import sys readline=sys.stdin.readline import heapq class Graph: def __init__(self,V,edges=False,graph=False,directed=False,weighted=False,inf=float("inf")): self.V=V self.directed=directed self.weighted=weighted self.inf=inf if graph: self.graph=graph self.edges=[] for i in range(self.V): if self.weighted: for j,d in self.graph[i]: if self.directed or not self.directed and i<=j: self.edges.append((i,j,d)) else: for j in self.graph[i]: if self.directed or not self.directed and i<=j: self.edges.append((i,j)) else: self.edges=edges self.graph=[[] for i in range(self.V)] if weighted: for i,j,d in self.edges: self.graph[i].append((j,d)) if not self.directed: self.graph[j].append((i,d)) else: for i,j in self.edges: self.graph[i].append(j) if not self.directed: self.graph[j].append(i) def SIV_DFS(self,s,bipartite_graph=False,cycle_detection=False,directed_acyclic=False,euler_tour=False,linked_components=False,lowlink=False,parents=False,postorder=False,preorder=False,subtree_size=False,topological_sort=False,unweighted_dist=False,weighted_dist=False): seen=[False]*self.V finished=[False]*self.V if directed_acyclic or cycle_detection or topological_sort: dag=True if euler_tour: et=[] if linked_components: lc=[] if lowlink: order=[None]*self.V ll=[None]*self.V idx=0 if parents or cycle_detection or lowlink or subtree_size: ps=[None]*self.V if postorder or topological_sort: post=[] if preorder: pre=[] if subtree_size: ss=[1]*self.V if unweighted_dist or bipartite_graph: uwd=[self.inf]*self.V uwd[s]=0 if weighted_dist: wd=[self.inf]*self.V wd[s]=0 stack=[(s,0)] if self.weighted else [s] while stack: if self.weighted: x,d=stack.pop() else: x=stack.pop() if not seen[x]: seen[x]=True stack.append((x,d) if self.weighted else x) if euler_tour: et.append(x) if linked_components: lc.append(x) if lowlink: order[x]=idx ll[x]=idx idx+=1 if preorder: pre.append(x) for y in self.graph[x]: if self.weighted: y,d=y if not seen[y]: stack.append((y,d) if self.weighted else y) if parents or cycle_detection or lowlink or subtree_size: ps[y]=x if unweighted_dist or bipartite_graph: uwd[y]=uwd[x]+1 if weighted_dist: wd[y]=wd[x]+d elif not finished[y]: if (directed_acyclic or cycle_detection or topological_sort) and dag: dag=False if cycle_detection: cd=(y,x) elif not finished[x]: finished[x]=True if euler_tour: et.append(~x) if lowlink: bl=True for y in self.graph[x]: if self.weighted: y,d=y if ps[x]==y and bl: bl=False continue ll[x]=min(ll[x],order[y]) if x!=s: ll[ps[x]]=min(ll[ps[x]],ll[x]) if postorder or topological_sort: post.append(x) if subtree_size: for y in self.graph[x]: if self.weighted: y,d=y if y==ps[x]: continue ss[x]+=ss[y] if bipartite_graph: bg=[[],[]] for tpl in self.edges: x,y=tpl[:2] if self.weighted else tpl if uwd[x]==self.inf or uwd[y]==self.inf: continue if not uwd[x]%2^uwd[y]%2: bg=False break else: for x in range(self.V): if uwd[x]==self.inf: continue bg[uwd[x]%2].append(x) retu=() if bipartite_graph: retu+=(bg,) if cycle_detection: if dag: cd=[] else: y,x=cd cd=self.Route_Restoration(y,x,ps) retu+=(cd,) if directed_acyclic: retu+=(dag,) if euler_tour: retu+=(et,) if linked_components: retu+=(lc,) if lowlink: retu=(ll,) if parents: retu+=(ps,) if postorder: retu+=(post,) if preorder: retu+=(pre,) if subtree_size: retu+=(ss,) if topological_sort: if dag: tp_sort=post[::-1] else: tp_sort=[] retu+=(tp_sort,) if unweighted_dist: retu+=(uwd,) if weighted_dist: retu+=(wd,) if len(retu)==1: retu=retu[0] return retu def Build_LCA(self,s,segment_tree=False): self.lca_segment_tree=segment_tree if self.lca_segment_tree: self.lca_euler_tour,self.lca_parents,depth=self.SIV_DFS(s,euler_tour=True,parents=True,unweighted_dist=True) self.lca_dfs_in_index=[None]*self.V self.lca_dfs_out_index=[None]*self.V for i,x in enumerate(self.lca_euler_tour): if x>=0: self.lca_dfs_in_index[x]=i else: self.lca_dfs_out_index[~x]=i self.ST=Segment_Tree(2*self.V,lambda x,y:min(x,y),self.V) lst=[None]*(2*self.V) for i in range(2*self.V-1): if self.lca_euler_tour[i]>=0: lst[i]=depth[self.lca_euler_tour[i]] else: lst[i]=depth[self.lca_parents[~self.lca_euler_tour[i]]] lst[2*self.V-1]=-1 self.ST.Build(lst) else: self.lca_parents,self.lca_depth=self.SIV_DFS(s,parents=True,unweighted_dist=True) self.lca_parents[s]=s self.lca_PD=Path_Doubling(self.V,self.lca_parents) self.lca_PD.Build_Next(self.V) def LCA(self,a,b): if self.lca_segment_tree: m=min(self.lca_dfs_in_index[a],self.lca_dfs_in_index[b]) M=max(self.lca_dfs_in_index[a],self.lca_dfs_in_index[b]) x=self.lca_euler_tour[self.ST.Fold_Index(m,M+1)] if x>=0: lca=x else: lca=self.lca_parents[~x] else: if self.lca_depth[a]>self.lca_depth[b]: a,b=b,a b=self.lca_PD.Permutation_Doubling(b,self.lca_depth[b]-self.lca_depth[a]) if a!=b: for k in range(self.lca_PD.k-1,-1,-1): if self.lca_PD.permutation_doubling[a][k]!=self.lca_PD.permutation_doubling[b][k]: a,b=self.lca_PD.permutation_doubling[a][k],self.lca_PD.permutation_doubling[b][k] a,b=self.lca_PD.permutation_doubling[a][0],self.lca_PD.permutation_doubling[b][0] lca=a return lca def Dijkstra(self,s,route_restoration=False): dist=[self.inf]*self.V dist[s]=0 queue=[(0,s)] if route_restoration: parents=[None]*self.V while queue: dx,x=heapq.heappop(queue) if dist[x]dx+dy: dist[y]=dx+dy if route_restoration: parents[y]=x heapq.heappush(queue,(dist[y],y)) if route_restoration: return dist,parents else: return dist class Segment_Tree: def __init__(self,N,f,e,lst=None,dynamic=False): self.f=f self.e=e self.N=N if dynamic: self.segment_tree=defaultdict(lambda:self.e) else: if lst==None: self.segment_tree=[self.e]*2*self.N else: assert len(lst)<=self.N self.segment_tree=[self.e]*self.N+[x for x in lst]+[self.e]*(N-len(lst)) for i in range(self.N-1,0,-1): self.segment_tree[i]=self.f(self.segment_tree[i<<1],self.segment_tree[i<<1|1]) def __getitem__(self,i): if type(i)==int: if -self.N<=i<0: return self.segment_tree[i+self.N*2] elif 0<=i1: i>>= 1 self.segment_tree[i]=self.f(self.segment_tree[i<<1],self.segment_tree[i<<1|1]) def Build(self,lst): for i,x in enumerate(lst,self.N): self.segment_tree[i]=x for i in range(self.N-1,0,-1): self.segment_tree[i]=self.f(self.segment_tree[i<<1],self.segment_tree[i<<1|1]) def Fold(self,L=None,R=None): if L==None: L=self.N else: L+=self.N if R==None: R=self.N*2 else: R+=self.N vL=self.e vR=self.e while L>=1 R>>=1 return self.f(vL,vR) def Fold_Index(self,L=None,R=None): if L==None: L=self.N else: L+=self.N if R==None: R=self.N*2 else: R+=self.N if L==R: return None x=self.Fold(L-self.N,R-self.N) while L>=1 R>>=1 while i>=1 r>>=1 if f(self.f(vl,vr)): return self.N v=self.e while True: while L%2==0: L>>=1 vv=self.f(v,self.segment_tree[L]) if f(vv): v=vv L+=1 else: while L>=1 r>>=1 if f(self.f(vl,vr)): return 0 v=self.e while True: R-=1 while R>1 and R%2: R>>=1 vv=self.f(self.segment_tree[R],v) if f(vv): v=vv else: while R