class Two_SAT: """2-SATを定義する. """ #※ i:変数 i が Trueの頂点, i+N:変数 i がFalseの頂点 #入力定義 def __init__(self,N): """N変数の2-SATを考える. """ self.N=N self.clause_number=0 self.adjacent_out=[set() for k in range(2*N)] #出近傍(vが始点) self.adjacent_in=[set() for k in range(2*N)] #入近傍(vが終点) #節の追加 def add_clause(self,X,F,Y,G): """(X=F) or (Y=G) という節を加える. X,Y:変数の名前 F,G:真偽値(True or False) """ assert 0<=X<self.N and 0<=Y<self.N F=bool(F);G=bool(G) (A,P)=(X,X+self.N) if F else (X+self.N,X) (B,Q)=(Y,Y+self.N) if G else (Y+self.N,Y) if not self.clause_exist(X,F,Y,G): self.clause_number+=1 #(X,not F)→(Y,G)を追加 self.adjacent_out[P].add(B) self.adjacent_in [B].add(P) #(Y,not G) → (X,F)を追加 self.adjacent_out[Q].add(A) self.adjacent_in [A].add(Q) #節を除く def remove_clause(self,X,F,Y,G): """(X=F) or (Y=G) という節を除く. X,Y:変数の名前 F,G:真偽値(True or False) """ assert 0<=X<self.N and 0<=Y<self.N F=bool(F);G=bool(G) (A,P)=(X,X+self.N) if F else (X+self.N,X) (B,Q)=(Y,Y+self.N) if G else (Y+self.N,Y) if not self.clause_exist(X,F,Y,G): return self.clause_number-=1 #(X,not F)→(Y,G)を除く self.adjacent_out[P].discard(B) self.adjacent_in [B].discard(P) #(Y,not G) → (X,F)を除く self.adjacent_out[Q].discard(A) self.adjacent_in [A].discard(Q) #グラフに節が存在するか否か def clause_exist(self,X,F,Y,G): """(X=F) or (Y=G) という節が存在するか? X,Y:変数の名前 F,G:真偽値(True or False) """ assert 0<=X<self.N and 0<=Y<self.N (A,P)=(X,X+self.N) if F else (X+self.N,X) (B,Q)=(Y,Y+self.N) if G else (Y+self.N,Y) return B in self.adjacent_out[P] #近傍 def neighbohood(self,v): pass #出次数 def out_degree(self,v): pass #入次数 def in_degree(self,v): pass #次数 def degree(self,v): pass #変数の数 def variable_count(self): return self.N #節の数 def clause_count(self): return self.clause_number #充足可能? def Is_Satisfy(self,Mode=0): """充足可能? Mode: 0(Defalt)---充足可能? 1 ---充足可能ならば,その変数の割当を変える.(不可能なときはNone) 2 ---充足不能の原因である変数を全て挙げる. """ from collections import deque N=self.N Group=[0]*(2*N) Order=[] for s in range(2*N): if Group[s]:continue S=deque([s]) Group[s]=-1 while S: u=S.pop() for v in self.adjacent_out[u]: if Group[v]:continue Group[v]=-1 S.append(u);S.append(v) break else: Order.append(u) K=0 for s in Order[::-1]: if Group[s]!=-1:continue S=deque([s]) Group[s]=K while S: u=S.pop() for v in self.adjacent_in[u]: if Group[v]!=-1:continue Group[v]=K S.append(v) K+=1 if Mode==0: for i in range(N): if Group[i]==Group[i+N]: return False return True elif Mode==1: T=[0]*N for i in range(N): if Group[i]>Group[i+N]: T[i]=1 elif Group[i]==Group[i+N]: return None return T elif Mode==2: return [i for i in range(N) if Group[i]==Group[i+N]] #================================================ import sys from itertools import product input=sys.stdin.readline N,M=map(int,input().split()) E=[] for _ in range(M): C=tuple(map(int,input().split())) E.append(C) K=[["*","*"]] G=Two_SAT(N+1) T=[0]*(N+1) Many=[] for i in range(1,N+1): T[i],*L=input()[:-1].split() T[i]=int(T[i]) L=tuple(L) if T[i]==1: L+=(-1,) G.add_clause(i,0,i,0) elif T[i]==2: pass else: Many.append(i) if T[i]==3 or T[i]==5: L+=(-1,) K.append(L) Lazy=[] for a,b,c in E: if a in Many or b in Many: Lazy.append((a,b,c)) else: if c==0: for s in [0,1]: for t in [0,1]: if K[a][s]==K[b][t]: G.add_clause(a,1-s,b,1-t) else: H=[x for x in K[a] if x in K[b]] if len(H)==0: #矛盾 G.add_clause(0,0,0,0) G.add_clause(0,1,0,1) elif len(H)==1: h=H[0] s=K[a].index(h) t=K[b].index(h) G.add_clause(a,s,a,s) G.add_clause(b,t,b,t) else: if K[a][0]==K[b][0]: G.add_clause(a,0,b,1) G.add_clause(a,1,b,0) else: G.add_clause(a,0,b,0) G.add_clause(a,1,b,1) Index={v:i for i,v in enumerate(Many)} Many_COLOR=[[(K[j][2*p],K[j][2*p+1]) for p in range(len(K[j])//2)] for j in Many] for U in product(*Many_COLOR): Flag=True for i in range(len(Many)): K[Many[i]]=U[i] A=[] for v in Many: if K[v][1]==-1 and not G.clause_exist(v,0,v,0): A.append((v,0,v,0)) G.add_clause(v,0,v,0) for a,b,c in Lazy: if a not in Many: a,b=b,a if c==0: for s in [0,1]: for t in [0,1]: if K[a][s]==K[b][t] and not G.clause_exist(a,1-s,b,1-t): A.append((a,1-s,b,1-t)) G.add_clause(a,1-s,b,1-t) else: H=[x for x in K[a] if x in K[b]] if len(H)==0: #矛盾 if not G.clause_exist(0,0,0,0): A.append((0,0,0,0)) G.add_clause(0,0,0,0) if not G.clause_exist(0,1,0,1): A.append((0,1,0,1)) G.add_clause(0,1,0,1) elif len(H)==1: h=H[0] s=K[a].index(h) t=K[b].index(h) if not G.clause_exist(a,s,a,s): A.append((a,s,a,s)) G.add_clause(a,s,a,s) if not G.clause_exist(b,t,b,t): A.append((b,t,b,t)) G.add_clause(b,t,b,t) else: if K[a][0]==K[b][0]: if not G.add_clause(a,0,b,1): A.append((a,0,b,1)) G.add_clause(a,0,b,1) if not G.add_clause(a,1,b,1): A.append((a,1,b,0)) G.add_clause(a,1,b,0) else: if not G.clause_exist(a,0,b,0): A.append((a,0,b,0)) G.add_clause(a,0,b,0) if not G.clause_exist(a,1,b,1): A.append((a,1,b,1)) G.add_clause(a,1,b,1) X=G.Is_Satisfy(1) if X!=None: S="" for i in range(1,N+1): S+=K[i][X[i]] print(S) exit(0) for B in A: G.remove_clause(*B) print("Fault")