結果
| 問題 | No.1917 LCMST |
| ユーザー |
 vwxyz
|
| 提出日時 | 2022-09-29 03:11:03 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 2,381 ms / 4,000 ms |
| コード長 | 58,858 bytes |
| コンパイル時間 | 549 ms |
| コンパイル使用メモリ | 91,196 KB |
| 実行使用メモリ | 301,772 KB |
| 最終ジャッジ日時 | 2024-06-02 01:52:44 |
| 合計ジャッジ時間 | 50,674 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge2 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 96 ms
94,132 KB |
| testcase_01 | AC | 96 ms
93,972 KB |
| testcase_02 | AC | 97 ms
94,312 KB |
| testcase_03 | AC | 150 ms
95,624 KB |
| testcase_04 | AC | 150 ms
95,576 KB |
| testcase_05 | AC | 154 ms
95,540 KB |
| testcase_06 | AC | 145 ms
95,568 KB |
| testcase_07 | AC | 146 ms
95,544 KB |
| testcase_08 | AC | 99 ms
94,072 KB |
| testcase_09 | AC | 2,381 ms
288,588 KB |
| testcase_10 | AC | 2,270 ms
288,108 KB |
| testcase_11 | AC | 2,216 ms
287,900 KB |
| testcase_12 | AC | 1,842 ms
288,864 KB |
| testcase_13 | AC | 761 ms
196,544 KB |
| testcase_14 | AC | 2,136 ms
301,772 KB |
| testcase_15 | AC | 510 ms
175,236 KB |
| testcase_16 | AC | 796 ms
200,820 KB |
| testcase_17 | AC | 1,158 ms
233,724 KB |
| testcase_18 | AC | 1,381 ms
259,096 KB |
| testcase_19 | AC | 565 ms
173,944 KB |
| testcase_20 | AC | 459 ms
165,268 KB |
| testcase_21 | AC | 592 ms
176,000 KB |
| testcase_22 | AC | 461 ms
165,580 KB |
| testcase_23 | AC | 444 ms
165,224 KB |
| testcase_24 | AC | 598 ms
176,700 KB |
| testcase_25 | AC | 457 ms
165,040 KB |
| testcase_26 | AC | 515 ms
168,960 KB |
| testcase_27 | AC | 533 ms
171,592 KB |
| testcase_28 | AC | 558 ms
173,704 KB |
| testcase_29 | AC | 2,154 ms
292,040 KB |
| testcase_30 | AC | 2,094 ms
292,040 KB |
| testcase_31 | AC | 2,133 ms
292,052 KB |
| testcase_32 | AC | 2,153 ms
292,136 KB |
| testcase_33 | AC | 2,134 ms
292,216 KB |
| testcase_34 | AC | 219 ms
165,272 KB |
| testcase_35 | AC | 220 ms
165,528 KB |
| testcase_36 | AC | 222 ms
165,232 KB |
| testcase_37 | AC | 2,142 ms
291,212 KB |
| testcase_38 | AC | 2,227 ms
291,960 KB |
| testcase_39 | AC | 2,175 ms
290,836 KB |
| testcase_40 | AC | 2,185 ms
292,096 KB |
| testcase_41 | AC | 2,231 ms
290,632 KB |
| testcase_42 | AC | 2,265 ms
290,936 KB |
| testcase_43 | AC | 265 ms
165,024 KB |
| testcase_44 | AC | 286 ms
165,152 KB |
ソースコード
from collections import defaultdict
from math import gcd
import sys
readline=sys.stdin.readline
import heapq
import random
from collections import defaultdict,deque
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
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 MIV_DFS(self,initial_vertices=None,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):
if initial_vertices==None:
initial_vertices=[s for s in range(self.V)]
seen=[False]*self.V
finished=[False]*self.V
if bipartite_graph:
bg=[None]*self.V
cnt=-1
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 bipartite_graph or unweighted_dist:
uwd=[self.inf]*self.V
if weighted_dist:
wd=[self.inf]*self.V
for s in initial_vertices:
if seen[s]:
continue
if bipartite_graph:
cnt+=1
bg[s]=(cnt,0)
if linked_components:
lc.append([])
if bipartite_graph or unweighted_dist:
uwd[s]=0
if weighted_dist:
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[-1].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 bipartite_graph:
bg[y]=(cnt,bg[x][1]^1)
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 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_=bg
bg=[[[],[]] for i in range(cnt+1)]
for tpl in self.edges:
i,j=tpl[:2] if self.weighted else tpl
if not bg_[i][1]^bg_[j][1]:
bg[bg_[i][0]]=False
for x in range(self.V):
if bg[bg_[x][0]]:
bg[bg_[x][0]][bg_[x][1]].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 SIV_BFS(self,s,bfs_tour=False,bipartite_graph=False,linked_components=False,parents=False,unweighted_dist=False,weighted_dist=False):
seen=[False]*self.V
seen[s]=True
if bfs_tour:
bt=[s]
if linked_components:
lc=[s]
if parents:
ps=[None]*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
queue=deque([s])
while queue:
x=queue.popleft()
for y in self.graph[x]:
if self.weighted:
y,d=y
if not seen[y]:
seen[y]=True
queue.append(y)
if bfs_tour:
bt.append(y)
if linked_components:
lc.append(y)
if parents:
ps[y]=x
if unweighted_dist or bipartite_graph:
uwd[y]=uwd[x]+1
if weighted_dist:
wd[y]=wd[x]+d
if bipartite_graph:
bg=[[],[]]
for tpl in self.edges:
i,j=tpl[:2] if self.weighted else tpl
if uwd[i]==self.inf or uwd[j]==self.inf:
continue
if not uwd[i]%2^uwd[j]%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 bfs_tour:
retu+=(bt,)
if bipartite_graph:
retu+=(bg,)
if linked_components:
retu+=(lc,)
if parents:
retu+=(ps,)
if unweighted_dist:
retu+=(uwd,)
if weighted_dist:
retu+=(wd,)
if len(retu)==1:
retu=retu[0]
return retu
def MIV_BFS(self,initial_vertices=None,bipartite_graph=False,linked_components=False,parents=False,unweighted_dist=False,weighted_dist=False):
if initial_vertices==None:
initial_vertices=[i for i in range(self.V)]
seen=[False]*self.V
if bipartite_graph:
bg=[None]*self.V
cnt=-1
if linked_components:
lc=[]
if parents:
ps=[None]*self.V
if unweighted_dist:
uwd=[self.inf]*self.V
if weighted_dist:
wd=[self.inf]*self.V
for s in initial_vertices:
if seen[s]:
continue
seen[s]=True
if bipartite_graph:
cnt+=1
bg[s]=(cnt,0)
if linked_components:
lc.append([s])
if unweighted_dist:
uwd[s]=0
if weighted_dist:
wd[s]=0
queue=deque([s])
while queue:
x=queue.popleft()
for y in self.graph[x]:
if self.weighted:
y,d=y
if not seen[y]:
seen[y]=True
queue.append(y)
if bipartite_graph:
bg[y]=(cnt,bg[x][1]^1)
if linked_components:
lc[-1].append(y)
if parents:
ps[y]=x
if unweighted_dist:
uwd[y]=uwd[x]+1
if weighted_dist:
wd[y]=wd[x]+d
if bipartite_graph:
bg_=bg
bg=[[[],[]] for i in range(cnt+1)]
for tpl in self.edges:
i,j=tpl[:2] if self.weighted else tpl
if not bg_[i][1]^bg_[j][1]:
bg[bg_[i][0]]=False
for x in range(self.V):
if bg[bg_[x][0]]:
bg[bg_[x][0]][bg_[x][1]].append(x)
retu=()
if bipartite_graph:
retu+=(bg,)
if linked_components:
retu+=(lc,)
if parents:
retu=(ps,)
if unweighted_dist:
retu+=(uwd,)
if weighted_dist:
retu+=(wd,)
if len(retu)==1:
retu=retu[0]
return retu
def Tree_Diameter(self,weighted=False):
def Farthest_Point(u):
dist=self.SIV_BFS(u,weighted_dist=True) if weighted else self.SIV_BFS(u,unweighted_dist=True)
fp=0
for i in range(self.V):
if dist[fp]<dist[i]:
fp=i
return fp,dist[fp]
u,d=Farthest_Point(0)
v,d=Farthest_Point(u)
return u,v,d
def SCC(self):
reverse_graph=[[] for i in range(self.V)]
for tpl in self.edges:
u,v=tpl[:2] if self.weighted else tpl
reverse_graph[v].append(u)
postorder=self.MIV_DFS(postorder=True)
scc_points=[]
seen=[False]*self.V
for s in postorder[::-1]:
if seen[s]:
continue
queue=deque([s])
seen[s]=True
lst=[]
while queue:
x=queue.popleft()
lst.append(x)
for y in reverse_graph[x]:
if not seen[y]:
seen[y]=True
queue.append(y)
scc_points.append(lst)
l=len(scc_points)
idx=[None]*self.V
for i in range(l):
for x in scc_points[i]:
idx[x]=i
scc_edges=set()
for tpl in self.edges:
u,v=tpl[:2] if self.weighted else tpl
if idx[u]!=idx[v]:
scc_edges.add((idx[u],idx[v]))
scc_edges=list(scc_edges)
return scc_points,scc_edges
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 Build_HLD(self,s):
self.hld_parents,size,self.hld_depth=self.SIV_DFS(s,parents=True,subtree_size=True,unweighted_dist=True)
stack=[s]
self.hld_tour=[]
self.hld_path_parents=[None]*self.V
self.hld_path_parents[s]=s
while stack:
x=stack.pop()
self.hld_tour.append(x)
max_size=0
max_size_y=None
for y in self.graph[x]:
if self.weighted:
y,d=y
if y==self.hld_parents[x]:
continue
if max_size<size[y]:
max_size=size[y]
max_size_y=y
for y in self.graph[x]:
if self.weighted:
y,d=y
if y==self.hld_parents[x]:
continue
if y!=max_size_y:
stack.append(y)
self.hld_path_parents[y]=y
if max_size_y!=None:
stack.append(max_size_y)
self.hld_path_parents[max_size_y]=self.hld_path_parents[x]
self.hld_tour_idx=[None]*self.V
for i in range(self.V):
self.hld_tour_idx[self.hld_tour[i]]=i
def HLD(self,a,b,edge=False):
L,R=[],[]
while self.hld_path_parents[a]!=self.hld_path_parents[b]:
if self.hld_depth[self.hld_path_parents[a]]<self.hld_depth[self.hld_path_parents[b]]:
R.append((self.hld_tour_idx[self.hld_path_parents[b]],self.hld_tour_idx[b]+1))
b=self.hld_parents[self.hld_path_parents[b]]
else:
L.append((self.hld_tour_idx[a]+1,self.hld_tour_idx[self.hld_path_parents[a]]))
a=self.hld_parents[self.hld_path_parents[a]]
if edge:
if self.hld_depth[a]!=self.hld_depth[b]:
retu=L+[(self.hld_tour_idx[a]+1,self.hld_tour_idx[b]+1)]+R[::-1]
else:
retu=L+R[::-1]
else:
if self.hld_depth[a]<self.hld_depth[b]:
retu=L+[(self.hld_tour_idx[a],self.hld_tour_idx[b]+1)]+R[::-1]
else:
retu=L+[(self.hld_tour_idx[a]+1,self.hld_tour_idx[b])]+R[::-1]
return retu
def Build_Hash(self,s,random_number=False,mod=(1<<61)-1,rerooting=False):
self.lower_hash=[None]*self.V
if random_number:
self.hash_random_number=random_number
else:
self.hash_random_number=[random.randint(1,10**10) for i in range(self.V)]
self.hash_mod=mod
parents,postorder,preorder=self.SIV_DFS(s,parents=True,postorder=True,preorder=True)
for x in postorder:
level=0
for y in self.graph[x]:
if self.weighted:
y,d=y
if y==parents[x]:
continue
h,l=self.lower_hash[y]
level=max(level,l+1)
ha=1
for y in self.graph[x]:
if self.weighted:
y,d=y
if y==parents[x]:
continue
h,l=self.lower_hash[y]
ha*=h+self.hash_random_number[l]
ha%=self.hash_mod
self.lower_hash[x]=(ha,level)
if rerooting:
self.upper_hash=[None]*self.V
self.upper_hash[s]=(1,-1)
for x in preorder:
children=[y for y,d in self.graph[x] if y!=parents[x]] if self.weighted else [y for y in self.graph[x] if y!=parents[x]]
if children:
l=len(children)
l_lst,r_lst=[None]*(l+1),[None]*(l+1)
l_lst[0],r_lst[l]=(1,-1),(1,-1)
for i in range(1,l+1):
h0,l0=l_lst[i-1]
h1,l1=self.lower_hash[children[i-1]]
l_lst[i]=(h0*(h1+self.hash_random_number[l1])%self.hash_mod,max(l0,l1))
for i in range(l-1,-1,-1):
h0,l0=r_lst[i+1]
h1,l1=self.lower_hash[children[i]]
r_lst[i]=(h0*(h1+self.hash_random_number[l1])%self.hash_mod,max(l0,l1))
for i in range(l):
if x==s:
ha,level=1,0
else:
ha,level=self.upper_hash[x]
h0,l0=l_lst[i]
h1,l1=r_lst[i+1]
ha*=h0*h1
level=max(level,l0+1,l1+1)
ha+=self.hash_random_number[level]
ha%=self.hash_mod
level+=1
self.upper_hash[children[i]]=(ha,level)
return
def Hash(self,root,subtree=False):
if subtree:
ha,level=self.lower_hash[root]
ha+=self.hash_random_number[level]
ha%=self.hash_mod
else:
h0,l0=self.lower_hash[root]
h1,l1=self.upper_hash[root]
ha=(h0*h1+self.hash_random_number[max(l0,l1)])%self.hash_mod
level=max(l0,l1)
return ha,level
def Build_Rerooting(self,s,f_transition,f_merge):
self.rerooting_s=s
self.rerooting_f_transition=f_transition
self.rerooting_f_merge=f_merge
parents,postorder,preorder=self.SIV_DFS(s,parents=True,postorder=True,preorder=True)
self.rerooting_lower_dp=[None]*self.V
for x in postorder:
self.rerooting_lower_dp[x]=self.rerooting_f_merge([self.rerooting_f_transition(self.rerooting_lower_dp[y]) for y in G.graph[x] if y!=parents[x]])
self.rerooting_upper_dp=[None]*self.V
for x in preorder:
children=[y for y in self.graph[x] if y!=parents[x]]
left_accumule_f=[None]*(len(children)+1)
right_accumule_f=[None]*(len(children)+1)
left_accumule_f[0]=self.rerooting_f_merge([])
for i in range(1,len(children)+1):
left_accumule_f[i]=self.rerooting_f_merge([left_accumule_f[i-1],self.rerooting_f_transition(self.rerooting_lower_dp[children[i-1]])])
right_accumule_f[len(children)]=self.rerooting_f_merge([])
for i in range(len(children)-1,-1,-1):
right_accumule_f[i]=self.rerooting_f_merge([right_accumule_f[i+1],self.rerooting_f_transition(self.rerooting_lower_dp[children[i]])])
for i in range(len(children)):
if parents[x]!=None:
self.rerooting_upper_dp[children[i]]=self.rerooting_f_merge([left_accumule_f[i],right_accumule_f[i+1],self.rerooting_f_transition(self.rerooting_upper_dp[x])])
else:
self.rerooting_upper_dp[children[i]]=self.rerooting_f_merge([left_accumule_f[i],right_accumule_f[i+1]])
def Rerooting(self,x):
if x==self.rerooting_s:
retu=self.rerooting_lower_dp[x]
else:
retu=self.rerooting_f_merge([self.rerooting_lower_dp[x],self.rerooting_f_transition(self.rerooting_upper_dp[x])])
return retu
def Centroid(self,root=0):
x=root
parents,size=self.SIV_DFS(x,parents=True,subtree_size=True)
while True:
for y in self.graph[x]:
if self.weighted:
y,d=y
if y==parents[x]:
continue
if size[y]*2>size[root]:
x=y
break
else:
for y in self.graph[x]:
if self.weighted:
y,d=y
if y==parents[x]:
continue
if size[root]<=2*size[y]:
return x,y
return x,None
def Centroid_Decomposition(self,edge=False,linked_point=False,point=False,tree=False):
if edge:
cd_edges_lst=[None]*self.V
if linked_point:
cd_linked_points=[None]*self.V
if point:
cd_points_lst=[None]*self.V
if tree:
cd_tree=[]*self.V
edges=self.edges
points=[i for i in range(self.V)]
prev_centroid=None
stack=[(edges,points,None,prev_centroid)] if linked_point else [(edges,points,prev_centroid)]
while stack:
if linked_point:
edges,points,lp,prev_centroid=stack.pop()
else:
edges,points,prev_centroid=stack.pop()
if len(points)==1:
centroid=points[0]
if edge:
cd_edges_lst[centroid]=[]
if linked_point:
cd_linked_points[centroid]=lp
if point:
cd_points_lst[centroid]=[centroid]
if tree and prev_centroid!=None:
cd_tree.append((prev_centroid,centroid))
continue
G=Graph(len(points),edges=edges,weighted=self.weighted)
centroid,_=G.Centroid()
if tree and prev_centroid!=None:
cd_tree.append((prev_centroid,points[centroid]))
parents,tour=G.SIV_DFS(centroid,parents=True,preorder=True)
dp=[None]*len(points)
edges_lst=[]
points_lst=[]
if linked_point:
linked_points=[]
for i,x in enumerate(G.graph[centroid]):
if G.weighted:
x,d=x
dp[x]=(i,0)
edges_lst.append([])
points_lst.append([points[x]])
if linked_point:
linked_points.append(points[x])
for x in tour[1:]:
for y in G.graph[x]:
if G.weighted:
y,d=y
if y==parents[x]:
continue
i,j=dp[x]
jj=len(points_lst[i])
edges_lst[i].append((j,jj,d) if G.weighted else (j,jj))
points_lst[i].append(points[y])
dp[y]=(i,jj)
centroid=points[centroid]
if edge:
cd_edges_lst[centroid]=edges
if linked_point:
cd_linked_points[centroid]=lp
if point:
cd_points_lst[centroid]=points
if linked_point:
for edges,points,lp in zip(edges_lst,points_lst,linked_points):
stack.append((edges,points,lp,centroid))
else:
for edges,points in zip(edges_lst,points_lst):
stack.append((edges,points,centroid))
retu=()
if edge:
retu+=(cd_edges_lst,)
if linked_point:
retu+=(cd_linked_points,)
if point:
retu+=(cd_points_lst,)
if tree:
retu+=(cd_tree,)
if len(retu)==1:
retu=retu[0]
return retu
def Bridges(self):
lowlink,preorder=self.MIV_DFS(lowlink=True,preorder=True)
order=[None]*self.V
for x in range(self.V):
order[preorder[x]]=x
bridges=[]
for e in self.edges:
if self.weighted:
x,y,d=e
else:
x,y=e
if order[x]<lowlink[y] or order[y]<lowlink[x]:
bridges.append((x,y))
return bridges
def Articulation_Points(self):
lowlink,parents,preorder=self.MIV_DFS(lowlink=True,parents=True,preorder=True)
order=[None]*self.V
for x in range(self.V):
order[preorder[x]]=x
articulation_points=[]
for x in range(self.V):
if parents[x]==None:
if len({y for y in self.graph[x] if parents[y]==x})>=2:
articulation_points.append(x)
else:
for y in self.graph[x]:
if parents[y]!=x:
continue
if order[x]<=lowlink[y]:
articulation_points.append(x)
break
return articulation_points
def TECCD(self):
lowlink,preorder=self.MIV_DFS(lowlink=True,preorder=True)
order=[None]*self.V
for x in range(self.V):
order[preorder[x]]=x
edges=[]
for e in self.edges:
if self.weighted:
x,y,d=e
else:
x,y=e
if order[x]>=lowlink[y] and order[y]>=lowlink[x]:
edges.append((x,y))
teccd=Graph(self.V,edges=edges).MIV_DFS(linked_components=True)
return teccd
def LCD(self):
lcd_points=self.MIV_DFS(linked_components=True)
lcd_edges=[[] for i in range(len(lcd_points))]
idx=[None]*self.V
for i in range(len(lcd_points)):
for j in range(len(lcd_points[i])):
idx[lcd_points[i][j]]=(i,j)
for tpl in self.edges:
if self.weighted:
x,y,d=tpl
else:
x,y=tpl
i,j0=idx[x]
i,j1=idx[y]
if self.weighted:
lcd_edges[i].append((j0,j1,d))
else:
lcd_edges[i].append((j0,j1))
return lcd_points,lcd_edges
def Dijkstra(self,s,route_restoration=False):
dist=[self.inf]*self.V
dist[s]=0
hq=[(0,s)]
if route_restoration:
parents=[None]*self.V
while hq:
dx,x=heapq.heappop(hq)
if dist[x]<dx:
continue
for y,dy in self.graph[x]:
if dist[y]>dx+dy:
dist[y]=dx+dy
if route_restoration:
parents[y]=x
heapq.heappush(hq,(dist[y],y))
if route_restoration:
return dist,parents
else:
return dist
def Bellman_Ford(self,s,route_restoration=False):
dist=[self.inf]*self.V
dist[s]=0
if route_restoration:
parents=[None]*self.V
for _ in range(self.V-1):
for i,j,d in self.edges:
if dist[j]>dist[i]+d:
dist[j]=dist[i]+d
if route_restoration:
parents[j]=i
if not self.directed and dist[i]>dist[j]+d:
dist[i]=dist[j]+d
if route_restoration:
parents[i]=j
negative_cycle=[]
for i,j,d in self.edges:
if dist[j]>dist[i]+d:
negative_cycle.append(j)
if not self.directed and dist[i]>dist[j]+d:
negative_cycle.append(i)
if negative_cycle:
is_negative_cycle=[False]*self.V
for i in negative_cycle:
if is_negative_cycle[i]:
continue
else:
queue=[i]
is_negative_cycle[i]=True
while queue:
x=queue.popleft()
for y,d in self.graph[x]:
if not is_negative_cycle[y]:
queue.append(y)
is_negative_cycle[y]=True
if route_restoration:
parents[y]=x
for i in range(self.V):
if is_negative_cycle[i]:
dist[i]=-self.inf
if route_restoration:
return dist,parents
else:
return dist
def Warshall_Floyd(self,route_restoration=False):
dist=[[self.inf]*self.V for i in range(self.V)]
for i in range(self.V):
dist[i][i]=0
if route_restoration:
parents=[[j for j in range(self.V)] for i in range(self.V)]
for i,j,d in self.edges:
if i==j:
continue
if dist[i][j]>d:
dist[i][j]=d
if route_restoration:
parents[i][j]=i
if not self.directed and dist[j][i]>d:
dist[j][i]=d
if route_restoration:
parents[j][i]=j
for k in range(self.V):
for i in range(self.V):
for j in range(self.V):
if dist[i][j]>dist[i][k]+dist[k][j]:
dist[i][j]=dist[i][k]+dist[k][j]
if route_restoration:
parents[i][j]=parents[k][j]
for i in range(self.V):
if dist[i][i]<0:
for j in range(self.V):
if dist[i][j]!=self.inf:
dist[i][j]=-self.inf
if route_restoration:
for i in range(self.V):
if dist[i][i]==0:
parents[i][i]=None
return dist,parents
else:
return dist
def BFS_01(self,s,route_restoration=False):
queue=deque([s])
seen=[False]*self.V
dist=[self.inf]*self.V
dist[s]=0
if route_restoration:
parents=[None]*self.V
while queue:
x=queue.popleft()
if seen[x]:
continue
seen[x]=False
for y,d in self.graph[x]:
if dist[y]>dist[x]+d:
dist[y]=dist[x]+d
if route_restoration:
parents[y]=x
if d:
queue.append(y)
else:
queue.appendleft(y)
if route_restoration:
return dist,parents
else:
return dist
def Distance_Frequency(self):
mod=206158430209
cnt=[0]*self.V
cd_edges,cd_points,cd_tree=self.Centroid_Decomposition(edge=True,point=True,tree=True)
CD=Graph(self.V,edges=cd_tree)
parents,tour=CD.SIV_DFS(cd_tree[0][0],parents=True,postorder=True)
for x in tour:
C=[0]*(len(cd_points[x])+1)
for y in CD.graph[x]:
if y==parents[x]:
continue
depth=Graph(len(cd_points[y]),edges=cd_edges[y],weighted=self.weighted).SIV_DFS(0,unweighted_dist=True)
CC=[0]*(max(depth)+2)
for d in depth:
CC[d+1]+=1
cnt[d+1]+=2
C[d+1]+=1
poly=NTT_Pow(CC,2)
for d,c in enumerate(poly):
if d<self.V:
cnt[d]-=c
while C and C[-1]==0:
C.pop()
poly=NTT_Pow(C,2)
for d,c in enumerate(poly):
if d<N:
cnt[d]+=c
for i in range(self.V):
cnt[i]//=2
return cnt
def Shortest_Path_Count(self,s,dist,mod=0):
cnt=[0]*self.V
cnt[s]=1
for x in sorted([x for x in range(self.V)],key=lambda x:dist[x]):
for y in self.graph[x]:
if self.weighted:
y,d=y
else:
d=1
if dist[x]+d==dist[y]:
cnt[y]+=cnt[x]
if mod:
cnt[y]%=mod
return cnt
def K_Shortest_Path_Routing(self,s,t,K,edge_unicursal=False,point_unicursal=False):
if point_unicursal:
if self.weighted:
dist,parents=self.Dijkstra(s,route_restoration=True)
else:
parents,dist=self.SIV_BFS(s,parents=True,unweighted_dist=True)
route=tuple(self.Route_Restoration(s,t,parents))
queue=[(dist[t],route,[dist[x] for x in route])]
set_queue=set()
set_queue.add(route)
retu=[]
while queue and K:
d,route,route_dist=heapq.heappop(queue)
retu.append((d,route,route_dist))
K-=1
set_route=set()
for i in range(len(route)-1):
x=route[i]
set_route.add(x)
if self.weighted:
edges=[(v,u,d) for u,v,d in self.edges if not u in set_route and not v in set_route]
else:
edges=[(v,u) for u,v in self.edges if not u in set_route and not v in set_route]
G_rev=Graph(self.V,edges=edges,directed=self.directed,weighted=self.weighted,inf=self.inf)
if self.weighted:
dist_rev,parents_rev=G_rev.Dijkstra(t,route_restoration=True)
else:
parents_rev,dist_rev=G_rev.SIV_BFS(t,parents=True,unweighted_dist=True)
for y in self.graph[x]:
if self.weighted:
y,d=y
else:
d=1
if y==route[i+1]:
continue
if dist_rev[y]==self.inf:
continue
tpl=route[:i+1]+tuple(self.Route_Restoration(t,y,parents_rev)[::-1])
if not tpl in set_queue:
heapq.heappush(queue,(route_dist[i]+d+dist_rev[y],tpl,route_dist[:i+1]+[route_dist[i]+d+dist_rev[y]-dist_rev[z] for z in tpl[i+1:]]))
set_queue.add(tpl)
elif edge_unicursal:
if self.weighted:
dist,parents=self.Dijkstra(s,route_restoration=True)
else:
parents,dist=self.SIV_BFS(s,parents=True,unweighted_dist=True)
route=tuple(self.Route_Restoration(s,t,parents))
queue=[(dist[t],route,[dist[x] for x in route])]
set_queue=set()
set_queue.add(route)
retu=[]
while queue and K:
d,route,route_dist=heapq.heappop(queue)
retu.append((d,route,route_dist))
K-=1
set_route=set()
for i in range(len(route)-1):
x=route[i]
y=route[i+1]
set_route.add((x,y,route_dist[i+1]-route_dist[i]))
if not self.directed:
set_route.add((y,x,route_dist[i+1]-route_dist[i]))
if self.weighted:
edges=[(v,u,d) for u,v,d in self.edges if not (u,v,d) in set_route]
else:
edges=[(v,u) for u,v in self.edges if not (u,v,d) in set_route]
G_rev=Graph(self.V,edges=edges,directed=self.directed,weighted=self.weighted,inf=self.inf)
if self.weighted:
dist_rev,parents_rev=G_rev.Dijkstra(t,route_restoration=True)
else:
parents_rev,dist_rev=G_rev.SIV_BFS(t,parents=True,unweighted_dist=True)
for y in self.graph[x]:
if self.weighted:
y,d=y
else:
d=1
if y==route[i+1]:
continue
if dist_rev[y]==self.inf:
continue
tpl=route[:i+1]+tuple(self.Route_Restoration(t,y,parents_rev)[::-1])
if not tpl in set_queue:
heapq.heappush(queue,(route_dist[i]+d+dist_rev[y],tpl,route_dist[:i+1]+[route_dist[i]+d+dist_rev[y]-dist_rev[z] for z in tpl[i+1:]]))
set_queue.add(tpl)
else:
if self.weighted:
dist,parents=self.Dijkstra(s,route_restoration=True)
else:
parents,dist=self.SIV_BFS(s,parents=True,unweighted_dist=True)
if dist[t]==self.inf:
return False
route_lst=[tuple(self.Route_Restoration(s,x,parents)) for x in range(self.V)]
if self.weighted:
edges_rev=[(j,i,d) for i,j,d in self.edges]
else:
edges_rev=[(j,i) for i,j in self.edges]
G_rev=Graph(self.V,edges=edges_rev,weighted=self.weighted,directed=self.directed,inf=self.inf)
if self.weighted:
dist_rev,parents_rev=G_rev.Dijkstra(t,route_restoration=True)
else:
parents_rev,dist_rev=G_rev.SIV_BFS(t,parents=True,unweighted_dist=True)
route_rev_lst=[]
for x in range(self.V):
route_rev_lst.append(tuple(self.Route_Restoration(t,x,parents_rev)[::-1]))
route=route_lst[t]
queue=[(dist[t],route,[dist[x] for x in route])]
set_queue=set()
set_queue.add(route)
retu=[]
while queue and K:
d,route,route_dist=heapq.heappop(queue)
retu.append((d,route,route_dist))
K-=1
for i in range(len(route)):
x=route[i]
for y in self.graph[x]:
if self.weighted:
y,d=y
else:
d=1
if i!=len(route)-1 and y==route[i+1]:
continue
tpl=route[:i+1]+route_rev_lst[y]
if not tpl in set_queue:
heapq.heappush(queue,(route_dist[i]+d+dist_rev[y],tpl,route_dist[:i+1]+[route_dist[i]+d+dist_rev[y]-dist_rev[z] for z in route_rev_lst[y]]))
set_queue.add(tpl)
return retu
def Euler_Path(self,s=None,t=None):
if self.directed:
indegree=[0]*self.V
outdegree=[0]*self.V
graph=[[] for x in range(self.V)]
for tpl in self.edges:
if self.weighted:
u,v,d=tpl
else:
u,v=tpl
indegree[v]+=1
outdegree[u]+=1
graph[v].append(u)
for x in range(self.V):
if indegree[x]+1==outdegree[x]:
if s==None:
s=x
elif s!=x:
return False
elif indegree[x]==outdegree[x]+1:
if t==None:
t=x
elif t!=x:
return False
elif indegree[x]!=outdegree[x]:
return False
if (s,t)==(None,None):
for x in range(self.V):
if graph[x]:
s=x
t=x
break
elif s==None:
s=t
elif t==None:
t=s
elif s==t:
for x in range(self.V):
if indegree[x]!=outdegree[x]:
return False
queue=[t]
euler_path=[]
while queue:
while graph[queue[-1]]:
queue.append(graph[queue[-1]].pop())
x=queue.pop()
euler_path.append(x)
for x in range(self.V):
if graph[x]:
return False
else:
degree=[0]*self.V
graph=[[] for x in range(self.V)]
use_count=[defaultdict(int) for x in range(self.V)]
for tpl in self.edges:
if self.weighted:
u,v,d=tpl
else:
u,v=tpl
degree[v]+=1
degree[u]+=1
graph[u].append(v)
graph[v].append(u)
for x in range(self.V):
if degree[x]%2:
if s==None and t!=x:
s=x
elif t==None and s!=x:
t=x
elif not x in (s,t):
return False
if s==None and t==None:
for x in range(self.V):
if graph[x]:
s=x
t=x
break
else:
s,t=0,0
elif s==None:
s=t
elif t==None:
t=s
elif s!=t:
if degree[s]%2==0 or degree[t]%2==0:
return False
queue=[t]
euler_path=[]
while queue:
while graph[queue[-1]]:
if use_count[queue[-1]][graph[queue[-1]][-1]]:
use_count[queue[-1]][graph[queue[-1]][-1]]-=1
graph[queue[-1]].pop()
else:
queue.append(graph[queue[-1]].pop())
use_count[queue[-1]][queue[-2]]+=1
x=queue.pop()
euler_path.append(x)
for x in range(self.V):
if graph[x]:
return False
if euler_path[0]!=s:
return False
return euler_path
def Route_Restoration(self,s,g,parents):
route=[g]
while s!=g:
if parents[g]==None:
route=[]
break
g=parents[g]
route.append(g)
route=route[::-1]
return route
def Negative_Cycls(self):
dist=[0]*self.V
for _ in range(self.V-1):
for i,j,d in self.edges:
dist[j]=min(dist[j],dist[i]+d)
for i,j,d in self.edges:
if dist[j]>dist[i]+d:
return True
return False
def Kruskal(self,maximize=False):
UF=UnionFind(self.V)
sorted_edges=sorted(self.edges,key=lambda x:x[2],reverse=maximize)
spnning_tree=[]
for i,j,d in sorted_edges:
if not UF.Same(i,j):
UF.Union(i,j)
spnning_tree.append((i,j,d))
return spnning_tree
def Max_Clique(self):
graph=[[False]*self.V for x in range(self.V)]
for x in range(self.V):
for y in self.graph[x]:
if self.weighted:
y,d=y
graph[x][y]=True
N0,N1=self.V//2,self.V-self.V//2
pop_count=[sum(bit>>i&1 for i in range(N1)) for bit in range(1<<N1)]
is_clique0=[True]*(1<<N0)
for j in range(N0):
for i in range(j):
if not graph[i][j]:
is_clique0[1<<i|1<<j]=False
for i in range(N0):
for bit in range(1<<N0):
if bit&1<<i:
is_clique0[bit]=is_clique0[bit]&is_clique0[bit^1<<i]
is_clique1=[True]*(1<<N1)
for j in range(N1):
for i in range(j):
if not graph[i+N0][j+N0]:
is_clique1[1<<i|1<<j]=False
for i in range(N1):
for bit in range(1<<N1):
if bit&1<<i:
is_clique1[bit]=is_clique1[bit]&is_clique1[bit^1<<i]
max_clique_bit=[bit if is_clique0[bit] else 0 for bit in range(1<<N0)]
for i in range(N0):
for bit in range(1<<N0):
if bit&1<<i and pop_count[max_clique_bit[bit]]<pop_count[max_clique_bit[bit^1<<i]]:
max_clique_bit[bit]=max_clique_bit[bit^1<<i]
dp=[(1<<N0)-1]*(1<<N1)
for j in range(N1):
for i in range(N0):
if not graph[j+N0][i]:
dp[1<<j]^=1<<i
for i in range(N1):
for bit in range(1<<N1):
if bit&1<<i:
dp[bit]&=dp[bit^1<<i]
bit0,bit1=0,0
for bit in range(1<<N1):
if is_clique1[bit] and pop_count[max_clique_bit[dp[bit]]]+pop_count[bit]>pop_count[bit0]+pop_count[bit1]:
bit0=max_clique_bit[dp[bit]]
bit1=bit
max_clique=[i for i in range(N0) if bit0&1<<i]+[i+N0 for i in range(N1) if bit1&1<<i]
return max_clique
def Cliques(self):
graph=[[False]*self.V for x in range(self.V)]
for x in range(self.V):
for y in self.graph[x]:
if self.weighted:
y,d=y
graph[x][y]=True
cliques=[]
points=[x for x in range(self.V)]
while points:
l=len(points)
min_degree,min_degree_point=self.inf,None
sum_degree=0
for x in points:
s=sum(graph[x][y] for y in points)
sum_degree+=s
if s<min_degree:
min_degree=s
min_degree_point=x
if min_degree**2>=sum_degree:
lst=points
else:
lst=[x for x in points if x==min_degree_point or graph[min_degree_point][x]]
l=len(lst)
is_clique=[True]*(1<<l)
for j in range(l):
for i in range(j):
if not graph[lst[i]][lst[j]]:
is_clique[1<<i|1<<j]=False
for i in range(l):
for bit in range(1<<l):
if bit&1<<i:
is_clique[bit]=is_clique[bit]&is_clique[bit^1<<i]
if min_degree**2>=sum_degree:
for bit in range(1<<l):
if is_clique[bit]:
cliques.append([lst[i] for i in range(l) if bit&1<<i])
else:
idx=lst.index(min_degree_point)
for bit in range(1<<l):
if is_clique[bit] and bit&1<<idx:
cliques.append([lst[i] for i in range(l) if bit&1<<i])
if min_degree**2>=sum_degree:
points=[]
else:
points=[x for x in points if x!=min_degree_point]
return cliques
def Ford_Fulkerson(self,s,t):
max_flow=0
residual_graph=[defaultdict(int) for i in range(self.V)]
if self.weighted:
for i,j,d in self.edges:
if not d:
continue
residual_graph[i][j]+=d
if not self.directed:
residual_graph[j][i]+=d
else:
for i,j in self.edges:
residual_graph[i][j]+=1
if not self.directed:
residual_graph[j][i]+=1
while True:
parents=[None]*self.V
parents[s]=s
seen=[False]*self.V
seen[s]=True
queue=deque([s])
while queue:
x=queue.popleft()
for y in residual_graph[x].keys():
if not seen[y]:
seen[y]=True
queue.append(y)
parents[y]=x
if y==t:
tt=t
while tt!=s:
residual_graph[parents[tt]][tt]-=1
residual_graph[tt][parents[tt]]+=1
if not residual_graph[parents[tt]][tt]:
residual_graph[parents[tt]].pop(tt)
tt=parents[tt]
max_flow+=1
break
else:
continue
break
else:
break
return max_flow
def BFS(self,s):
seen=[False]*self.V
seen[s]=True
queue=deque([s])
while queue:
x=queue.popleft()
for y in self.graph[x]:
if self.weighted:
y,d=y
if not seen[y]:
seen[y]=True
queue.append(y)
return
def DFS(self,s):
seen=[False]*self.V
finished=[False]*self.V
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)
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)
elif not finished[x]:
finished[x]=True
return
class UnionFind:
def __init__(self,N,label=None,f=None,weighted=False):
self.N=N
self.parents=[None]*self.N
self.size=[1]*self.N
self.roots={i for i in range(self.N)}
self.label=label
if self.label!=None:
self.label=[x for x in label]
self.f=f
self.weighted=weighted
if self.weighted:
self.weight=[0]*self.N
def Find(self,x):
stack=[]
while self.parents[x]!=None:
stack.append(x)
x=self.parents[x]
if self.weighted:
w=0
for y in stack[::-1]:
self.parents[y]=x
w+=self.weight[y]
self.weight[y]=w
else:
for y in stack[::-1]:
self.parents[y]=x
return x
def Union(self,x,y,w=None):
root_x=self.Find(x)
root_y=self.Find(y)
if root_x==root_y:
if self.weighted:
if self.weight[y]-self.weight[x]==w:
return True
else:
return False
else:
if self.size[root_x]<self.size[root_y]:
x,y=y,x
root_x,root_y=root_y,root_x
if self.weighted:
w=-w
self.parents[root_y]=root_x
self.size[root_x]+=self.size[root_y]
self.roots.remove(root_y)
if self.label!=None:
self.label[root_x]=self.f(self.label[root_x],self.label[root_y])
if self.weighted:
self.weight[root_y]=w+self.weight[x]-self.weight[y]
def Size(self,x):
return self.size[self.Find(x)]
def Same(self,x,y):
return self.Find(x)==self.Find(y)
def Label(self,x):
return self.label[self.Find(x)]
def Weight(self,x,y):
root_x=self.Find(x)
root_y=self.Find(y)
if root_x!=root_y:
return None
return self.weight[y]-self.weight[x]
def Roots(self):
return list(self.roots)
def Link_Components_Count(self):
return len(self.roots)
def __str__(self):
linked_components=defaultdict(list)
for x in range(self.N):
linked_components[self.Find(x)].append(x)
return "\n".join(f"{r}: {m}" for r,m in linked_components.items())
N=int(readline())
M=10**5
cnt=[0]*(M+1)
for a in map(int,readline().split()):
cnt[a]+=1
ans=0
for a in range(M+1):
ans+=a*max(0,cnt[a]-1)
edges=[]
for d in range(1,M+1):
lst=[a for a in range(d,M+1,d) if cnt[a]]
if lst:
x=lst[0]
for y in lst[1:]:
edges.append((x,y,x*y//gcd(x,y)))
G=Graph(M+1,edges=edges,weighted=True)
for a,b,d in G.Kruskal():
ans+=d
print(ans)