges of the graph. This method does not add or +remove any edges, but randomly sets the weights of the existing edges. + +The arguments that may be passed in C<%options> are as follows. If neither of +the C or C options is given, then this +method does not change the current edge weights of the graph. + +=over + +=item startEdgeWeight + +If this option is given, then it must be an integer that is greater than zero. +In this case this will be the first edge weight used in the generated graph, and +the other edge weights will be determined by adding increments of the +following C option. + +=item edgeWeightIncrement + +If this option is given, then it must be an integer that is greater than zero. +Note that this is only used if C is also given and is greater +than zero, and in that case increments of this number will be added to the +C and used for the weights of the edges. If the +C is given, is an integer greater than 0, and this option is +undefined, then an edge weight increment of 1 will be used. + +=item edgeWeightRange + +If this is given, then it must be a reference to an array of two or three +numbers. In this case the weights of the edges will be a random number from the +first number to the second number with increments of the optional third number +(the increment defaults to 1). In fact the elements of this array are passed +directly to the C function (see L). Note that the C option takes precedence over this +option, and if it is given and is an integer greater than one, then it will be +used instead of this option. + +=item edgeCount + +The number of edges in the graph for which to set random weights. If this +option is not given, then the number of edges in the graph will be computed, and +all the weights of all existing edges will be randomized. If the number of edges +in the graph is already known, then set this option to that value for efficiency +so that this method does not need to compute that number again. + +=back + +=head2 isEqual + + $graph->isEqual($other); + +This method returns 1 if the graph object represented by C<$graph> and the graph +represented by C<$other> are the same. This does not necessarily return 1 for +isomorphic graphs. The two graphs must literally have the same adjacency matrix. + +=head2 isIsomorphic + + $graph->isIsomorphic($other); + +This method returns 1 if the graph represented by C<$graph> and the graph +represented by C<$other> are isomorphic. + +WARNING: This method uses a brute force approach and compares the first graph to +all possible permutations of the other graph, and so should not be used for +graphs with a large number of vertices (probably no more than 8). + +=head2 image + + $graph->image(%options); + +Constructs and returns a L object via the L macro that +provides a pictorial representation of the graph. The returned object may be +inserted into the problem via the C method (see L), L, or using the PGML image syntax +(for example, C<< [!alt text!]{$graph->image} >>). + +Note that the C, C, and C options that +can be set when the C object is created affect how the +graph is displayed. See the L function for details on those +options. Note that those options can also be set as properties of the object +anytime after construction and before this method is called (for example, +C<< $self->{bipartiteLayout = 1; >>). If none of those layout options are used, +then the vertices of the graph will be evenly spaced around the perimeter of a +circle with the vertex at index 0 on the right. + +The following options can be set via the C<%options> argument. + +=over + +=item width + +This is the width of the image. Default is 250. + +Note that the width option is not honored if the C is used. Note +that the width can still be set via the L C method, or +via the width option for the PGML image syntax. + +=item height + +This is the height of the image. Default is the value of the width option above. + +Note that the height option is not honored if the C is used. Note +that the height can still be set via the L C method, or +via the height option for the PGML image syntax. + +=item showLabels + +If this is 1, then vertex labels will be shown. If this is 0, then vertex labels +will not be shown. Default is 1 (so labels must be explicitly hidden by setting +this to 0). + +=item showWeights + +If this is 1, then edge weights will be shown. If this is 0, then edge weights +will not be shown. Default is 0. + +Note that the display of edge weights often does not work well as it can be +unclear which edge a given weight belongs to in the image. Particularly if a +graph has a large number of edges. Graphs that are created via the +C function using the row and column C<$size> argument (and +hence are displayed using the grid layout) do work quite well for this. + +=back + +=head2 gridLayoutImage + + $graph->gridLayoutImage(%options); + +This method is not intended to be used externally. It is called by the L +method if the C property is set for the graph object. If this method +is called directly and the C property is set, then it will still +work. Otherwise an exception will be thrown. It accepts the same options as the +L method, but does not honor the C and C options. + +=head2 bipartiteLayoutImage + + $graph->bipartiteLayoutImage(%options); + +This method is not intended to be used externally. It is called by the L +method if the C property is set for the graph object. If this +method is called directly and the C property is set, then it +will still work. Otherwise an exception will be thrown. Note that an exception +will also be thrown if the C property is set to 1, and the +graph is not bipartite. It accepts the same options as the L method. + +=head2 wheelLayoutImage + + $graph->wheelLayoutImage(%options); + +This method is not intended to be used externally. It is called by the L +method if the C property is set for the graph object. If this +method is called directly and the C property is set, then it will +still work. Otherwise an exception will be thrown. It accepts the same options +as the L method. + +=head2 copy + + $copy = $graph->copy; + +This method returns a copy of C<$graph>. This is an exact copy with all +properties duplicated to the returned object. + +=head2 shuffle + + $shuffledGraph = $graph->shuffle($permuteLabels); + +This method returns a randomization of C<$graph> obtained by permuting the +vertices. The edges that connected the vertices in the original graph will still +connect the permuted vertices in the resulting graph. In other words the +returned graph is isomorphic to the original. + +If the optional C<$permuteLabels> argument is provided and is true, then the +labels will also be permuted with the vertices. Otherwise the labels will remain +in the same order that they were in the original graph, but the shuffled graph +will still have the same labels. + +If the C property is set to 1 for the original graph, then the +shuffled graph will also have also have that property set to 1. However, the +array form of the C property will not be preserved. + +The C property is not preserved in the shuffled graph in any case. + +The C property is preserved in the shuffled graph, and its value +will be permuted so that the vertex it marked as the hub of the wheel is still +the hub. This means that the hub of the wheel will still be displayed in the +center when the L method is used to display the graph. If that is not +desired, then delete the C property for the result. For example, + + $shuffledGraph = $graph->shuffle; + delete $shuffledGraph->{wheelLayout}; + +=head2 nearestNeighborPath + + ($path, $weight) = $graph->nearestNeighborPath($vertex); + +This is an implementation of the nearest neighbor algorithm. It attempts to find +the shortest path starting at the vertex indexed by C<$vertex> that visits all +vertices in the graph exactly once and returns to the starting vertex. Note that +if such a path is not possible, then the algorithm will go as far as it can, but +the path that is returned may not visit all vertices and may not be a circuit. +This method will always succeed for complete graphs. + +The method returns a list whose first entry is a reference to an array +containing the path found, and whose second entry is the total weight of that +path. + +=head2 kruskalGraph + + ($tree, $treeWeight, $treeWeights) = $graph->kruskalGraph($vertex); + +This is an implementation of Kruskal's algorithm. It attempts to find a minimum +spanning tree or forest for the graph. Note that if the graph is connected, then +the result will be a tree, and otherwise it will be a forest consisting of +minimal spanning trees for each component. + +The method returns a list with three entries. The first entry is a +C object representing the tree or forest found. The +second entry is the total weight of that tree or forest. The last entry is a +reference to an array containing the weights of the edges in the tree or forest +in the order that they are added by the algorithm. + +=head2 hasEulerCircuit + + $hasEulerCircuit = $graph->hasEulerCircuit; + ($hasEulerCircuit, $message) = $graph->hasEulerCircuit; + +In scalar context this method returns 1 if the graph has an Euler circuit, and 0 +otherwise. In list context this method returns 1 if the graph has an Euler +circuit, and if the graph does not have an Euler circuit it returns a list whose +first entry is 0 and whose second entry is a message stating the reason that the +graph does not have an Euler circuit. + +=head2 hasEulerTrail + + $hasEulerTrail = $graph->hasEulerTrail; + ($hasEulerTrail, $message) = $graph->hasEulerTrail; + +In scalar context this method returns 1 if the graph has an Euler trail, and 0 +otherwise. In list context this method returns 1 if the graph has an Euler +trail, and if the graph does not have an Euler trail it returns a list whose +first entry is 0 and whose second entry is a message stating the reason that the +graph does not have an Euler trail. + +=head2 pathIsEulerTrail + + $pathIsEulerTrail = $graph->pathIsEulerTrail(@path); + ($pathIsEulerTrail, $message) = $graph->pathIsEulerTrail(@path); + +The C<@path> argument should contain a list of vertex indices in the graph +representing a path in the graph. In scalar context this method returns 1 if the +path forms an Euler trail in the graph, and 0 otherwise. In list context this +method returns 1 if the path forms an Euler trail in the graph, and if the path +does not form an Euler trail in the graph it returns a list whose first entry is +0 and whose second entry is a message stating the reason that the path does not +form an Euler trail. + +=head2 pathIsEulerCircuit + + $pathIsEulerCircuit = $graph->pathIsEulerCircuit(@path); + ($pathIsEulerCircuit, $message) = $graph->pathIsEulerCircuit(@path); + +The C<@path> argument should contain a list of vertex indices in the graph +representing a path in the graph. In scalar context this method returns 1 if the +path forms an Euler circuit in the graph, and 0 otherwise. In list context this +method returns 1 if the path forms an Euler circuit in the graph, and if the +path does not form an Euler circuit in the graph it returns a list whose first +entry is 0 and whose second entry is a message stating the reason that the path +does not form an Euler circuit. + +=head2 hasCircuit + + $hasCircuit = $graph->hasCircuit; + +This method returns 1 if the graph has a circuit, and 0 otherwise. + +=head2 isTree + + $isTree = $graph->isTree; + ($isTree, $message) = $graph->isTree; + +In scalar context this method returns 1 if the graph is a tree, and 0 otherwise. +In list context this method returns 1 if the graph is a tree, and if the graph +is not a tree it returns a list whose first entry is 0 and whose second entry is +a message stating the reason that the graph is not a tree. + +=head2 isForest + + $isForest = $graph->isForest; + ($isForest, $message) = $graph->isForest; + +In scalar context this method returns 1 if the graph is a forest, and 0 +otherwise. In list context this method returns 1 if the graph is a forest, and +if the graph is not a forest it returns a list whose first entry is 0 and whose +second entry is a message stating the reason that the graph is not a forest. + +=head2 isBipartite + + $isBipartite = $graph->isBipartite; + +This method returns 1 if the graph is bipartite, and 0 otherwise. + +=head2 bipartitePartition + + ($upper, $lower) = $graph->bipartitePartition; + +If the graph is bipartite, then this method returns a list containing two +entries that form a partition of the vertices of the graph into two sets for +which no two vertices in the same set have an edge connecting them. If the graph +is not bipartite, then this method returns undefined. + +=head2 dijkstraPath + + ($distance, @path) = $graph->dijkstraPath($start, $end); + +This is an implementation of Dijkstra's algorithm for finding the shortest path +between nodes in a weighted graph. + +The C<$start> and C<$end> arguments are required and should be indices of two +vertices in the graph representing the start vertex and end vertex for which to +find the shortest path between. + +The return value will be a list whose first entry is the shortest distance from +the start vertex to the end vertex, and whose remaining entries are the indices +of the vertices that form the shortest path from the start vertex to the end +vertex. + +=head2 sortedEdgesPath + + ($sortedEdgesPath, $edgeWeights) = $graph->sortedEdgesPath; + +This is an implementation of the sorted edges algorithm for finding the shortest +Hamiltonian circuit in a graph. That is a path that visits each vertex in the +graph exactly once. The return value will be a list with two entries The first +entry is the resulting sorted edges graph, and the second entry is a reference +to an array containing the weights of the edges in the path in the order that +they are chosen by the algorithm. Note that the returned graph will contain a +Hamiltonian circuit from the original graph if one exists. In any case the graph +will contain all edges chosen in the algorithm. + +=head2 chromaticNumber + + $graph->chromaticNumber; + +This method returns the chromatic number of the graph. That is the minimum +number of colors required to color the vertices such that no two adjacent +vertices share the same color. + +=head2 kColoring + + $graph->kColoring($k); + +The argument C<$k> is required and is the desired number of colors for which to +find a k-coloring. That is a coloring of the graph consisting of at most k +colors such that no two adjacent vertices share the same color. If such a +coloring is possible, then this method will return a list with the number of +entries equal to the number of vertices in the graph, and whose i-th entry will +be an integer from 1 to k where the integers represent a coloring for the vertex +with index i in the graph and such that the list forms a k-coloring. If a +k-coloring is not possible, then this method returns undefined. + +=head1 EdgeSet Context + +A context for edge sets is provided. To activate the context Call + + Context('EdgeSet'); + +Then C and +C objects can be constructed with + + $edge = Compute('{B, E}'); + $edgeSet = Compute('{{A, B}, {A, E}, {A, F}, {E, F}}'); + +or by using the C or C methods described before (those methods +can also be used outside of this context). Note that the vertices must be added +as strings to the context and marked as being vertices with the C +flag before constructing the C or C. For example, by calling + + Context()->strings->add( + map { $_ => { isVertex => 1, caseSensitive => 1 } } 'A' .. 'F' + ); + +If it is prefered that the vertices not be case sensitive, then remove +C<< caseSensitive => 1 >> from the above call. + +If the vertices in the C or C belong to a C +object that has already been constructed, then the L +method can instead be used to add the vertices to the context. + +=cut diff --git a/macros/math/SimpleGraphCatalog.pl b/macros/math/SimpleGraphCatalog.pl new file mode 100644 index 000000000..c90fd2c71 --- /dev/null +++ b/macros/math/SimpleGraphCatalog.pl @@ -0,0 +1,10735 @@ + +=head1 NAME + +SimpleGraphCatalog.pl - A catalog of all simple graphs with fewer than 8 +vertices. + +=head1 DESCRIPTION + +This macro defines the C<$graphCatalog> hash. It has 8 keys (0, 1, ..., 8). the +value of each key is a reference to an array containing all possible graphs (up +to isomorphism) with that number of vertices. + +Graphs are represented as a reference to an array of arrays which are suitable +for passing to L to create a +C object. + +For example, C<$graphCatalog{3}[3]> is + + [ [ 0, 1, 1 ], [ 1, 0, 1 ], [ 1, 1, 0 ] ] + +=cut + +BEGIN { strict->import; } + +sub _GraphCatalog_init { } + +%main::graphCatalog = ( + 1 => [ [ [0] ] ], + 2 => [ [ [ 0, 0 ], [ 0, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ] ], + 3 => [ + [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], + [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 1, 0, 0 ] ], + [ [ 0, 0, 1 ], [ 0, 0, 1 ], [ 1, 1, 0 ] ], + [ [ 0, 1, 1 ], [ 1, 0, 1 ], [ 1, 1, 0 ] ], + ], + 4 => [ + [ [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], + [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 1, 0, 0, 0 ] ], + [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 1, 1, 0, 0 ] ], + [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 1 ], [ 0, 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