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Complex.py
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Complex.py
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from sympy import *
import random
class Complex:
def __init__(self):
self.master = []
self.D = []
self.H = []
self.deadsimps = []
self.tor = [[[0,2],[3,5]],[[0,1],[4,5]],[[1,2],[3,4]]]
def complexify(self,s,p = 0): #generates the complex of simplex s when p is left as 0. Otherwise a subcomplex of faces with vertices < p never removed, is generated
if len(s)==1:
return [s]
else:
l=[]
for i in range(p,len(s)):
n=len(s)-i-1
ss=s.copy()
ss.remove(s[n])
l.append(ss)
l=l+self.complexify(ss,i)
return l + [s]
def isLess(self,s,t): #lexicographical ordering boolean function for simplices. if s < t, returns true
n=min(len(s),len(t))
for i in range(0,n):
if s[i]<t[i]:
return True
elif s[i]>t[i]:
return False
if len(s)<len(t):
return True
return False
def MakeLex(self,l): #quicksort implimented with lexicographic ordering that also preferences size of simplices
if len(l)<=1:
return l
p=random.randrange(0,len(l))
piv=l[p]
a=[]
b=[]
for i in range(0,len(l)):
y = [len(l[i])]+l[i]
z =[len(piv)]+piv
if self.isLess(y,z) and i!=p:
a.append(l[i])
if self.isLess(z,y) and i!=p:
b.append(l[i])
l=self.MakeLex(a)+[piv]+self.MakeLex(b)
return l
def choose(self,n,k): #standard combination function
if 0<=k<=n:
num=1
den=1
for t in range(1,min(k,n-k)+1):
num*=n
den*=t
n-=1
return num//den
else:
return 0
def j(self,gen,n): #returns the number of n dimensional simplices that will be generated from the generating set gen
sum=0
for i in range(0,len(gen)):
sum=sum+gen[i][0]*self.choose(gen[i][1]+1,n+1)
return sum
def par(self,gen): #returns partition generated from generating set gen
P=[]
N=[]
for i in range(0,self.j(gen,0)):
N.append(i)
for t in range(0,len(gen)):
n=gen[t]
for k in range(0,n[0]):
P.append(N[0:n[1]+1])
N=N[n[1]+1:len(N)]
return P
def M(self,gen): #generates the full master list of simplices from gen
P=self.par(gen)
l=[]
for i in range(0,len(P)):
l=l+self.complexify(P[i],0)
return self.MakeLex(l)
def h(self,s): #converts a simplice to string and returns its hash value
str1 = ','.join(str(e) for e in s)
return hash(str1)
def location(self,s,x = 'r'): #uses hash table H to find quickly find the relative ('r') or absolute ordering of a simplex
k = self.h(s)%len(self.H)
if type(self.H[k][0]) == list:
for i in range(0,len(self.H[k])):
if self.H[k][i][2] == s:
if x == 'r':
return self.H[k][i][0]
else:
return self.H[k][i][1]
else:
if x == 'r':
return self.H[k][0]
else:
return self.H[k][1]
def setGenData(self,s,r = [],boundaryrel =None,steps = 0):
self.master.clear()
self.D.clear()
self.deadsimps.clear()
self.master = self.M(s)
self.D=[zeros(1,self.j(s,0))]
d = s[len(s)-1][1]
#initializes boundary matrix D
self.D = self.D + [zeros(self.j(s,i),self.j(s,i+1)) for i in range(0,d)]
self.H.clear()
self.H = [0 for i in range(0,2*len(self.master))]
c = 0
for i in range(0,len(self.master)): #fills H with simplex positions
n = self.h(self.master[i])%len(self.H)
flag = 0
if len(self.master[i]) > len(self.master[i-1]):
c = 0
if self.H[n] == 0:
flag = 1
self.H[n] = [c,i,self.master[i],self.master[i]]
else:
if type(self.H[n][0]) == list:
flag = 2
self.H[n].append([c,i,self.master[i],self.master[i]])
else:
flag = 3
self.H[n] = [self.H[n],[c,i,self.master[i],self.master[i]]]
#if master[i] == [18,19,20,21,22,23,24,25]:
# print("H(n)",H[n])
# input(flag)
c = c+1
if not (boundaryrel == None):
a = self.getBoundary(boundaryrel[0])
b = self.getBoundary(boundaryrel[1])
if type(steps) == list:
x = self.coupleSimps(a,b,steps[0])
else:
x = self.coupleSimps(a,b,steps)
stepind = 1
for simp in boundaryrel[2:len(boundaryrel)]:
bound = self.getBoundary(simp)
if type(steps) == list:
x = self.coupleSimps(x,bound,steps[stepind])
stepind+=1
else:
x = self.coupleSimps(x,bound,steps)
r = r + x
self.initializeRelations(r)
self.makeBoundary(s)
self.pruneSimps()
def boundarydecomp(self,s,p = 0): #takes simplex s, and recursively calls a rightbound. Will update all boundaries of the COMPLEX of s (s and all faces)
if type(s) == int:
s = [s]
dim = len(s) - 1
m = self.D[dim]
for x in range(0,p):
if len(s) <= 1:
break
ss = s.copy()
ss.remove(s[dim-x])
pos = self.location(self.lowestOrderRelation(ss),'r')
if self.lowestOrderRelation(ss) != ss:
self.deadsimps.append(ss)
if (dim-x)%2 ==0:
m[pos,self.location(s,'r')] = m[pos,self.location(s,'r')] + 1
else:
m[pos,self.location(s,'r')] = m[pos,self.location(s,'r')] - 1
for y in range(p,len(s)):
if len(s) <= 1:
break
ss = s.copy()
ss.remove(s[dim-y])
pos = self.location(self.lowestOrderRelation(ss),'r')
if self.lowestOrderRelation(ss) != ss:
self.deadsimps.append(ss)
if (dim-y)%2 == 0:
m[pos,self.location(s,'r')] = m[pos,self.location(s,'r')] + 1
else:
m[pos,self.location(s,'r')] = m[pos,self.location(s,'r')] - 1
self.boundarydecomp(self.lowestOrderRelation(ss),y)
def makeBoundary(self,s):
P = self.par(s)
for i in range(0,len(P)):
self.boundarydecomp(P[i],0)
def initializeRelations(self,Rel):#gets the full set of relations, and updates H with relation information for each simplex
global H,deadsimps
Rel = self.getAllRelations(Rel)
for a in Rel:
for j in range(1,len(a)):
self.deadsimps+=self.complexify(a[j])
for i in range(0,len(Rel)):
s = self.MakeLex(Rel[i])
lor = s[0]
for u in range(0,len(s)):
k = self.h(s[u])%len(self.H)
#print("Simp",s[u],"hash",k,"table element",H[k],"\n")
if type(self.H[k][0]) == list:
for x in range(0,len(self.H[k])):
if self.H[k][x][2] == s[u]:
lor2 = self.lowestOrderRelation(s[u])
if self.isLess(lor2,lor):
self.H[k][x][3] = lor2
l = self.h(s[0])%len(self.H)
b = self.H[l]
if type(b[0]) == list:
for y in range(0,len(b)):
if b[y][2] == s[0]:
b[y][3] = lor2
else:
b[3] = lor2
lor = lor2
else:
self.H[k][x][3] = lor
break
else:
lor2 = self.lowestOrderRelation(s[u])
if self.isLess(lor2,lor):
self.H[k][3] = lor2
l = self.h(s[0])%len(self.H)
b = self.H[l]
if type(b[0]) == list:
for y in range(0,len(b)):
if b[y][2] == s[0]:
b[y][3] = lor2
else:
b[3] = lor2
lor = lor2
else:
self.H[k][3] = lor
def lowestOrderRelation(self,s): #finds the lowest order relation of simplex s, using H
k = self.h(s)%len(self.H)
if type(self.H[k][0]) == list:
for i in range(0,len(self.H[k])):
if self.H[k][i][2] == s:
if self.H[k][i][3] == s:
return s
else:
return self.lowestOrderRelation(self.H[k][i][3])
else:
if self.H[k][3] == s:
return s
else:
return self.lowestOrderRelation(self.H[k][3])
def getAllRelations(self,Rel):#generates all lower order relations from a relation set Rel
fullRel = []
for i in range(0,len(Rel)):
lowerRels = []
for u in range(0,len(Rel[i])):
lowerRels.append(self.complexify(Rel[i][u],0))
for k in range(0,len(lowerRels[0])):
newRel = []
for x in range(0,len(lowerRels)):
newRel.append(lowerRels[x][k])
fullRel.append(newRel)
return Rel+fullRel
def pruneSimps(self):
global D
d = self.MakeLex(self.deadsimps)
currentdim = 0
i = 0
for x in d:
if len(x)-1>currentdim:
currentdim = len(x)-1
i = 0
p = self.location(x)
self.D[currentdim].col_del(p - i)
if currentdim < len(self.D) - 1:
self.D[currentdim + 1].row_del(p - i)
i = i + 1
def coupleSimps(self,s1,s2,parity = 0): #takes two lists of simplices and parity as either an integer or list, and will combine simps based on steps by parity
start = s1[0] #This function is useful for easily specifying relations over a set of ordered simplices (see makeHole fuction as an example)
a = 0
if type(parity) == int:
b = parity
else:
b = parity[0]
rels = []
for i in range(0,len(s1)):
if type(s1[a][0]) == int:
r = [s1[a],s2[b%len(s2)]]
rels.append(r)
elif type(s1[a][0]) == list:
rels.append(s1[a])
rels[len(rels)-1].append(s2[b%len(s2)])
a += 1
if parity == 0:
b+= 1
elif type(parity) == list:
b+= parity[(i+1)%len(parity)]
else:
b += parity
return rels
def getBoundary(self,s): #returns a list representing the boundary of simplex s
b = []
for a in s:
ss = s.copy()
ss.remove(a)
b.append(ss)
b.reverse()
return b
def makeHole(self,k): #takes dimension k, and set generation data to construct a complex with a k-dimensional hole
S = [[2,k]]
P = self.par(S)
R = self.coupleSimps(self.getBoundary(P[0]),self.getBoundary(P[1]))
self.setGenData(S,R)
def ker(self,k,mat): #takes matrix boundary list, and dimension k argument
return Matrix([a.T for a in mat[k].nullspace()]).T
def Betti(self,k,m = None): #returns codimension of m[k+1] in the the kernel of m[k]
if m == None:
m = self.D
if k>=len(m):
return 0
mm = self.ker(k,m)
if k +1>= len(m):
imd = 0
else:
imd = len(m[k+1].rref()[1])
return mm.shape[1] - imd
def BettiAll(self):
return [self.Betti(k,self.D) for k in range(0,len(self.D))]