曾有两次 [最短路树]

$曾有两次$

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$\color{red}{正解部分}$

以 $1$ 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 $u$ 到 $1$ 的最短路长是什么,

一定是 $u \rightarrow u子树内节点 \ v\ -^{非树边}\rightarrow w \rightarrow 1$,
最短路长为 $deps_v-deps_u+W_{非树边}+deps_w$,

于是考虑使用 树链剖分 和 线段树 维护 $deps_v+deps_w+W_{非树边}$ 的 最小值 $minn$, $minn - deps_u$ 即为答案 .

非树边 $v \rightarrow w$ 作用的范围为 $u$ 到 $v$ 的 树上路径 上的点 (除 $lca$), 如下图所示黄色部分,

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$\color{red}{实现部分}$

• 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;
}