import bisect import copy import decimal import fractions import heapq import itertools import math import random import sys import time from collections import Counter,deque,defaultdict from functools import lru_cache,reduce from heapq import heappush,heappop,heapify,heappushpop,_heappop_max,_heapify_max def _heappush_max(heap,item): heap.append(item) heapq._siftdown_max(heap, 0, len(heap)-1) def _heappushpop_max(heap, item): if heap and item < heap[0]: item, heap[0] = heap[0], item heapq._siftup_max(heap, 0) return item from math import gcd as GCD read=sys.stdin.read readline=sys.stdin.readline readlines=sys.stdin.readlines write=sys.stdout.write class Graph: def __init__(self,V,edges=None,graph=None,directed=False,weighted=False,inf=float("inf")): self.V=V self.directed=directed self.weighted=weighted self.inf=inf if graph!=None: 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_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[k][a]!=self.lca_PD.permutation_doubling[k][b]: a,b=self.lca_PD.permutation_doubling[k][a],self.lca_PD.permutation_doubling[k][b] a,b=self.lca_PD.permutation_doubling[0][a],self.lca_PD.permutation_doubling[0][b] lca=a return lca class Path_Doubling: def __init__(self,N,permutation,lst=None,f=None,e=None): self.N=N self.permutation=permutation self.lst=lst self.f=f self.e=e def Build_Next(self,K=None): if K==None: K=self.N self.k=K.bit_length() self.permutation_doubling=[[None]*self.N for k in range(self.k)] for n in range(self.N): self.permutation_doubling[0][n]=self.permutation[n] if self.lst!=None: self.doubling=[[None]*self.N for k in range(self.k)] for n in range(self.N): self.doubling[0][n]=self.lst[n] for k in range(1,self.k): for n in range(self.N): if self.permutation_doubling[k-1][n]!=None: self.permutation_doubling[k][n]=self.permutation_doubling[k-1][self.permutation_doubling[k-1][n]] if self.f!=None: self.doubling[k][n]=self.f(self.doubling[n][k-1],self.doubling[k-1][self.permutation_doubling[k-1][n]]) def Permutation_Doubling(self,N,K): if K<0 or 1<>k&1 and N!=None: N=self.permutation_doubling[k][N] return N def Doubling(self,N,K): if K<0: return self.e retu=self.e for k in range(self.k): if K>>k&1: if self.permutation_doubling[k][N]==None: return None retu=self.f(retu,self.doubling[k][N]) N=self.permutation_doubling[k][N] return N,retu def Bisect(self,x,is_ok): if not is_ok(x): return -1,None K=0 for k in range(self.k-1,-1,-1): if K|1<