HDU 4871 Shortest-path tree

先用dijkstra把最短路树建出来，然后就是树的质心分治了。

经过k个点的路径，要么全在子树上，要么经过根结点，因此可以分治。

如果分治的时候选点不好会变成O(n^2)，比较极端的情况是比如树是一条链。

选择质心可以保证最大子树结点不超过n/2，每次递归至少减半，递归层数是O(logn)的。

找子树质心是O(n)的树形dp，枚举到根结点的路径是O(n)的。

把经过根节点并且路径上有c个结点的最长路径以及方案保存到一个map，对于一条新增的路径去查找k-c个点的路径，就可以更新答案了。

如果用的unorder_map，那么查找是O(1)的，因此分治复杂度是T(n) = 2*T(n/2) + O(n) ≈ O(nlogn)。

总体复杂度是O(mlogn + nlogn)

/*********************************************************
*            ------------------                          *
*   author AbyssFish                                     *
**********************************************************/
#include<cstdio>
#include<iostream>
#include<string>
#include<cstring>
#include<queue>
#include<vector>
#include<stack>
#include<map>
#include<set>
#include<algorithm>
#include<cmath>
#include<numeric>
#include<climits>
#include<unordered_map>
using namespace std;

const int maxn = 30000+2;
const int maxm = 60000*2+3;

typedef long long ll;
#define sInt 4
int hd[maxn], nx[maxm], to[maxm], we[maxm], ec;
#define eachedge int i = hd[u]; ~i; i = nx[i]
void init_g(int n){ memset(hd+1,0xff,sInt*n); ec = 0; }
void add_edge(int u,int v,int c)
{
    to[ec] = v;
    we[ec] = c;
    nx[ec] = hd[u];
    hd[u] = ec++;
}

int n,m,k;

int di[maxn], fe[maxn];
typedef pair<int,int> pii;
#define dist first
#define ver second
priority_queue<pii,vector<pii>,greater<pii> > q;

void dijkstra()
{
    memset(di+1,0x3f,sInt*n);

    di[1] = 0; fe[1] = -1;
    q.push(pii(0,1));
    while(!q.empty()){
        pii x = q.top(); q.pop();
        if(x.dist != di[x.ver]) continue;
        int u = x.ver;
        for(int i = hd[u]; ~i; i = nx[i]){
            int v = to[i];
            if(di[v] > di[u] + we[i]){
                di[v] = di[u]+we[i];
                fe[v] = i;
                q.push(pii(di[v],v));
            }
            else if(di[v] == di[u] + we[i] && to[fe[v]^1] > u){
                fe[v] = i;
            }
        }
    }
}


void rewrite(int u,int i)
{
    nx[i] = hd[u];
    hd[u] = i;
}

void build_tree()
{
    dijkstra();
    init_g(n);
    for(int v = 2; v <= n; v++){
        int e = fe[v];
        int u = to[e^1];
        rewrite(u,e);
        rewrite(v,e^1);
    }
}

bool vis_c[maxn];
int tr_sz[maxn];

void cal_tr_sz(int u,int f)
{
    int &c = tr_sz[u];
    c = 1;
    for(eachedge){
        int v = to[i];
        if(v == f || vis_c[v]) continue;
        c += tr_sz[v];
        cal_tr_sz(v,u);
    }
}

int block_size;
int best, centroid;
void findCentroid(int u,int f)
{
    int mx = 0;
    for(eachedge){
        int v = to[i];
        if(v == f || vis_c[v]) continue;
        findCentroid(v,u);
        mx = max(mx, tr_sz[v]);
    }
    mx = max(mx,block_size-tr_sz[u]);
    if(best > mx){
        best = mx;
        centroid = u;
    }
}

typedef unordered_map<int,pii> cont;
typedef cont::iterator con_it;
//key 经过的点数，value(最长距离,方案数)
cont prv;
cont tmp;


void update(cont &res,int c, const pii &np)
{
    con_it it = res.find(c);
    if(it != res.end()){
        pii &p = it->second;
        if(p.dist == np.dist) p.ver += np.ver;
        else if(p.dist < np.dist) {
            p = np;
        }
    }
    else {
        res.insert(make_pair(c,np));
    }
}

void enum_path(int u,int f,int c,int d,cont &res)
{
    if(c >= k) return;
    update(res,c,pii(d,1));

    for(eachedge){
        int v = to[i];
        if(v == f || vis_c[v]) continue;
        enum_path(v,u,c+1,d+we[i],res);
    }
}

int ans;
ll cnt;

void divide(int rt)
{
    cal_tr_sz(rt,-1);
    best = INT_MAX;
    block_size = tr_sz[rt];
    findCentroid(rt,-1);
    int u = centroid;
    vis_c[u] = true;

    for(eachedge){
        if(!vis_c[to[i]]) divide(to[i]);
    }

    prv.clear();
    prv.insert(make_pair(1,pii(0,1)));
    for(eachedge){
        if(vis_c[to[i]]) continue;
        tmp.clear();
        enum_path(to[i],u,1,we[i],tmp);
        con_it it, it2;
        for(it = tmp.begin(); it != tmp.end(); it++){
            int c = it->first;
            pii &p = it->second;
            if((it2 = prv.find(k-c)) != prv.end()){
                ll dis = it2->second.dist + p.dist;
                if(dis > ans){
                    ans = dis; cnt = p.ver * it2->second.ver;
                }
                else if(dis == ans) cnt += p.ver * it2->second.ver;
            }
        }

        for(it = tmp.begin(); it != tmp.end(); it++){
            int c = it->first+1;
            update(prv,c,it->second);
        }
    }

    vis_c[u] = false;
}

void solve()
{
    build_tree();
    ans = cnt = 0;
    divide(1);
    printf("%d %lld
",ans,cnt);
}


void init()
{
    scanf("%d%d%d",&n,&m,&k);
    init_g(n);
    for(int i = 0; i < m; i++){
        int a,b,c; scanf("%d%d%d",&a,&b,&c);
        add_edge(a,b,c);
        add_edge(b,a,c);
    }
}


//#define LOCAL
int main()
{
#ifdef LOCAL
    freopen("in.txt","r",stdin);
#endif
    int T; scanf("%d",&T);
    while(T--){
        init();
        solve();
    }
    return 0;
}