(* ::Package:: *)

BeginPackage["SimplicialComplexes`",{"GraphUtilities`","Combinatorica`"}]

VList::usage="hello";
AbsValueBoundaryMap::usage="";
BoundaryMap::usage="";
MonomialBasis;
BoundaryMapList;
Duplicates;
OneSkeleton;
BoundaryMapMatrixRepresentation;
BettiOne;
BettiTwoSurfaces;
EulerCharacteristic;
Genus;
ActualV;
CountSharedEdges;
ValidComplex;
ValidComplexCoarseStart;
SubComplex;
GraphCutPointQ;
BowTieQ;
CapBowTie;
AddPseudoManifoldTwoSimplex;
RandomComplexes;
(*CyclesInOneComplex*)
EdgeInTriangleQ;
EdgeOpenQ;
PlotComp;
HashRules;
CoolPlot;
Manip;
DynamicPlot;
Meets;
SkeletonGraph;
ComplexesEquivalentQ;
ValenceProfile;
VertexValenceEquivalentQ;
AnnotateBins;
BinEquivalentComplexes;
MakeAnnotatedBins;
AppendToAnnotatedBins;
BinAccordingToList;
CorrespondBins;
SortAnnotatedBinsByDecreasingPopulation;
MakeDictionary;
EdgeMeets;
RandomDiagSwap;
RandomSwapEnsemble;
DiagSwapAtKthEdgemeet;
UnnormalizedSwapMarkov;
NormalizeRowsAndTranspose;
MarkovComponent;
PermToPermMatrix;
PermMatrices;
DictionaryToPermMatrix;
OrderedCountMatrix;
OrderedMarkovMatrix;
SizeOfAutomorphismGroups;
DistanceBetweenComplexes;
DescendantQ;
MakeSubdivisionTree;
DrawSubdivisionTree;
TreeTake;
MinDistFromPrimeBySubdivision;
ValidateLutz;
ValidateSingleLutzComplex;
OrderedTriangleEdges;
ComplexToOrientedGraph;
MakeFlowers;
MakeFlowers2;
GlueFlowers;
LongVertexNames;
VerticesInTriangle;
SimplexFromLongVertexNamesAndGroups;
ComplexFromLongVertexNamesAndGroups;
GroupsToComplexViaLongvertexnames;
GroupsToComplexViaHash;
GroupsToComplexViaHashWithStartingSubset;
BuildRandomUnorientedTrivGraph;
BuildManyRandomUnorientedTrivGraphs;
BuildManyRandomUnorientedTrivGraphsLeaveDupes;
DupesQ;
SplitGraphs;
BinEquivalentUnorientedTrivGraphs;
BinEquivalentUnorientedTrivGraphsAccordingToList;
MakeAndTallyRandomTrigraphs;
UnorderedPairsToUnorientedGraph;
ComplexToUnorderedPairs;
ApplyCyclicOrderingToGraph;
ConnectedIncreasingFlowerArrangements;
TreeFindValidGraphOrderings;
TreeFindValidGraphOrderings2;
v;



SubdivisionTreeGenus0;
SubdivisionTreeGenus1;


Begin["`Private`"]


(* ncrule redefines NonCommutativeMultiply to allow commuting of numbers
singletonrule gets rid of the NonCommutativeMultiply head on single arguments
errule writes terms in increasing order
rev rule gets rid of the minus sign, writing the term in decreasing order if necessary
mbrule is for the monomial basis. writes terms in increasing order and drops the minus sign *)
ncrule=NonCommutativeMultiply[a___,x_?NumberQ,b___]:>   x*NonCommutativeMultiply[a,b];
singletonrule=NonCommutativeMultiply[a_]:> a;
errule=v[n_]**v[m_]/;n>m:> -v[m]**v[n];
revrule=-v[n_]**v[m_]:> v[m]**v[n];
mbrule=v[n_]**v[m_]/;n>m:> v[m]**v[n];

VList[nodes_]:=Module[{n=nodes,l},{
	l=Table[v[j],{j,1,n}];
	Return[l,Module]
}];

(*computes the boundary map, but ignoring the signs on terms. used to invalidate
a complex when >2 instances of an edge are present but the regular boundary map
passed inspection because terms cancelled out. *)

AbsValueBoundaryMap[compl_] := Module[{complex=compl,z1,simplex,z2},{
	z1=0;
	Do[{
		simplex=complex[[j]];
		Do[
			z1=z1+(Delete[simplex,k]/.List->NonCommutativeMultiply/.ncrule);
			,{k,1,Dimensions[simplex][[1]]}]
		},{j,1,Dimensions[complex][[1]]}];
	z2=z1/.mbrule;
	Return[z2,Module]
}];

(*the legit boundary map*)

BoundaryMap[compl_] := Module[{complex=compl, z1,simplex,z2},{
	z1=0;
	Do[{
		simplex=complex[[j]];
		Do[
			z1=z1+(ReplacePart[simplex,k->(-1)^(k+1)]/.List->NonCommutativeMultiply/.ncrule/.singletonrule);
			,{k,1,Dimensions[simplex][[1]]}]
		},{j,1,Dimensions[complex][[1]]}];
	z2=z1/.errule;
	Return[z2,Module]
}];

(*writes sorted basis systematically for expansions of complexes*)

MonomialBasis[compl_]:= Module[{complex=compl,z1,simplex,z2,z3},{
	(* make list of all monomials in boundary *)
	z1={};
	Do[{
		simplex=complex[[j]];
		Do[
			z1=Append[z1,Delete[simplex,k]];
			,{k,1,Dimensions[simplex][[1]]}]
		},{j,1,Dimensions[complex][[1]]}];
	(* remove duplicate monomials *)
	z2=Union[z1];

	(* turn list of lists into list of products *)
	Do[{
		z2[[l]]=(z2[[l]]/.List->NonCommutativeMultiply/.mbrule/.singletonrule)
		},{l,1,Dimensions[z2][[1]]}];
	z3=Union[z2];
	Return[z3,Module]
}];

(* writes the boundary map as a list of ordered pairs of edges in the boundary *)
BoundaryMapList[comp_]:=Module[{c=comp,BM,BMlist},{
	BM=BoundaryMap[c];
	BMlist=((BM/.Plus->List)/.revrule)/.NonCommutativeMultiply->List;
	Return[BMlist,Module];
}];

(*checks complexes for duplicates*)
Duplicates[compl_]:=Module[{complex=compl},{
	If[
		Dimensions[Union[Map[Union,complex]]][[1]]!=Dimensions[complex][[1]],
		Return[True, Module],Return[False, Module]
	];
}];

(*computes the 1-skeleton of a complex*)
OneSkeleton[compl_]:=Module[{complex=compl,z1,pairs},{
	z1={};
	Do[{
		pairs=Subsets[complex[[j]],{2}];
		z1=Join[z1,pairs];
		},{j,1,Dimensions[complex][[1]]}];
	Return[z1,Module]
}];

(*computes matrix representation of the boundary map so Mathematica's linalg routines can be used out of the box on them*)
BoundaryMapMatrixRepresentation[compl_]:=Module[{complex=compl,z},{ 
	z=Table[Coefficient[BoundaryMap[{complex[[j]]}],MonomialBasis[complex]],{j,Dimensions[complex][[1]]}];
	Return[z,Module]
}];

BettiOne[compl_]:=Module[{complex=compl,A,B,betti},{
	A=Transpose[BoundaryMapMatrixRepresentation[complex]];
	B=Transpose[BoundaryMapMatrixRepresentation[MonomialBasis[complex]/. Plus->List/.NonCommutativeMultiply-> List]];
	betti=Dimensions[NullSpace[B]][[1]]-MatrixRank[A];
	Return[betti, Module]
}];

BettiTwoSurfaces[compl_]:=Module[{complex=compl,A,betti},{
	A=Transpose[BoundaryMapMatrixRepresentation[complex]];
	betti=Dimensions[NullSpace[A]][[1]];
	Return[betti, Module]
}];

EulerCharacteristic[compl_]:=Module[{complex=compl,z},{
	z=Dimensions[complex][[1]]-Dimensions[MonomialBasis[complex]][[1]]+Dimensions[Union[Flatten[complex]]][[1]];
	Return[z,Module]
}];

Genus[compl_]:= Module[{complex=compl,z},{
	z=Dimensions[complex][[1]]-Dimensions[MonomialBasis[complex]][[1]]+Dimensions[Union[Flatten[complex]]][[1]];
	Return[(2-z)/2,Module]
}];

ActualV[compl_]:=Module[{complex=compl},{
	Return[Dimensions[Union[Flatten[complex]]][[1]],Module]
}];

(*counts shared edges in a complex*)
CountSharedEdges[compl_]:=Module[{complex=compl,count},{
	count=0;
	Do[{
		Do[{
			If[Dimensions[Intersection[complex[[j]],complex[[k]]]][[1]]==2,count=count+1]
			},{k, j+1, Dimensions[complex][[1]]}]
		},{j,1,Dimensions[complex][[1]]}];
	Return[count,Module]
}];

(*determines whether a complex is a valid triangulation of a pseudomanifold*)
ValidComplex[compl_]:=Module[{complex=compl,mb,BM,j,absvalueBM}, {

	If[Duplicates[complex]==True,Return["died in duplicates",Module]];

	mb=MonomialBasis[complex];
	BM=BoundaryMap[complex];
	Do[
		If[Abs[Coefficient[BM,mb[[j]]]]>1,Return["died in regular", Module]]
		,{j,1,Dimensions[mb][[1]]}];
	absvalueBM=AbsValueBoundaryMap[complex];
	Do[{
		If[Coefficient[absvalueBM,mb[[j]]]>2,Return["died in abs value", Module]]
		},{j,1,Dimensions[mb][[1]]}];

	Return[True, Module]
}];

(* same as validcomplex, except tests first whether each triangle has three distinct vertices *)
ValidComplexCoarseStart[compl_]:=Module[{complex=compl,mb,BM,j,absvalueBM}, {

	If[Union[Map[Length,Map[Union,complex]]]!={3},Return["died in stupid",Module]];

	If[Duplicates[complex]==True,Return["died in duplicates",Module]];

	mb=MonomialBasis[complex];
	BM=BoundaryMap[complex];
	Do[
		If[Abs[Coefficient[BM,mb[[j]]]]>1,Return["died in regular", Module]]
		,{j,1,Dimensions[mb][[1]]}];
	absvalueBM=AbsValueBoundaryMap[complex];
	Do[{
		If[Coefficient[absvalueBM,mb[[j]]]>2,Return["died in abs value", Module]]
		},{j,1,Dimensions[mb][[1]]}];

	Return[True, Module]
}];


(* makes list L containing all of the simplices that contain $vertex$*) 
SubComplex[vert_,compl_]:= Module[{vertex=vert,complex=compl,L},{
	L={};
	Do[{
		If[MemberQ[complex[[j]],vertex]==True,
		AppendTo[L,complex[[j]]]]
		},{j,1,Dimensions[complex][[1]]}];
	Return[L, Module]
}];

(* tests whether $vertex$ is a cut point of the complex *) 
(* returns a list {Boolean, L} *)
GraphCutPointQ[vert_,compl_]:=Module[{vertex=vert,complex=compl,L,mm,LL,grph},{
	L=SubComplex[vertex,complex];
	(*at this point, L is the subcomplex of $complex$ containing $vertex$*)
	mm=OneSkeleton[L];
	LL={};
	Do[{
		If[MemberQ[mm[[j]],vertex]==False,AppendTo[LL,mm[[j]]]]
		},{j,1,Dimensions[mm][[1]]}];
	(*LL is the subcomplex of $L$ consisting of simplices NOT containing $vertex$*)
	grph=Table[LL[[j]]/.v[a_]:>a/.({x_,y_}:>(x->y)),{j,1,Dimensions[LL][[1]]}];
	If[ Dimensions[WeakComponents[grph]][[1]]>1,Return[{True,L},Module]];
	Return[{False,L},Module]
}];

(* tests the whole complex for the presence of BowTies - jumps out when it hits the first one *)
BowTieQ[compl_]:=Module[{complex=compl,vert},{
	vert=Union[Flatten[complex]];
	Do[{ If[GraphCutPointQ[vert[[i]],complex][[1]]==True,Return[True,Module]]
		},{i,1,Dimensions[vert][[1]]}];
	Return[False,Module]
}];

(* If the last simplex added resulted in a bow-tie at the new vertex, cap
it by adding a second new simplex that bridges the bow-tie. *)
CapBowTie[vert_,newsimpl_,compl_]:= Module[{vertex=vert,newsimplex=newsimpl,attemptedcomplex=compl,gcp,subcomp,subcompBM,subcompBMlist,candidateedges,group1,group2,edgepairs,pair,thirdedge,candidatecap,temp},
{
	(* set gcp to the Boolean output of GraphCutPointQ (True if $vertex$ is a cutpoint)*)
	(* set subcomp to the set of simplices containing $vertex$ *)
	{gcp, subcomp}=GraphCutPointQ[vertex,attemptedcomplex];

	If[gcp==True,{
	
		subcompBM=BoundaryMap[subcomp];
		subcompBMlist=((subcompBM/.Plus->List)/.revrule)/.NonCommutativeMultiply->List;
		(* from the boundary of subcomp, make list of edges containing $vertex$ *)
		candidateedges=SubComplex[vertex,subcompBMlist];
		(* group1 = candidate edges that are from the new simplex *)
		group1=Intersection[candidateedges,((BoundaryMap[{newsimplex}]/.Plus->List)/.revrule)/.NonCommutativeMultiply->List];
		(* group2 = all other candidate edges *)
		group2= Complement[candidateedges, group1];
		(* list of the 4 possible combinations of pairs of edges (one from group1 and one from group2), randomly ordered *)
		edgepairs=RandomChoice[Permutations[{{group1[[1]],group2[[1]]},{group1[[1]],group2[[2]]},{group1[[2]],group2[[1]]},{group1[[2]],group2[[2]]}}]];

		Do[{

			pair=edgepairs[[j]];
			thirdedge=Flatten[{Complement[pair[[1]],{vertex}],Complement[pair[[2]],{vertex}]}];
			(*Print["thirdedge ",thirdedge,", EdgeOpen ",EdgeOpenQ[thirdedge,attemptedcomplex]];*)
			If[EdgeOpenQ[thirdedge,attemptedcomplex]==True,{
				candidatecap=Append[Reverse[pair[[1]]],Complement[pair[[2]],{vertex}][[1]] ];
				(*Print["edges ",pair,", candidatecap ",candidatecap];*)
				temp=Append[attemptedcomplex,candidatecap];
				If[ValidComplex[temp]==True,{
					(*Print["valid complex"];*)
					Return[{True,temp},Module];
					}];
				}];
			},{j,1,4}];
		(* If the Do loop is exited without finding a possible way to cap the bowtie, then the *)
		(* bowtie is excised and the remaining complex is returned. *)

		Return[{False,Delete[attemptedcomplex,-1]},Module];
		}]; (* end if cut point *)

	(* If the vertex was not a cut point, then just return the attemptedcomplex without alteration *)
	Return[{Null,attemptedcomplex},Module]
}];

(*the atom of the complex-growing process, this function adds a valid simplex to a complex*)
AddPseudoManifoldTwoSimplex[compl_,vertexlist_]:=Module[{complex=compl,VL=vertexlist,BM,BMlist,Vincomplex,Vinboundary,candidateV,EdgeLegal,VertexLegal,RandomEdge,RandomV,NewSimplex,temp,NewComplex,newedgelist,newedgepos,newVlist,newVpos}, {

	BM=BoundaryMap[complex];
	BMlist=((BM/.Plus->List)/.revrule)/.NonCommutativeMultiply->List;

	(* take interior points out of the candidate vertices *)
	Vincomplex=Union[Flatten[complex]];
	Vinboundary=Union[Flatten[BMlist]];
	candidateV=Complement[VL,Complement[Vincomplex,Vinboundary]];

	EdgeLegal=False;
	VertexLegal=False;
	newedgelist=BMlist;
	(*Print["adding new simplex"];*)
	While[(EdgeLegal==False || VertexLegal==False) &&Dimensions[newedgelist][[1]]>0 ,
		newedgepos=RandomInteger[{1,Dimensions[newedgelist][[1]]}];
		RandomEdge=newedgelist[[newedgepos]];
		EdgeLegal=EdgeOpenQ[RandomEdge,complex];
		(*Print["edgelist=",newedgelist];*)
		(*Print["got first edge ", RandomEdge, EdgeLegal];*)

		If[EdgeLegal==True,{
			newVlist=Complement[candidateV,RandomEdge];
			newVpos=RandomInteger[{1,Dimensions[newVlist][[1]]}];
			RandomV=newVlist[[newVpos]];
			VertexLegal=And[EdgeOpenQ[{RandomEdge[[1]],RandomV},complex],EdgeOpenQ[{RandomEdge[[2]],RandomV},complex]];
			(*Print["in if, got first randomV= ",RandomV, VertexLegal];*)
			(*Print["newVlist=",newVlist,"newVpos=",newVpos];*)

			(* There can in fact be bad vertices in the boundary *)
			While[VertexLegal==False && Dimensions[newVlist][[1]]>1,
				newVlist=Delete[newVlist,newVpos];
				newVpos=RandomInteger[{1,Dimensions[newVlist][[1]]}];
				(*Print["newVlist=",newVlist,"newVpos=",newVpos];*)
				RandomV=newVlist[[newVpos]];
				VertexLegal=And[EdgeOpenQ[{RandomEdge[[1]],RandomV},complex],EdgeOpenQ[{RandomEdge[[2]],RandomV},complex]]; 
				(*Print["randomV=",RandomV,VertexLegal];*)
				]; (*end while *)

			(* If VertexLegal is still False after all vertices, try another edge *)
			If[VertexLegal==False,{
				EdgeLegal=False;
				newedgelist=Delete[newedgelist,newedgepos]
				}];

			},{newedgelist=Delete[newedgelist,newedgepos]}]; (*end if*)
		];  (*end while edgelegal*)

	(* If an acceptable edge/vertex combination was found, append new simplex and test validcomplex.
	Otherwise, return Null. *)
	If[EdgeLegal==True,{
		NewSimplex=Append[Reverse[RandomEdge],RandomV];
		temp=Append[complex,NewSimplex];
		If[ValidComplex[temp]==True, NewComplex=temp,NewComplex=complex,NewComplex=complex];
		NewComplex=CapBowTie[RandomV,NewSimplex,NewComplex][[2]];
		}, NewComplex=Null];

	Return[NewComplex,Module]
}];

(*generates complexes by randomly adding simplices to the seed*)
(* turning $seed$ into a string and then back into an expresion is a workaround for importing
symbolic seed. couldn't figure out how to import variable names, such as v[1]. seems that
our Global`v[1] can't mix with the Private`v[n] generated by the package. thought we could
fix that by passing in a vertex list made in Global, but that didn't work...??*)

RandomComplexes[seed_,size_,universe_]:=Module[{complex=ToString[seed],sz=size,universesize=universe,VL,temp,tries}, {
	(*complex={{v[1],v[2],v[3]}};*)
	complex=ToExpression[complex];
	VL=VList[universesize];
	tries=0;
	While[Dimensions[complex][[1]]<sz && ToString[BoundaryMap[complex]]!= "0" && tries<10,
		temp=AddPseudoManifoldTwoSimplex[complex,VL];
		(* Null means that AddPseudoManifoldTwoSimplex couldn't find a legal edge/vertex combination - return complex *)
		If[temp== Null,Return[complex,Module]];
		(* If the loop runs 10 times without the size of the complex changing, quit *)
		If[Dimensions[temp][[1]]==Dimensions[complex][[1]],tries++,tries=0];
		complex=temp;
		];
	Return[complex,Module]
}];

(*counts cycles in a 1-simplex. NOT IMPLEMENTED - might error?*)
CyclesInOneComplex[compl_]:=Module[{complex=compl,g,z}, {
	g=complex/.  revrule /. Plus->List/.NonCommutativeMultiply-> List/.v[a_]:> a;
	z=Dimensions[ExtractCycles[FromUnorderedPairs[g]]][[1]];
	Return[z,Module]
}];

(*determines whether an edge belongs to a triangle*)
EdgeInTriangleQ[edg_,triangl_]:=Module[{edge=edg,triangle=triangl,x,y},{
	Return[MapThread[MemberQ,{{triangle,triangle},edge}]/.{x_,y_}-> And[x,y],Module]
}];

(*determines whether an edge in a 2-complex is available to attach another triangle to. 
returns true if the edge is in the boundary of one other triangle*)
EdgeOpenQ[edg_,compl_]:=Module[{edge=edg,complex=compl,count},{ 
	count=0;
	Do[{
		If[EdgeInTriangleQ[edge,complex[[j]]]==True,count++];
		If[count>1,Return[False,Module]]},{j,Dimensions[complex][[1]]}];
	Return[True,Module]
}];

(*gives a stick plot of the 1-skeleton*)
PlotComp[complex_]:= GraphPlot3D[Union[Map[Union,OneSkeleton[complex]/.{v[a_]->a}]]/.{x_,y_}->  x->y, VertexLabeling->True, ImageSize->400]

(*renames our vertices to conform to Mathematica's conventions in plotting routines. 
vertices are renamed according to their order of appearance in the one-skeleton.*)
HashRules[compl_]:=Module[{complex=compl,cc,AlreadySeen,rules},{
	cc=Flatten[complex];
	AlreadySeen={};
	rules={};
	Do[{
		If[MemberQ[AlreadySeen,cc[[j]]]==False,{AppendTo[AlreadySeen,cc[[j]]],AppendTo[rules,cc[[j]]->Position[AlreadySeen,cc[[j]]][[1,1]] ] } ]
		},{j,1,Dimensions[cc][[1]]}];
	Return[rules,Module]
}];

(*High-level 3d plot of the complexes*)

CoolPlot[compl_]:=Module[{complex=compl,skeleton,hrules,i,vv,L,BM,BMlist,boundary},{
	skeleton=Union[Map[Union,OneSkeleton[complex]]];
	hrules=HashRules[skeleton];
	skeleton=skeleton/.hrules;
	i=complex/.hrules;
	vv=GraphCoordinates3D[skeleton/.{x_,y_}:>   x->y];
	L=Table[
		Text[Framed[hrules[[j,1]][[1]],{Background->RGBColor[1,1,0.8],FrameStyle->RGBColor[0.94,0.85,0.36],FrameMargins->Automatic}],hrules[[j,2]]]
		,{j,1,Dimensions[hrules][[1]]}];
	BM=BoundaryMap[complex];
	BMlist=((BM/.Plus->List)/.revrule)/.NonCommutativeMultiply->List;
	boundary=BMlist/.Append[hrules,0->{{}}];
	Graphics3D[{Opacity[.9],GraphicsComplex[vv,{L,Yellow, Polygon[i],Thickness[0.01],Black,Line[boundary]}]},ImageSize->400]
	(*Graphics3D[{Opacity[.9],GraphicsComplex[vv,{Polygon[i],Thickness[0.01],Line[boundary],L}]},ImageSize->400]*)
}];

Manip[vert_,hashcompl_,vsliders_,labls_]:=Manipulate[Module[{vertices=vert,hashcomplex=hashcompl,labels=labls},{
	Graphics3D[{
		Opacity[.9],GraphicsComplex[vertices,{Polygon[hashcomplex[[Range[n]]]],labels}]},PlotRange->{{-4,4},{-4,4},{-4,4}}, ImageSize->800,Ticks->Automatic,Axes->True,AxesLabel->{x,y,z}]
		}],
	vsliders,{n,1,Dimensions[hashcompl][[1]],1},SaveDefinitions->True
];

DynamicPlot[compl_,verts_]:=Module[{complex=compl,skeleton,hrules,vinits=verts,hashcomplex,symbolichashcomplex,vertices,vsliders,vsl,L,m},{
	skeleton=Union[Map[Union,OneSkeleton[complex]]];
	hrules=HashRules[skeleton];
	skeleton=skeleton/.hrules;
	hashcomplex=complex/.hrules;
	symbolichashcomplex=hashcomplex/.a_?NumberQ:> v[a];
	vertices=Union[Map[Union,Flatten[symbolichashcomplex]]];
	If[vinits==Null,vinits=GraphCoordinates3D[skeleton/.{x_,y_}:>   x->y]];
	vsliders=Table[{{v[n],vinits[[n]]},{-4,-4,-4},{4,4,4}},{n,1,Dimensions[vertices][[1]]}];
	vsl=vsliders/.{a___}:> a;
	SetAttributes[Manip,SequenceHold];
	L=Table[Text[Framed[hrules[[j,2]],{Background->RGBColor[1,1,0.8],FrameStyle->RGBColor[0.94,0.85,0.36],FrameMargins->Automatic}],hrules[[j,2]]],{j,1,Dimensions[hrules][[1]]}];
	m=Manip[vertices,hashcomplex,vsl,L];
	Return[m ,Module]
}];

(* $Meets$ is a function that, when given a complex, outputs a list of the form {n (i,j), i, j}, 
where n (i,j) is the number of vertices in the intersection of the simplices in the ith and jth 
positions in the complex, resp.*)    

Meets[compl_]:=Module[{c=compl,z},{
	z=Table[{Dimensions[Intersection[c[[i]],c[[j]]]][[1]],i,j},{i,1,Dimensions[c][[1]]-1},{j,i+1,Dimensions[c][[1]]}];
	z=Flatten[z,1];
	Return[z,Module]
}];

(*SkeletonGraph[dim,complex] forms the graph of the connections in the complex of level $dim$. 
Define level to mean the size of the intersections. Level 2 means intersection on an edge, level 1 means at a vertex... *)

SkeletonGraph[dimn_,compl_] :=Module[{ d=dimn,c=compl,a,b,z,zz,siz}, {
	z=Cases[Meets[c],{d,_,_}][[All,{2,3}]];
	zz=FromUnorderedPairs[z];
	Return[zz,Module]
}];

(* ComplexesEquivalentQ determines whether two complexes are equivalent. 
It checks the adjacency graphs of levels up to siz's limit for equivalence qua graphs.
siz=1 means connectedness of simplices at vertices; siz=2 means connectedness of simplices at edges*)

ComplexesEquivalentQ[complex1_,complex2_]:=Module[{c1=complex1, c2=complex2,siz},{
	(*If[Dimensions[c1][[1]]!= Dimensions[c2][[1]],Return[False,Module]];*)
	Do[If[IsomorphicQ[SkeletonGraph[siz,c1],SkeletonGraph[siz,c2]]==False,Return[False,Module]],{siz,1,2}];
	Return[True,Module]
}];

(* calculates the number of occurence of each vertex in a complex. returns a sorted list. *)

ValenceProfile[complex_]:=Module[{c1=complex,valence,verts},{
	verts=Union[Flatten[c1]];
	valence=Table[Count[Flatten[c1],verts[[i]]],{i,1,Dimensions[verts][[1]]}];
	valence=Sort[valence];
	Return[valence,Module]
}];

(* determines whether two complexes have identical vertex valence profiles. *)

VertexValenceEquivalentQ[complex1_,complex2_]:=Module[{c1=complex1, c2=complex2,answer},{
	answer=ValenceProfile[c1]==ValenceProfile[c2];
	Return[answer,Module]
}];

(*The function annotates bins in the second file with the complexes from the first. The form out will be {{complex, complex's bincount},...}.*)

AnnotateBins[complxs_,bin_]:=Module[{complexes=complxs,bins=bin,annotatedbins},{
	annotatedbins=Table[{complexes[[bins[[k,1]]]],Dimensions[bins[[k]]][[1]]},{k,1,Dimensions[bins][[1]]}];
	Return[annotatedbins,Module]
}];

(*This bins complexes that are equivalent according to the ComplexesEquivalentQ criterion.
Output: a list of bins, where each bin is a list of positions of complexes in L.
Dynamically displays {iterator j (which runs through the list of complexes L), number of bins found so far} during evaluation.*)

BinEquivalentComplexes[lista_] :=Module[{L=lista,bins,done,j},{
	bins={{1}};
	done=0;
	Monitor[
		Do[{
			Do[{
				If[ComplexesEquivalentQ[L[[bins[[i,1]]]],L[[j]]]==True,
					{AppendTo[bins[[i]],j],done=j,Break[]}];
				},{i,1,Dimensions[bins][[1]]}];
			If[done!=j,bins=Join[bins,{{j}}]];
			},{j,2,Dimensions[L][[1]]}];
		,{j,Dimensions[bins]}];
	Return[bins,Module]
}];

MakeAnnotatedBins[listofcomplexes_]:=Module[{data=listofcomplexes,bins,annotatedbins},{
	bins=BinEquivalentComplexes[data];
	annotatedbins=AnnotateBins[data,bins];
	Return[annotatedbins,Module]
}];

AppendToAnnotatedBins[listofcomplexes_,annotatedbins_]:=Module[{L=listofcomplexes,bins=annotatedbins,done},{
	done=0;
	Monitor[
	Do[{
		Do[
			If[ComplexesEquivalentQ[bins[[i,1]],L[[j]]]==True,
				{bins=ReplacePart[bins,{i,2}->bins[[i,2]]++],done=j,Break[]}];
			,{i,1,Dimensions[bins][[1]]}];
		If[done!=j,bins=Join[bins,{L[[j]],1}]];
		},{j,1,Dimensions[L][[1]]}];
		,j];
	Return[bins,Module]
}];

(* bins complexes in $newlist$ into classes defined by the exemplars in $standardlist$. 
Returns a list of counts in the order defined by $standardlist$. *)
BinAccordingToList[standardlist_,newlist_]:=Module[{stdlist=standardlist,L=newlist,bins},{
	(* initialize list of counts to zeroes *)
	bins=PadLeft[{},Dimensions[stdlist][[1]]];
	Monitor[
		Do[{
			Do[
				If[ComplexesEquivalentQ[stdlist[[bin]],L[[i]]]==True,{bins=ReplacePart[bins,bin->bins[[bin]]+1];Break[]}]
				,{bin,1,Dimensions[stdlist][[1]]}];
			},{i,1,Dimensions[L][[1]]}];
		,i];
	Return[bins,Module]
}];


CorrespondBins[lista_,listb_] :=Module[{Lb=lista,La=listb}, {
	Monitor[
	Do[{ 
		Do[{
			If[ComplexesEquivalentQ[Lb[[j,1]],La[[i,1]]]==True,{
				Lb[[j]]=Append[Lb[[j]],La[[i]][[2]]],Break[]}];
			},{j,1,Dimensions[Lb][[1]]}];
		},{i,1,Dimensions[La][[1]]}];
	,i];
	Return[Lb,Module]
}];

SortAnnotatedBinsByDecreasingPopulation[annotatedbins_]:=Module[{ab=annotatedbins},
	{ab=Reverse[SortBy[ab,Last]];
	Return[ab,Module]
}];

(* Takes an annotated bin of triangulations and sorts it according to the 
Markov swap probability of the triangulation, going from most frequent to least frequent.
Returns a list of lists of the form {exemplar complex, original index, markov frequency}.
The markov frequencies are not normalized. *)
MakeDictionary[annotatedbins_]:=Module[{ab=annotatedbins,n,mm,vv,tab,vvv},{
	n=Dimensions[ab][[1]];
	mm=NormalizeRowsAndTranspose[UnnormalizedSwapMarkov[ab]];
	vv=Eigenvectors[mm];
	(*This code of Meli's picks out the eigenvector of the Markon transition matrix that has all positive components.*)
	vvv=Select[vv,Map[Abs,#]==#&][[1]];
	tab=Table[{ab[[k,1]],k,vvv[[k]]},{k,n}];
	tab=Reverse[SortBy[tab,Last]];
	Return[tab,Module]
}];


(********** FUNCTIONS FOR SWAPS (BISTELLAR MOVES AS IN LUTZ) ************)

(* $EdgeMeets$ is a function that, when given a complex, outputs a list {i, j} of pairs of positions of simplices in the complex that meet on an edge.*)

EdgeMeets[compl_]:=Module[{c=compl,z}, {
	z={};
	Do[
		If[Dimensions[Intersection[c[[i]],c[[j]]]][[1]]==2,AppendTo[z,{i,j}]],{i,1,Dimensions[c][[1]]-1}
		,{j,i+1,Dimensions[c][[1]]}];
	Return[z,Module]
}];

RandomDiagSwap[compl_]:=Module[{c=compl ,edgemeets,z,commonverts,rest,did,donow,newsplx1,newsplx2,temp}, {
	edgemeets=EdgeMeets[c];
	z=RandomInteger[{1,Dimensions[edgemeets][[1]]}];
	commonverts=Intersection[c[[edgemeets[[z]][[1]]]],c[[edgemeets[[z]][[2]]]]];
	rest=Complement[Flatten[Union[c[[edgemeets[[z]][[1]]]],c[[edgemeets[[z]][[2]]]]]],commonverts];

	If[MemberQ[c[[edgemeets[[z]][[1]]]],rest[[1]]],
		{newsplx1=ReplacePart[c[[edgemeets[[z]][[1]]]],Position[c[[edgemeets[[z]][[1]]]],commonverts[[1]]][[1,1]]->rest[[2]]],did=2}
		,{newsplx1=ReplacePart[c[[edgemeets[[z]][[1]]]],Position[c[[edgemeets[[z]][[1]]]],commonverts[[1]]][[1,1]]->rest[[1]]],did=1}];

	donow=3-did;

	newsplx2=ReplacePart[c[[edgemeets[[z]][[2]]]],Position[c[[edgemeets[[z]][[2]]]],commonverts[[2]]][[1,1]]->rest[[donow]]];
	c=Delete[c,{{edgemeets[[z]][[1]]} ,{edgemeets[[z]][[2]]}}];
	c=Join[c,{newsplx1,newsplx2}];
	Return[c,Module]
}];

RandomSwapEnsemble[compl_,size_]:=Module[{c=compl,s=size,z,i,j,temp}, {
	z={c};
	i=1;j=0;
	While[i<s&&j<4 s,
		temp=RandomDiagSwap[z[[-1]]];
		If[ValidComplex[temp]== True,AppendTo[z,temp]];
		i=Dimensions[z][[1]];
		j++;
		];
	Return[z,Module]
}];

(*The following function can be run in tandem with EdgeMeets which will provide valid k.*)

DiagSwapAtKthEdgemeet[compl_,k_]:=Module[{c=compl ,z=k,edgemeets,commonverts,rest,did,donow,newsplx1,newsplx2,temp},
	edgemeets=EdgeMeets[c];
	commonverts=Intersection[c[[edgemeets[[z]][[1]]]],c[[edgemeets[[z]][[2]]]]];
	rest=Complement[Flatten[Union[c[[edgemeets[[z]][[1]]]],c[[edgemeets[[z]][[2]]]]]],commonverts];

	If[MemberQ[c[[edgemeets[[z]][[1]]]],rest[[1]]],
		{newsplx1=ReplacePart[c[[edgemeets[[z]][[1]]]],Position[c[[edgemeets[[z]][[1]]]],commonverts[[1]]][[1,1]]->rest[[2]]],did=2}
		,{newsplx1=ReplacePart[c[[edgemeets[[z]][[1]]]],Position[c[[edgemeets[[z]][[1]]]],commonverts[[1]]][[1,1]]->rest[[1]]],did=1}];

	donow=3-did;

	newsplx2=ReplacePart[c[[edgemeets[[z]][[2]]]],Position[c[[edgemeets[[z]][[2]]]],commonverts[[2]]][[1,1]]->rest[[donow]]];
	c=Delete[c,{{edgemeets[[z]][[1]]} ,{edgemeets[[z]][[2]]}}];
	c=Join[c,{newsplx1,newsplx2}];
	Return[c,Module]
];

(*Here's the (transpose of the unnormalized--should normalize ROWS) Markov matrix. The position (i,j) counts the number of ways that exemplar i goes to exemplar j. *)

UnnormalizedSwapMarkov[annotatedbins_]:=Module[{ab=annotatedbins,mark,ems,sk},
	mark=DiagonalMatrix[Table[0,{Range[Dimensions[ab][[1]]]}]];
	Monitor[
	Do[{
		ems=EdgeMeets[ab[[j,1]]];

		(*The loop in j goes through the exemplary complexes in the annotated bin and makes a list of the edgemeets. *)

		Do[{
			sk=DiagSwapAtKthEdgemeet[ab[[j,1]],k];

			(*The valid swaps are then compared with the full list of the binning's exemplars to determine which bin we've swapped to. 
			If sk is an invalid swap, then the next edgemeet is swapped on and the process is repeated. 
			If the swapped complex matches an exemplar, then the matrix $mark$ is incremented by one in the position corresponding to 
			the initial and terminal positions of the exemplar swapped.*)

			If[ValidComplex[sk]==True,
				Do[{
					If[ComplexesEquivalentQ[sk,ab[[l,1]]]==True,{mark=ReplacePart[mark,{j,l}->1+mark[[j,l]]],Break[]}];
					},{l,1,Dimensions[ab][[1]]}]
				]
			},{k,1,Dimensions[ems][[1]]}]
		},{j,1,Dimensions[ab][[1]]}];
		,j];
	Return[mark,Module]
];

NormalizeRowsAndTranspose[m_]:=Module[{mark=m,n,k,sums,divisors},{
	n=Dimensions[mark][[1]];
	sums=Map[Total,mark];
	divisors[x_]:=Piecewise[{{x,x!=0},{1,x==0}}];
	Do[
		mark=ReplacePart[mark,k->(1/divisors[sums[[k]]]) mark[[k]]]
		,{k,1,n}];
	mark=Transpose[mark];
	Return[mark,Module]
}];


MarkovComponent[ab_,j_,l_]:=Module[{abin=ab,jj=j,ll=l,m,comp},
	m=0;
	Do[{
		comp=DiagSwapAtKthEdgemeet[abin[[jj,1]],k];
		If[ValidComplex[comp]==True,
		If[ComplexesEquivalentQ[comp,abin[[ll,1]]],m=m+1]]
		},{k,1,Dimensions[EdgeMeets[abin[[ll,1]]]][[1]]}];
	Return[m,Module]
];

PermToPermMatrix[perm_]:=Module[{p=perm,n,L,pairs,assignments,permmat},{
	n=Dimensions[p][[1]];
	L=PadLeft[{{}},{n,n}];
	pairs=Table[{p[[k]],k},{k,Dimensions[p][[1]]}];
	assignments=Table[pairs[[k]]->1,{k,n}];
	permmat=ReplacePart[L,assignments];
	Return[permmat,Module]
}];

PermMatrices[dim_]:=Module[{n=dim,p,L,pairs,assignments,permats},
	p=Permutations[Range[n]];
	L=PadLeft[{{}},{n,n}];
	pairs=Table[Table[{p[[k,j]], p[[1,j]]},{j,n}],{k,Dimensions[p][[1]]}];
	assignments=Table[Table[pairs[[j,k]]-> 1,{k,n}],{j,Dimensions[p][[1]]}];
	permats=Table[ReplacePart[L,assignments[[k]] ],{k,1,Dimensions[p][[1]]}];
	Return[permats,Module]
];

DictionaryToPermMatrix[dictionarybin_]:=Module[{db=dictionarybin,permmat},{
	permmat=PermToPermMatrix[db[[All,2]]];
	Return[permmat,Module]
}];

(*The following functions take annotated bins and compute the Counts and Markov transition
matrices in bases ordered according to the frequency of occurrence of the bins.
The frequencies are chosen to be decreasing.*)

OrderedCountMatrix[annotatedbins_]:=Module[{ab=annotatedbins,n,countmatrix,mm,vv,vvv,tab,permmat,orderedcount},{
	n=Dimensions[ab][[1]];
	countmatrix=UnnormalizedSwapMarkov[ab];
	mm=NormalizeRowsAndTranspose[countmatrix];
	vv=Eigenvectors[mm];
	(*This code of Meli's picks out the eigenvector of the Markov transition matrix that has all positive components.*)
	vvv=Select[vv,Map[Abs,#]==#&][[1]];
	tab=Table[{ab[[k]][[1]],k,vvv[[k]]},{k,n}];
	tab=Reverse[SortBy[tab,Last]];
	permmat=DictionaryToPermMatrix[tab];
	orderedcount=Transpose[permmat].countmatrix.permmat;
	Return[orderedcount,Module]
}];

OrderedMarkovMatrix[annotatedbins_]:=Module[{ab=annotatedbins,n,mm,vv,vvv,tab,permmat,orderedmarkov},{
	n=Dimensions[ab][[1]];
	mm=NormalizeRowsAndTranspose[UnnormalizedSwapMarkov[ab]];
	vv=Eigenvectors[mm];
	(*This code of Meli's picks out the eigenvector of the Markov transition matrix that has all positive components.*)
	vvv=Select[vv,Map[Abs,#]==#&][[1]];
	tab=Table[{ab[[k]][[1]],k,vvv[[k]]},{k,n}];
	tab=Reverse[SortBy[tab,Last]];
	permmat=DictionaryToPermMatrix[tab];
	orderedmarkov=Transpose[permmat].mm.permmat;
	Return[orderedmarkov,Module]
}];

SizeOfAutomorphismGroups[height_,complx_]:=Module[{h=height,c=complx,siz},
	siz=Dimensions[Automorphisms[SkeletonGraph[h,c]]][[1]];
	Return[siz,Module]
]

(*The following code is not feasible as it stands. It checks too many renamings and so is impossibly slow. 
A pair of 6-simplex complexes' distance (at either height, (1=vertex meets, 2=edgemeets)) will take about 100 seconds. *)

DistanceBetweenComplexes[cplx1_,cplx2_,height_]:=Module[{c1=cplx1,c2=cplx2,h=height,,m1,m2,d,permutedm1,wrongs,distances,dist},
	m1=ToAdjacencyMatrix[SkeletonGraph[h,c1]];
	m2=ToAdjacencyMatrix[SkeletonGraph[h,c2]];
	d=Dimensions[m1][[1]];
	permutedm1=Table[Inverse[PermMatrices[d][[j]]].m1.PermMatrices[d][[j]],{j,1,d!}];
	wrongs=Table[permutedm1[[k]][[i,j]]!=m2[[i,j]],{k,d!},{i,d},{j,d}];
	distances=Map[Count[Flatten[#],True] &,wrongs];
	dist=Min[distances];
	Return[dist,Module]
];

(************* THE FOLLOWING FUNCTIONS ARE FOR DETERMINING GENEALOGY **************)

(* determines whether candidate complex is a descendant of parent complex
by barycentric subdivision. takes two complexes and returns a boolean. *)
DescendantQ[parnt_, cand_] := Module[{parent=parnt,candidate=cand,vertices,valencies,possibleverts,subcomp,BMlist,mergedsimplex,coarsecomplex,foundmatch},{
	vertices=Union[Flatten[candidate]];
	If[Dimensions[Union[Flatten[parent]]][[1]]!=Dimensions[vertices][[1]]-1,Return[False,Module]];
	valencies=Table[Count[Flatten[candidate],vertices[[i]]],{i,1,Dimensions[vertices][[1]]}];
	possibleverts=vertices[[Flatten[Position[valencies,3]]]];
	(* check each vertex with valence 3. *)
	foundmatch=False;
	Do[{
		subcomp=SubComplex[possibleverts[[i]],candidate];
		BMlist=((BoundaryMap[subcomp]/.Plus->List)/.revrule)/.NonCommutativeMultiply->List;
		mergedsimplex=Join[BMlist[[1]],Complement[Union[Flatten[BMlist]],BMlist[[1]]]];
		coarsecomplex=Append[Complement[candidate,subcomp],mergedsimplex];
		(*Print[coarsecomplex];*)
		If[ValidComplex[coarsecomplex]==True,{
			If[ComplexesEquivalentQ[parent,coarsecomplex]==True,Return[True,Module]]}
			];
		},{i,1,Dimensions[possibleverts][[1]]}];
	(* if the Do loop finishes without returning from the module, no coarsenings were found
	that made a complex equivalent to the parent, and False is returned. *)
	Return[False, Module]
}];

(* Builds a geneaology tree of triangulations related by barycentric subdivision. *)
(* takes list {t[g,n],t[g,n+1],t[g,n+2],...} where g is genus, n is number of vertices,
and t[g,n] is a list of exemplars of the unique triangulation classes. *)
(* returns a list {adjmat, vertexcoords, vertexlabels, pairs}. adjmat is the adjacency matrix of the tree graph.
vertexcoords is a list of manual vertex coordinates {x,y}. vertexlabels is a list of strings that label nodes. *)
MakeSubdivisionTree[triangulations_]:=Module[{t=triangulations,i,j,generation,vertexcoords,vertexlabels,minlevel,maxlevel,numverts,adjmat,pairs,
level,parents,children,miny},{
	vertexcoords={};
	vertexlabels={};
	minlevel=ActualV[t[[1,1]]];
	maxlevel=ActualV[t[[-1,1]]];
	numverts=Total[Map[Dimensions,t][[All,1]]];
	adjmat=PadLeft[{{}},{numverts,numverts}];
	(*pairs={};*)

	(* initialize tree by making first generation *)
	Do[{
		AppendTo[vertexlabels,"v"<>ToString[minlevel]<>"."<>ToString[i]];
		AppendTo[vertexcoords,{i,0}];
		},{i,1,Dimensions[t[[1]]][[1]]}];
	
	(* build the rest of the tree *)
	Do[{
		level=generation+minlevel-1;
		parents=t[[generation-1]];
		children=t[[generation]];
		miny=Min[vertexcoords[[All,2]]];
		Do[{
			AppendTo[vertexlabels,"v"<>ToString[level]<>"."<>ToString[i]];
			AppendTo[vertexcoords,{i,miny-1}];
			Do[{
				If[DescendantQ[parents[[j]],children[[i]]]==True,{
					adjmat=ReplacePart[adjmat,
					{Flatten[Position[vertexlabels,"v"<>ToString[level-1]<>"."<>ToString[j]]][[1]],
					Flatten[Position[vertexlabels,"v"<>ToString[level]<>"."<>ToString[i]]][[1]]}->1];
					(*AppendTo[pairs,"v"<>ToString[level-1]<>"."<>ToString[j]->"v"<>ToString[level]<>"."<>ToString[i]]*)
					}]
				},{j,1,Dimensions[parents][[1]]}]
			},{i,1,Dimensions[children][[1]]}];
		},{generation,2,Dimensions[t][[1]]}];
	Return[{adjmat,vertexcoords,vertexlabels},Module]
}];

(* Draws a graphical representation of the geneaology tree computed by MakeSubdivisionTree. 
takes a tree data structure exactly as returned from MakeSubdivisionTree. Returns two graph objects,
one drawn with my coordinates (this one places the primes wrong), and one drawn with mathca's treeplot routine. 
neither one is really satisfactory, especially for trees with many generations. *)
DrawSubdivisionTree[treedata_]:=Module[{tree=treedata,adjmat,vertexcoords,vertexlabels,fancycoords,g,g2},{
	{adjmat,vertexcoords,vertexlabels}=tree;
	fancycoords=Table[i->(vertexcoords/.{a_,b_}:>{Automatic,b})[[i]],{i,1,Dimensions[vertexcoords][[1]]}];
	g=GraphPlot[adjmat,
		VertexCoordinateRules->fancycoords,VertexRenderingFunction->({Text[Framed[vertexlabels[[#2]],{Background->RGBColor[1,1,0.8],FrameStyle->RGBColor[0.94,0.85,0.36],FrameMargins->Automatic}],#1]}&),
		ImageSize->100 Abs[Min[vertexcoords[[All,2]]]]
		];
	g2=TreePlot[adjmat,
		VertexRenderingFunction->({Text[Framed[vertexlabels[[#2]],{Background->RGBColor[1,1,0.8],FrameStyle->RGBColor[0.94,0.85,0.36],FrameMargins->Automatic}],#1]}&),
		ImageSize->100 Abs[Min[vertexcoords[[All,2]]]]
		];
	Return[{g,g2},Module]
}];

(* takes a subset of treedata corresponding to the generation range. 
treedata is of the form output by MakeSubdivisionTree.
generation range is of the form {n,n+1,n+2,...}. *)
TreeTake[treedata_,generationrange_]:=Module[{tree=treedata,gens=generationrange,adjmat,coords,labels,newlabels,temp,positions,newcoords,newadjmat},{
	(* check that the generation range is a set of increasing sequential integers *)
	If[Table[i,{i,IntegerPart[First[gens]],First[gens]+Dimensions[gens][[1]]-1}]!=gens,{Message[TreeTake::nonsequential];Return[$Failed,Module]}];
	{adjmat,coords,labels}=tree;
	newlabels={};
	(* find the labels which correspond to gens *)
	Do[{
		temp=Flatten[Map[StringCases[#,RegularExpression["v"<>ToString[gen]<>".+"]]&,labels]];
		newlabels=Join[newlabels,temp];
		If[Dimensions[temp][[1]]==0,Message[TreeTake::generationrange,gens]];
		},{gen,First[gens],Last[gens]}];
	(* get the positions of the new labels in the big label list *)
	positions=Flatten[Map[Position[labels,#]&,newlabels]];
	newcoords=coords[[positions]];
	(* make a new adjacency matrix which is the submatrix of adjmat corresponding to gens *)
	newadjmat=Table[adjmat[[i,j]],{i,First[positions],Last[positions]},{j,First[positions],Last[positions]}];
	Return[{newadjmat,newcoords,newlabels},Module]
}];
TreeTake::nonsequential="Sequential increasing integers needed for generation range. Example: {4,5,6}";
TreeTake::generationrange="Generation range `1` is out of range of the tree data.";

(* uses Mathematica's GraphDistanceMatrix function to calculate the minimum distance between every pair of
nodes in the tree. returns the distances that apply to $generation$, in the order they appear in the matrix (dictionary order). *)
MinDistFromPrimeBySubdivision[treedata_,generation_]:=Module[{tree=treedata,gen=generation,gdmT,labels,primes,mylabels,mindisttoprime,positions},{
	gdmT=Transpose[GraphDistanceMatrix[tree[[1]]]];
	labels=tree[[3]];
	primes={};
	Do[
		If[Total[gdmT[[j]]/.Infinity->0]==N[0],{AppendTo[primes,j]}]
		,{j,1,Dimensions[gdmT][[1]]}];
	mylabels=Flatten[Map[StringCases[#,RegularExpression["v"<>ToString[gen]<>".+"]]&,labels]];
	If[Dimensions[mylabels][[1]]==0,{Message[MinDistFromPrimeBySubdivision::genmissing,gen],Return[{},Module]}];
	positions=Flatten[Map[Position[labels,#]&,mylabels]];
	mindisttoprime=IntegerPart[Table[Min[gdmT[[positions[[j]],primes]]],{j,1,Dimensions[positions][[1]]}]];
	Return[mindisttoprime,Module]
}];
MinDistFromPrimeBySubdivision::genmissing="Generation `1` is not present in the tree."


(* takes a string downloaded as text from lutz's website, and imported as follows:
lutzstring=ToString[Import["manifolds_lex_d2_n9_o1_g1.txt"]];
This string represents a list of complexes written in a funny order. 
Returns a list of complexes with vertices reordered to make them valid. *)
ValidateLutz[lutzstring_]:=Module[{ll=lutzstring,t1,l,temp,t2,validlist,comp,valid,k,edges,badsimplex},{
	t1="{"<>StringReplace[lutzstring,{RegularExpression["m.+="]->"","["->"{","]"->"}","\n\n"->",","\n"->"",WhitespaceCharacter->""}]<>"}";
	t2=ToExpression[t1]/.a_Integer:>v[a];
	validlist={};
	Do[{
		comp=t2[[cc]];
		valid={comp[[1]]};
		k=2;
		While[Dimensions[valid][[1]]< Dimensions[comp][[1]],{
			edges=OneSkeleton[valid];
			If[Dimensions[Intersection[edges,Subsets[comp[[k]],{2}]]][[1]]>0,
				{
				If[ValidComplex[temp=Append[valid,comp[[k]]]]==True,
				valid=temp,
				AppendTo[valid,{comp[[k]][[1]],comp[[k]][[3]],comp[[k]][[2]]}],
				AppendTo[valid,{comp[[k]][[1]],comp[[k]][[3]],comp[[k]][[2]]}]];
				k++
				},
				{
				badsimplex=comp[[k]];
				comp=Delete[comp,k];
				comp=Append[comp,badsimplex];
				}];
			}];
		AppendTo[validlist,valid];
		},{cc,1,Dimensions[t2][[1]]}];
	Return[validlist,Module]
}];

(* takes a complex written {{v[1],v[2],v[3]},...} but with the vertices unordered. returns 
a valid complex. *)
ValidateSingleLutzComplex[lutzcomplex_]:=Module[{l,temp,comp=lutzcomplex,validlist,valid,k,edges,badsimplex},{
	validlist={};
		valid={comp[[1]]};
		k=2;
		While[Dimensions[valid][[1]]< Dimensions[comp][[1]],{
			edges=OneSkeleton[valid];
			If[Dimensions[Intersection[edges,Subsets[comp[[k]],{2}]]][[1]]>0,
				{
				If[ValidComplex[temp=Append[valid,comp[[k]]]]==True,
				valid=temp,
				AppendTo[valid,{comp[[k]][[1]],comp[[k]][[3]],comp[[k]][[2]]}],
				AppendTo[valid,{comp[[k]][[1]],comp[[k]][[3]],comp[[k]][[2]]}]];
				k++
				},
				{
				badsimplex=comp[[k]];
				comp=Delete[comp,k];
				comp=Append[comp,badsimplex];
				}];
			}];
		AppendTo[validlist,valid];
	Return[validlist[[1]],Module];
}];


(****************************************************************************)
(********* The following functions are for trivalent graphs *****************)
(****************************************************************************)


(* gives the edges of the triangle in order. takes a triangle {v[1],v[2],v[3]} and 
returns {{v[1],v[2]},{v[2],v[3]},{v[3],v[1]}} *)
OrderedTriangleEdges[triang_]:=Module[{t=triang},{
	Return[{{t[[1]],t[[2]]},{t[[2]],t[[3]]},{t[[3]],t[[1]]}},Module]
}];

(* write edgemeets in correct cyclical order. vertex names are lost. triangles are renamed with numbers in their 
order of appearance in the complex. returns groups data structure {{1,{2,3,4}}, ...}. only valid for oriented complexes! *)
ComplexToOrientedGraph[compl_]:=Module[{c=compl,grps,petals,i,edges},{
	grps={};
	Do[{
		edges=OrderedTriangleEdges[c[[k]]];
		petals={};
		Do[{
			If[Dimensions[Intersection[edges[[i]],c[[j]]]][[1]]==2,AppendTo[petals,j]],
			},{i,1,3},{j,1,Dimensions[c][[1]]}];
		petals=DeleteCases[petals,k];
		AppendTo[grps,{k,petals}];
	},{k,1,Dimensions[c][[1]]}];
	Return[grps,Module];
}];

(* make flowers for each triangle, each flower having totally separate vertex numbers
 v[middleguy][trianglename][trianglevertex (1,2,3)]. generates list idents containing identifications
used to make the flowers *)
MakeFlowers[groupscompl_]:=Module[{gr=groupscompl,idents,rawverts,tflower,middleguyvlist,petalvlist,middleguyedges,a1,a2,},{
	idents={};
	rawverts={};
	Do[{
		tflower=k;
		(* make vertices for the four triangles in the flower *)
		middleguyvlist=Table[v[tflower[[1]]][tflower[[1]]][i],{i,1,3}];
		petalvlist=Table[v[tflower[[1]]][n][i],{n,tflower[[2]]},{i,1,3}];
		(* collect all the vertices in rawverts list *)
		rawverts=Join[rawverts,Append[petalvlist,middleguyvlist]];
		(* all the edges on the middle guy *)
		middleguyedges=OrderedTriangleEdges[middleguyvlist];
		Do[{
			(* go through all edges on middle guy *)
			a1=middleguyedges[[petal]];
			(* use arbitrary edge on petal (target) *)
			a2={petalvlist[[petal,2]],petalvlist[[petal,1]]};
			idents=Join[idents,{{a1[[1]],a2[[1]]},{a1[[2]],a2[[2]]}}];
			},{petal,1,3}]
		},{k,gr}];
	Return[idents,Module];
}];

(* the same as MakeFlowers, except also adds each new vertex identified with itself. 
this function is used by GroupsToComplexViaHashWithStartingSubset.*)
MakeFlowers2[groupscompl_]:=Module[{gr=groupscompl,idents,rawverts,tflower,middleguyvlist,petalvlist,middleguyedges,a1,a2},{
	idents={};
	rawverts={};
	Do[{
		tflower=k;
		(* make vertices for the four triangles in the flower *)
		middleguyvlist=Table[v[tflower[[1]]][tflower[[1]]][i],{i,1,3}];
		petalvlist=Table[v[tflower[[1]]][n][i],{n,tflower[[2]]},{i,1,3}];
		(* collect all the vertices in rawverts list *)
		rawverts=Join[rawverts,Append[petalvlist,middleguyvlist]];
		(* all the edges on the middle guy *)
		middleguyedges=OrderedTriangleEdges[middleguyvlist];
		Do[{
			(* go through all edges on middle guy *)
			a1=middleguyedges[[petal]];
			(* use arbitrary edge on petal (target) *)
			a2={petalvlist[[petal,2]],petalvlist[[petal,1]]};
			idents=Join[idents,{{a1[[1]],a2[[1]]},{a1[[2]],a2[[2]]}}];
			},{petal,1,3}]
		},{k,gr}];
	idents=Join[idents,Table[{i,i},{i,Flatten[rawverts]}]];
	Return[idents,Module];
}];



(* glue flowers. generates list globalfloweridents containing identifications used to glue flowers *)
GlueFlowers[groupscompl_]:=Module[{gr=groupscompl,globalfloweridents,idents,t2,t1idents,t2idents,t1tip,t2tip,floweridents,floweridents2,t1},{
	globalfloweridents={};
	idents=MakeFlowers[gr];
	Do[{
		Do[{
			t2=flower[[2,petal]];
			If[Cases[gr[[All,1]],t2]=={},Continue[]];
			t1=flower[[1]];
			(* vertices glueing t1 and t2 in flower where t1 is center *)
			t1idents=Cases[idents,{v[t1][t1][_],v[t1][t2][_]}];
			(* vertices glueing t1 and t2 in flower where t2 is center *)
			t2idents=Cases[idents,{v[t2][t2][_],v[t2][t1][_]}];
			(* the third vertices in t1 *)
			t1tip=Flatten[{Complement[Table[v[t1][t1][i],{i,1,3}],Flatten[t1idents]],Complement[Table[v[t2][t1][i],{i,1,3}],Flatten[t2idents]]}];
			(* the third vertices in t2 *)
			t2tip=Flatten[{Complement[Table[v[t2][t2][i],{i,1,3}],Flatten[t2idents]],Complement[Table[v[t1][t2][i],{i,1,3}],Flatten[t1idents]]}];
			(* try glueing whole thing in one orientation *)
			(*floweridents={t1tip,t2tip,{t1idents[[1,2]],t2idents[[1,2]]},{t1idents[[2,2]],t2idents[[2,2]]}};*)
			(* other orientation. I think this one's just always right.*)
			floweridents2={t1tip,t2tip,{t1idents[[1,2]],t2idents[[2,2]]},{t1idents[[2,2]],t2idents[[1,2]]}};
			(* add to list of identifications. use orientation 2 *)
			globalfloweridents=Join[globalfloweridents,floweridents2];
		},{petal,1,3}]
	},{flower,gr}];
	idents=Join[idents,globalfloweridents];
	idents=Union[Table[Reverse[Sort[idents[[j]]]],{j,1,Dimensions[idents][[1]]}]];
	Return[idents,Module];
}];

(* name vertices according to the triangles they touch. takes the list of identifications, as rules.
ex: {v[1][1][1]->v[1][2][2],...} *)
LongVertexNames[identsgraph_]:=Module[{ids=identsgraph,lvns},{
	lvns=Map[Union,WeakComponents[ids]/.v[_][m_][_]:> m];
	Return[lvns,Module];
}];

VerticesInTriangle[longvertexnames_,triang_]:=Module[{
	lvns=longvertexnames,trian=triang,vv},{vv=Cases[longvertexnames,{___,trian,___}];
Return[vv,Module]}];

(* writes the name of the three vertices of a triangle, given the groups gr={{1,{2,3,4}},...} 
and the long vertex names, which are the list of vertices named according to which triangles
they touch. *)
SimplexFromLongVertexNamesAndGroups[position_,longvertnames_,groups_]:=Module[{
	lvns=longvertnames, pos=position,grps=groups,vv,vvv,vec,rulz,orderpos,z,triangle,p,n},{
	vv=VerticesInTriangle[lvns,pos];
	vvv=Map[DeleteCases[#,pos]&,vv];
	vec=grps[[pos,2]]/.n_Integer->p[n];
	rulz=Table[vec[[k]]->k,{k,1,3}];
	orderpos=(Flatten[Table[Intersection[vvv[[k]],vvv[[1+Mod[k+1,3]]]],{k,1,3}]]/.n_Integer->p[n])/.rulz;
	z=SignaturePermutation[orderpos];
	(* hash vertex names *)
	triangle=Flatten[Map[Position[lvns,# ]&,vv]] /.n_Integer->v[n];
	(* reverse list if an odd permutation of flower representation *)
	If[z==-1,triangle=Reverse[triangle]];
	Return[triangle,Module]
	}];

(* writes a complex in our usual form C={{v[1],v[2],v[3]},...}, given the groups and the long vertex names *)
ComplexFromLongVertexNamesAndGroups[longverts_,groups_]:=Module[{lvns=longverts,gr=groups,newcomp,triangle},{
	newcomp={};
	Do[{
		triangle=SimplexFromLongVertexNamesAndGroups[k,lvns, gr];
		AppendTo[newcomp,triangle];
	},{k,1,Dimensions[gr][[1]]}];
	Return[newcomp,Module];
}];

(* builds the complex {{v[1],v[2],v[3]},...} from groups {{1,{2,,4}},...}.
Must be a boundaryless complex! *)
GroupsToComplexViaLongvertexnames[groups_]:=Module[{gr=groups,allidentsgraph,lvn,newcomp},{
	allidentsgraph=Apply[Rule,GlueFlowers[gr],1];
	lvn=LongVertexNames[allidentsgraph];
	newcomp=ComplexFromLongVertexNamesAndGroups[lvn,gr];
	Return[newcomp,Module];
}];

(* takes the ordered triv graph in groups form gr={{1,{2,3,4}},...} *)
GroupsToComplexViaHash[groups_]:=Module[{gr=groups,triangles,tflower,allidents,vertexequivalenceclasses,HashV,newc,n,deletepositions},{

	(* write the complex in long form, as {{v[1][1][1],v[1][1][2],v[1][1][3]},...}, with
	all vertex names still unidentified.  *)
	triangles={};
	Do[{
		AppendTo[triangles,Join[{Table[v[flower[[1]]][flower[[1]]][i],{i,1,3}]},Table[v[flower[[1]]][n][i],{n,flower[[2]]},{i,1,3}]]][[1]];
		},{flower,gr}];
	triangles=Flatten[triangles,1];

	(* get the list of vertex identifications. also add each vertex identified with
	itself, so they appear in equivalenceclasses even if complex is partial. *)
	allidents=Join[GlueFlowers[gr],Table[{i,i},{i,Flatten[triangles]}]];
	(* bin identified vertices *)
	vertexequivalenceclasses=Map[Union,WeakComponents[Apply[Rule,allidents,1]]];
	(* hash the raw vertices according to the postion of their equivalence class *)
	HashV[vertex_]:=Flatten[Position[Map[MemberQ[#,vertex]&,vertexequivalenceclasses],True]]/.{n_}->v[n];
	newc=Map[HashV,triangles,{2}];

	(* take all the duplicates out. makes sense to do this, because the trivalent graph doesn't allow for duplicates anyway *)
	For[i=1,i<=Dimensions[newc][[1]]-1,i++,{
		deletepositions={};
		For[j=i+1,j<=Dimensions[newc][[1]],j++,{
			If[Sort[newc[[i]]]==Sort[newc[[j]]],AppendTo[deletepositions,{j}]];
		}];
		newc=Delete[newc,deletepositions];
		}];
	Return[newc,Module];
}];

(* takes the ordered triv graph in groups form gr={{1,{2,3,4}},...}. Same as 
GroupsToComplexViaHash, but starts with a pre-built set of identifications. *)
GroupsToComplexViaHashWithStartingSubset[partialgr_,subsetidentlist_]:=Module[{gr=partialgr,idents=subsetidentlist,newf,tflower,allidents,vertexequivalenceclasses,HashV,newc,n,deletepositions,middleguyedges,petalvlist,middleguyverts,t1,t2,t1idents,t2idents,t1tip,t2tip,floweridents2,triangles,i,j},{

	newf=Last[partialgr];
	idents=subsetidentlist;

	idents=Join[idents,MakeFlowers2[{newf}]];

	(* make idents between new flower and everything previous *)
	Do[{
		t1=newf[[1]];
		t2=newf[[2,petal]];
		If[Cases[gr[[All,1]],t2]=={},Continue[]];
		
		(* identifications glueing t1 and t2 in flower where t1 is center *)
		t1idents=Cases[idents,{v[t1][t1][_],v[t1][t2][_]}];
	
		(* identification glueing t1 and t2 in flower where t2 is center *)
		t2idents=Cases[idents,{v[t2][t2][_],v[t2][t1][_]}];

		(* the third vertices in t1 *)
		t1tip=Flatten[{Complement[Table[v[t1][t1][i],{i,1,3}],Flatten[t1idents]],Complement[Table[v[t2][t1][i],{i,1,3}],Flatten[t2idents]]}];
		(* the third vertices in t2 *)
		t2tip=Flatten[{Complement[Table[v[t2][t2][i],{i,1,3}],Flatten[t2idents]],Complement[Table[v[t1][t2][i],{i,1,3}],Flatten[t1idents]]}];

		(*glue whole thing in orientation 2. I think this one's just always right.*)
		floweridents2={t1tip,t2tip,{t1idents[[1,2]],t2idents[[2,2]]},{t1idents[[2,2]],t2idents[[1,2]]}};
		(* add to list of identifications. use orientation 2 *)
		idents=Join[idents,floweridents2];
	},{petal,1,3}];

	(* bin identified vertices *)
	vertexequivalenceclasses=Map[Union,WeakComponents[Apply[Rule,idents,1]]];
	(* hash the raw vertices according to the postion of their equivalence class *)
	HashV[vertex_]:=Flatten[Position[Map[MemberQ[#,vertex]&,vertexequivalenceclasses],True]]/.{n_}->v[n];
	
	triangles={};
	Do[{
		AppendTo[triangles,Join[{Table[v[flower[[1]]][flower[[1]]][i],{i,1,3}]},Table[v[flower[[1]]][n][i],{n,flower[[2]]},{i,1,3}]]][[1]];
		},{flower,gr}];
	triangles=Flatten[triangles,1];

	newc=Map[HashV,triangles,{2}];

	(* take all the duplicates out. makes sense to do this, because the trivalent graph doesn't allow for duplicates anyway *)
	For[i=1,i<=Dimensions[newc][[1]]-1,i++,{
		deletepositions={};
		For[j=i+1,j<=Dimensions[newc][[1]],j++,{
			If[Sort[newc[[i]]]==Sort[newc[[j]]],AppendTo[deletepositions,{j}]];
		}];
		newc=Delete[newc,deletepositions];
		}];
	Return[{newc,idents},Module];
}];


(* pick pairs of balls out of hat. numtriangles must be even. output is in the form of unordered pairs {{1,2},...}  *)
BuildRandomUnorientedTrivGraph[numtriangles_]:=Module[{numt=numtriangles,balllist,distinct,indices,sparray,out},{
	If[EvenQ[numt]==False,{Message[BuildRandomUnorientedTrivGraph::noteven],Return[False,Module]}];
	balllist=Flatten[Table[{i,i,i},{i,1,numt}]];
	sparray={};
	While[Dimensions[balllist][[1]]>0,{
		(*Print["balllistbefore=",balllist];*)
		distinct=False;
		While[distinct==False,{
			indices=Table[RandomInteger[{1,Dimensions[balllist][[1]]}],{k,1,2}];
			If[Dimensions[Union[balllist[[indices]]]][[1]]<2,
				{distinct=False,
				(*Print["pick=",balllist[[indices]],"distinct=false"]*),If[Dimensions[Union[balllist]][[1]]==1,{Message[BuildRandomUnorientedTrivGraph::bottleneck],Return[False,Module]}]
				},
				distinct=True];
		}];
		out=balllist[[indices]];
		balllist=Delete[balllist,Partition[indices,1]];
		(*Print["out=",out];
		Print["balllistafter=",balllist];*)
		sparray=Append[sparray,out];
	}];
	Return[sparray,Module];
}];
BuildRandomUnorientedTrivGraph::noteven="The input number of triangles must be even.";
BuildRandomUnorientedTrivGraph::bottleneck="The last two balls in the hat were the same.";

(* make a list of numgraphs unoriented random trivalent graphs, each with numtiangl nodes. does not use graphs with multiple
connections between the same two triangles, i.e. with Dimensions[sparray] != Dimensions[Union[Map[Sort,sparray]]] *)
BuildManyRandomUnorientedTrivGraphs[numgraphs_,numtriangl_]:=Module[{numg=numgraphs,numt=numtriangl,trivgraphlist,n,sparray,k},{
	trivgraphlist=Table[Null,{numg}];
	k=1;
	While[k<numg+1,{
		sparray=BuildRandomUnorientedTrivGraph[numt];
		If[
			Dimensions[sparray][[1]]==Dimensions[Union[Map[Sort,sparray]]][[1]],
			{trivgraphlist[[k]]=sparray; k++}]; 
		}]; 
	Return[trivgraphlist,Module];
}];

(* the same as BuildManyRandomUnorientedTrivGraphs, except does not weed out duplicates (graphs with 
two edges connecting the same two nodes) *)
BuildManyRandomUnorientedTrivGraphsLeaveDupes[numgraphs_,numtriangl_]:=Module[{numg=numgraphs,numt=numtriangl,trivgraphlist,n,sparray,k},{
	trivgraphlist=Table[Null,{numg}];
	k=1;
	While[k<=numg,{
		sparray=BuildRandomUnorientedTrivGraph[numt];
		If[sparray==False,,{trivgraphlist[[k]]=sparray; k++},{trivgraphlist[[k]]=sparray; k++}];
		}]; 
	Return[trivgraphlist,Module];
}];

(* tests a triraph graph, written as unordered pairs, for duplicates (two edges connecting the same two nodes)  *)
DupesQ[unordpairs_]:=Module[{graph=unordpairs},
	If[Length[graph]!=Length[Union[Map[Sort,graph]]],
			Return[True,Module], 
			Return[False,Module]]
];

(* Divides a trigraph, written as unordered pairs, into n pieces, each connected-increasing. *)
SplitGraphs[unorderedpairs_,numberofdivisions_]:=Module[{graph=unorderedpairs,n=numberofdivisions,grps,p,pieces,piecesgrps},{
	grps=UnorderedPairsToUnorientedGraph[graph];
	graph=Apply[Rule,graph,1];
	pieces=MinCut[graph,n];
	piecesgrps=Table[Flatten[Map[Cases[grps,{#,_}]&,p],1],{p,pieces}];
	Return[Map[ConnectedIncreasingFlowerArrangements,piecesgrps],Module];
}];

MakeAndTallyRandomTrigraphs[num_, numtriangl_]:=Module[{n=num, numt=numtriangl,goodcomps,unordpairs,gg,grp},{
	goodcomps={};
	Do[
		{
		unordpairs=BuildManyRandomUnorientedTrivGraphsLeaveDupes[1,numt][[1]];
		If[DupesQ[unordpairs]==False,{
		grp=UnorderedPairsToUnorientedGraph[unordpairs];
		gg=TreeFindValidGraphOrderings2[grp];
		goodcomps=Join[Map[ApplyCyclicOrderingToGraph[grp,#]&, gg],goodcomps];
	}];
	If[IntegerQ[i/10], {i,numt,Length[goodcomps], goodcomps}>>"tally"<>ToString[numt]<>".csv"];
	},{i,n}];
	Return[{n,goodcomps},Module];
}];

(* Bins a list of unordered trivalent graphs into isomorphism groups.
The trivalent graph should be given as a list of unordered pairs {{1,2},...} *)
BinEquivalentUnorientedTrivGraphs[lista_] :=Module[{L=lista,bins,done,j,g,binexemplar},{
	bins={{1}};
	done=0;
	Monitor[
		Do[{
			Do[{
	g=FromUnorderedPairs[L[[j]]];
	binexemplar=FromUnorderedPairs[L[[bins[[i,1]]]]];
				If[IsomorphicQ[binexemplar,g]==True,
					{AppendTo[bins[[i]],j],done=j,Break[]}];
				},{i,1,Dimensions[bins][[1]]}];
			If[done!=j,bins=Join[bins,{{j}}]];
			},{j,2,Dimensions[L][[1]]}];
		,{j,Dimensions[bins]}];
	Return[bins,Module]
}];


(* Bin unordered trivalent graphs against a dictionary list.
Takes two lists of combinatorica graph objects. *)
BinEquivalentUnorientedTrivGraphsAccordingToList[standardlist_,newlist_]:=Module[{stdlist=standardlist,L=newlist,bins},{
	(* initialize list of counts to zeroes *)
	bins=PadLeft[{},Dimensions[stdlist][[1]]];
	Monitor[
		Do[{
			Do[
				If[IsomorphicQ[stdlist[[bin]],L[[i]]]==True,{bins=ReplacePart[bins,bin->bins[[bin]]+1];Break[]}]
				,{bin,1,Dimensions[stdlist][[1]]}];
			},{i,1,Dimensions[L][[1]]}];
		,i];
	Return[bins,Module]
}];

(* makes group data structure gr={{1,{2,3,4}},...} from list of 
unordered pairs (such as generated by BuildRandomUnorientedTrivGraph) *)
UnorderedPairsToUnorientedGraph[trivgraph_]:=Module[{pairs=trivgraph,triangles,gr,connectedpairs,petals},{
	triangles=Union[Flatten[pairs]];
	gr={};
	Do[{
		connectedpairs=Cases[pairs,{___,triangles[[i]],___}];
		petals=Sort[Complement[Flatten[connectedpairs],{triangles[[i]]}]];
		AppendTo[gr,{triangles[[i]],petals}]
	},{i,triangles}];
	Return[gr,Module];
}]

ComplexToUnorderedPairs[compl_] :=Module[{ c=compl,a,b,z,zz,siz}, {
	z=Cases[Meets[c],{2,_,_}][[All,{2,3}]];
	Return[z,Module];
}];

(* takes an unoriented graph in groups format gr={{1,{2,3,4}},...} and an ordering list {1,-1,..}. 
the ordering list should be the same length as the number of triangles, and have -1 in positions that 
are to be reversed. *)
ApplyCyclicOrderingToGraph[unorderedgraph_,orderinglist_]:=Module[{unord=unorderedgraph,orders=orderinglist,ordpetals,ordgraph},{
	ordpetals=MapAt[Reverse,unord[[All,2]],Position[orders,-1]];
	(*ordgraph=ReplacePart[unord,{i_,2}->ordpetals[[i]]];*)
	unord[[All,2]]=ordpetals;
	Return[unord,Module];
}]

(* Rearranges the order of flowers in the groups structure so that each flower is meli-connected to
something ahead of it in the list. This is necessary for the tree, which builds partial complexes. *)
ConnectedIncreasingFlowerArrangements[groups_]:=Module[{grps=groups,sofar,deletedsofar,petalssofar,nextindex},{
	sofar={grps[[1]]};
	deletedsofar=grps[[Range[2,Dimensions[grps][[1]]]]];
	While[deletedsofar!={},{
		petalssofar=Union[Flatten[sofar[[All,2]]]];
		If[(g=Flatten[Map[Position[deletedsofar[[All,1]],#]&,petalssofar]])=={}, Return[$Failed,Module]];
		nextindex=First[g];
		AppendTo[sofar,deletedsofar[[nextindex]]];
		deletedsofar=Delete[deletedsofar,nextindex];
	}];
	Return[sofar,Module];
}];

(* tree algorithm for finding orderings that give valid complexes. takes graph in flower structure *)
TreeFindValidGraphOrderings[groups_]:=Module[{gr=groups,test,goodones,connectedgr,partial,maybe1,c1,maybe2,c2,ordercomp},{
	test=0;
	goodones={{1}};
	connectedgr=ConnectedIncreasingFlowerArrangements[gr];
	Timing[
		Do[{
			partial=connectedgr[[Range[k]]];
			Monitor[
					Do[{
						(* tack on a +1 and see if it's valid *)
						maybe1=Append[orderlist,1];
						c1=ApplyCyclicOrderingToGraph[partial,maybe1];
						If[ValidComplex[GroupsToComplexViaHash[c1]]==True,AppendTo[goodones,maybe1]];
						(* tack on a -1 and see if it's valid *)
						maybe2=Append[orderlist,-1];
						c2=ApplyCyclicOrderingToGraph[partial,maybe2];
						If[ValidComplex[GroupsToComplexViaHash[c2]]==True,AppendTo[goodones,maybe2]];
						(* in any case, kill the original orderlist *)
						(*Print[k];
						Print[orderlist];
						Print[ValidComplex[GroupsToComplexViaHash[c1]],ValidComplex[GroupsToComplexViaHash[c2]]];*)
						goodones=DeleteCases[goodones,orderlist];
						(*Print["------------"];*)
						test++;
						},{orderlist,goodones}];
					,{k,test}];
				},{k,2,Dimensions[gr][[1]]}];
			];
	(*Print[goodones];*)
	(*ordercomp=ApplyCyclicOrderingToGraph[connectedgr,goodones[[1]]]*);
	If[goodones=={},Message[TreeFindValidGraphOrderings::novalidcomps]];
	Return[goodones,Module];
}];
TreeFindValidGraphOrderings::novalidcomps="The graph does not represent any valid complexes.";

(* new version of tree that saves partial ident lists as it goes. about 10% faster than first tree. *)
TreeFindValidGraphOrderings2[groups_]:=Module[{gr=groups,test,goodones,connectedgr,partialgr,maybe1,
	c1,maybe2,c2,ordercomp,idents,partialidents,orderlist,tempgoodones,tempidents,maybecomp,
	maybeidents,badpositions,n,k,goodonessize},{

	test=0;
	gr=ConnectedIncreasingFlowerArrangements[gr];
	goodones={{1}};
	partialidents={MakeFlowers2[{gr[[1]]}]};
	(*goodonessize={1};*)
	Do[{  (* loop that goes through the flowers *)
		partialgr=gr[[Range[k]]];
		tempgoodones=Table[Null,{2 Dimensions[goodones][[1]]}];
		tempidents=Table[Null,{2 Dimensions[goodones][[1]]}];
		n=1;
		Monitor[
			Do[{  (* loop that goes through the goodones *)
		
				orderlist=goodones[[pos]];
				idents=partialidents[[pos]];
				(* tack on a +1 and see if it's valid *)
				maybe1=Append[orderlist,1];
				c1=ApplyCyclicOrderingToGraph[partialgr,maybe1];
				{maybecomp,maybeidents}=GroupsToComplexViaHashWithStartingSubset[c1,idents];
				If[ValidComplex[maybecomp]==True,
					{tempgoodones[[n]]=maybe1,
					tempidents[[n]]=maybeidents,
					n++}];
	
				(* tack on a -1 and see if it's valid *)
				maybe2=Append[orderlist,-1];
				c2=ApplyCyclicOrderingToGraph[partialgr,maybe2];
				{maybecomp,maybeidents}=GroupsToComplexViaHashWithStartingSubset[c2,idents];
				If[ValidComplex[maybecomp]==True,
					{tempgoodones[[n]]=maybe2,
					tempidents[[n]]=maybeidents,
					n++}];

				test++;
			},{pos,Dimensions[goodones][[1]]}];
		,{k,Length[goodones]}];
		goodones=DeleteCases[tempgoodones,Null];
		partialidents=DeleteCases[tempidents,Null];
		(*AppendTo[goodonessize,Length[goodones]];*)
	},{k,2,Dimensions[gr][[1]]}];
		
	If[goodones=={},Message[TreeFindValidGraphOrderings::novalidcomps]];
	Return[goodones,Module];
}];
TreeFindValidGraphOrderings::novalidcomps="The graph does not represent any valid complexes.";


(***** Pre-set variables available for export **********)
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EndPackage[ ]