In the past few days, my brain has been rusted when I write USACO water questions... I am greedy when I come up, and I am WA.
In fact, this is a thing called the shortest path tree, because it can guarantee that there is only one shortest path, so all the shortest paths are combined into one tree. A tree rooted at 1, and on this tree, the path occupied by the sprite for each point is the
first spfa that leads to the father. Find the shortest path dis[] and the shortest path tree, and select it into the shortest path tree. Mark the edge for deletion
, cut the tree, put it on the segment tree to maintain the minimum value,
and then look at the edge (u, v, w) that has not been deleted, and set it on the tree like this:
Then consider this The contribution of the edge, observe the two x points and the v point (for convenience, the v point is also included in x). If you delete the way to father, you can go from 1 to 1→lca→u→v→x, so this (u,v,w) can contribute to point x on the path from lca (excluding lca) to v A path of length dis u +w(u→v)+(dis[v]-dis[x])(v→x). Because the shortest path is required, -dis[x] is fixed, min(dis[u]+dis[v]+w) is required, and this value of these points can be updated directly on the line segment tree.
#include<iostream>
#include<cstdio>
#include<queue>
#include<cstring>
using namespace std;
const int N=100005,M=400005,inf=2e9+7;
int n,m,h[N],cnt,dis[N],x[M],y[M],z[M],bl[N];
int fa[N],de[N],hs[N],si[N],fr[N],id[N],rl[N],tmp;
bool v[N],mk[M];
struct qwe
{
int ne,to,va;
}e[M];
struct xds
{
int l,r,mn,lz;
}t[N<<2];
int read()
{
int r=0,f=1;
char p=getchar();
while(p>'9'||p<'0')
{
if(p=='-')
f=-1;
p=getchar();
}
while(p>='0'&&p<='9')
{
r=r*10+p-48;
p=getchar();
}
return r*f;
}
void add(int u,int v,int w)
{//if(!w)cerr<<u<<" "<<v<<endl;
cnt++;
e[cnt].ne=h[u];
e[cnt].to=v;
e[cnt].va=w;
h[u]=cnt;
x[cnt]=u,y[cnt]=v,z[cnt]=w;
}
void jia(int u,int v)
{
cnt++;
e[cnt].ne=h[u];
e[cnt].to=v;
h[u]=cnt;
}
void dfs1(int u,int fat)
{
si[u]=1;
de[u]=de[fat]+1;
for(int i=h[u];i;i=e[i].ne)
if(e[i].to!=fat)
{
dfs1(e[i].to,u);
si[u]+=si[e[i].to];
if(si[hs[u]]<si[e[i].to])
hs[u]=e[i].to;
}
}
void dfs2(int u,int top)
{
fr[u]=top;
id[u]=++tmp;
rl[tmp]=u;
if(!hs[u])
return;
dfs2(hs[u],top);
for(int i=h[u];i;i=e[i].ne)
if(e[i].to!=hs[u]&&e[i].to!=fa[u])
dfs2(e[i].to,e[i].to);
}
int lca(int u,int v)
{
for(;fr[u]!=fr[v];de[fr[u]]>de[fr[v]]?u=fa[fr[u]]:v=fa[fr[v]]);
return de[u]<de[v]?u:v;
}
void pd(int ro)
{
if(t[ro].lz!=inf)
{
t[ro<<1].mn=min(t[ro<<1].mn,t[ro].lz);
t[ro<<1].lz=min(t[ro<<1].lz,t[ro].lz);
t[ro<<1|1].mn=min(t[ro<<1|1].mn,t[ro].lz);
t[ro<<1|1].lz=min(t[ro<<1|1].lz,t[ro].lz);
t[ro].lz=inf;
}
}
void build(int ro,int l,int r)
{
t[ro].l=l,t[ro].r=r,t[ro].mn=t[ro].lz=inf;
if(l==r)
return;
int mid=(l+r)>>1;
build(ro<<1,l,mid);
build(ro<<1|1,mid+1,r);
}
void update(int ro,int l,int r,int w)
{//cerr<<t[ro].l<<" "<<t[ro].r<<" "<<l<<" "<<r<<endl;
if(t[ro].l==l&&t[ro].r==r)
{
t[ro].mn=min(t[ro].mn,w);
t[ro].lz=min(t[ro].lz,w);
return;
}
pd(ro);
int mid=(t[ro].l+t[ro].r)>>1;
if(r<=mid)
update(ro<<1,l,r,w);
else if(l>mid)
update(ro<<1|1,l,r,w);
else
update(ro<<1,l,mid,w),update(ro<<1|1,mid+1,r,w);
t[ro].mn=min(t[ro<<1].mn,t[ro<<1|1].mn);
}
void gai(int u,int v,int w)
{//cerr<<u<<" "<<v<<endl;
while(fr[u]!=fr[v])
{
if(de[fr[u]]<de[fr[v]])
swap(u,v);
update(1,id[fr[u]],id[u],w);
u=fa[fr[u]];
}
if(u!=v)
{
if(de[u]>de[v])
swap(u,v);
update(1,id[u]+1,id[v],w);
}
}
int ques(int ro,int p)
{
if(t[ro].l==t[ro].r)
return t[ro].mn;
pd(ro);
int mid=(t[ro].l+t[ro].r)>>1;
if(p<=mid)
return ques(ro<<1,p);
else
return ques(ro<<1|1,p);
}
int main()
{
n=read(),m=read();
for(int i=1;i<=m;i++)
{
int x=read(),y=read(),z=read();
add(x,y,z),add(y,x,z);
}
for(int i=1;i<=n;i++)
dis[i]=inf;
deque<int>q;
v[1]=1,dis[1]=0,q.push_back(1);
while(!q.empty())
{
int u=q.front();
q.pop_front();
v[u]=0;
for(int i=h[u];i;i=e[i].ne)
if(dis[e[i].to]>dis[u]+e[i].va)
{
dis[e[i].to]=dis[u]+e[i].va;
fa[e[i].to]=u,bl[e[i].to]=i;
if(!v[e[i].to])
{
v[e[i].to]=1;
if(q.empty()||dis[e[i].to]>dis[q.front()])
q.push_back(e[i].to);
else
q.push_front(e[i].to);
}
}
}
memset(h,0,sizeof(h));
cnt=0;
for(int i=2;i<=n;i++)
{
jia(fa[i],i),jia(i,fa[i]);
mk[bl[i]]=1;
}
dfs1(1,0);
dfs2(1,1);
build(1,1,n);
for(int i=1;i<=2*m;i++)
if(!mk[i])
gai(lca(x[i],y[i]),y[i],dis[x[i]]+dis[y[i]]+z[i]);//,cerr<<x[i]<<" "<<y[i]<<" "<<z[i]<<endl;//cerr<<"ok"<<endl;
for(int i=2;i<=n;i++)
{
int ans=ques(1,id[i]);
printf("%d\n",ans==inf?-1:ans-dis[i]);
}
return 0;
}