以 号节点 为 根节点 建立 最短路树, 一个节点删边时一定是删去 父边,
考虑 删去父边后当前节点 到 的最短路长是什么,
一定是 ,
最短路长为 ,
于是考虑使用 树链剖分 和 线段树 维护 的 最小值 , 即为答案 .
非树边 作用的范围为 到 的 树上路径 上的点 (除 ), 如下图所示黄色部分,
- 注意重边, 自环 .
#include<bits/stdc++.h>
#define reg register
#define se second
typedef long long ll;
int read(){
char c;
int s = 0, flag = 1;
while((c=getchar()) && !isdigit(c))
if(c == '-'){ flag = -1, c = getchar(); break ; }
while(isdigit(c)) s = s*10 + c-'0', c = getchar();
return s * flag;
}
const ll inf = 1e17 + 5;
const int maxn = 1e6 + 10;
int N;
int M;
int num0;
int Tmp_1;
int dfs_tim;
int Mp[maxn];
int Fe[maxn];
int Fk[maxn];
int top[maxn];
int dep[maxn];
int dfn[maxn];
int son[maxn];
int head[maxn];
int size[maxn];
ll Dis[maxn];
bool vis[maxn];
struct Edge{ int nxt, to, w; } edge[maxn << 1], E[maxn];
void Add(int from, int to, int w){ edge[++ num0] = (Edge){ head[from], to, w }; head[from] = num0; }
struct Segment_Tree{
struct Node{ int l, r; ll min_v, tg; } T[maxn << 2];
void Push_down(const int &k){
T[k].min_v = std::min(T[k].min_v, T[k].tg);
if(T[k].l == T[k].r) return T[k].tg=inf, void();
T[k<<1].tg = std::min(T[k<<1].tg, T[k].tg), T[k<<1|1].tg = std::min(T[k<<1|1].tg, T[k].tg);
T[k].tg = inf;
}
void Push_up(const int &k){
if(T[k<<1].tg != inf) Push_down(k<<1);
if(T[k<<1|1].tg != inf) Push_down(k<<1|1);
T[k].min_v = std::min(T[k<<1].min_v, T[k<<1|1].min_v);
}
void Build(int k, int l, int r){
T[k].l = l, T[k].r = r, T[k].min_v = T[k].tg = inf;
if(l == r) return ;
int mid = l+r >> 1; Build(k<<1, l, mid), Build(k<<1|1, mid+1, r);
}
void Modify(int k, const int &ql, const int &qr, const ll &aim){
int l = T[k].l, r = T[k].r;
if(T[k].tg != inf) Push_down(k);
if(r < ql || l > qr) return ;
if(ql <= l && r <= qr){ T[k].tg = std::min(T[k].tg, aim); Push_down(k); return ; }
Modify(k<<1, ql, qr, aim), Modify(k<<1|1, ql, qr, aim); Push_up(k);
}
ll Query(int k, const int &ql, const int &qr){
int l = T[k].l, r = T[k].r;
if(T[k].tg != inf) Push_down(k);
if(r < ql || l > qr) return inf;
if(ql <= l && r <= qr) return T[k].min_v;
return std::min(Query(k<<1, ql, qr), Query(k<<1|1, ql, qr));
}
} seg_t;
void Dij(){
typedef std::pair<ll, int> pr;
std::priority_queue <pr, std::vector<pr>, std::greater<pr> > Q;
for(reg int i = 2; i <= N; i ++) Dis[i] = inf; Q.push(pr(0, 1));
while(!Q.empty()){
int ft = Q.top().se; Q.pop();
if(vis[ft]) continue ; vis[ft] = 1;
for(reg int i = head[ft]; i; i = edge[i].nxt){
int to = edge[i].to;
if(Dis[to] > Dis[ft] + edge[i].w){
Fk[to] = ft, Fe[to] = Mp[i];
Dis[to] = Dis[ft] + edge[i].w;
Q.push(pr(Dis[to], to));
}
}
}
num0 = 1;
for(reg int i = 1; i <= N; i ++) head[i] = vis[i] = 0;
for(reg int i = 2; i <= N; i ++) Add(Fk[i], i, E[Fe[i]].w), Add(i, Fk[i], E[Fe[i]].w), vis[Fe[i]] = 1;
}
void DFS_1(int k, int fa){
size[k] = 1, dep[k] = dep[fa] + 1;
for(reg int i = head[k]; i; i = edge[i].nxt){
int to = edge[i].to;
if(to == fa) continue ;
DFS_1(to, k); size[k] += size[to];
if(size[to] > size[son[k]]) son[k] = to;
}
}
void DFS_2(int k, int tp){
top[k] = tp, dfn[k] = ++ dfs_tim;
if(son[k]) DFS_2(son[k], tp);
for(reg int i = head[k]; i; i = edge[i].nxt){
int to = edge[i].to;
if(to == son[k] || to == Fk[k]) continue ;
DFS_2(to, to);
}
}
void Modify(int x, int y, ll d){
while(top[x] != top[y]){
if(dep[top[x]] < dep[top[y]]) std::swap(x, y);
seg_t.Modify(1, dfn[top[x]], dfn[x], d);
x = Fk[top[x]];
}
if(dfn[x] < dfn[y]) std::swap(x, y);
if(dfn[y]+1 > dfn[x]) return ;
seg_t.Modify(1, dfn[y]+1, dfn[x], d);
}
int main(){
read(); N = read(); M = read();
seg_t.Build(1, 1, N); Tmp_1 = 0;
num0 = 1;
for(reg int i = 1; i <= M; i ++){
int u = read(), v = read(), w = read();
if(u == v) continue ;
Add(u, v, w), Add(v, u, w);
E[++ Tmp_1] = (Edge){ u, v, w };
Mp[num0] = Mp[num0^1] = Tmp_1;
}
M = Tmp_1; Dij(); DFS_1(1, 0); DFS_2(1, 1);
for(reg int i = 1; i <= Tmp_1; i ++){
if(vis[i]) continue ;
int u = E[i].nxt, v = E[i].to;
Modify(u, v, Dis[u] + Dis[v] + E[i].w);
}
printf("0 ");
for(reg int i = 2; i <= N; i ++){
ll min_d = seg_t.Query(1, dfn[i], dfn[i]);
if(min_d == inf) min_d = -1;
else min_d -= Dis[i];
printf("%lld ", min_d);
}
return 0;
}