[HDU - 6005 ]-Pandaland(最小環，MST+LCA)
鏈接：
題面：
思路：
將二維平面上的節點轉給一個一般圖中的節點，可以發現這是一個平面圖。
分析可得性質：一個圖的最小環（sum{cost} is minimal）一定是該圖最小生成樹上添一個邊。
於是我們可以對給定的圖求出最小生成樹的森林（因爲給定的圖不一定聯通，所以應該是一個森林）、。
然後我們枚舉不在森林中的邊，用樹的lca算法可以求出該邊的兩點在樹上的路徑距離。
該邊的兩點在樹上的路徑再填上該邊後，一定組成一個環的。
於是我們可以對所有這種環取出最小值，該值就是圖的最小環權值。
代碼:
#include <iostream>
#include <cstdio>
#include <algorithm>
#include <bits/stdc++.h>
#define ALL(x) (x).begin(), (x).end()
#define sz(a) int(a.size())
#define rep(i,x,n) for(int i=x;i<n;i++)
#define repd(i,x,n) for(int i=x;i<=n;i++)
#define pii pair<int,int>
#define pll pair<long long ,long long>
#define gbtb ios::sync_with_stdio(false),cin.tie(0),cout.tie(0)
#define MS0(X) memset((X), 0, sizeof((X)))
#define MSC0(X) memset((X), '\0', sizeof((X)))
#define pb push_back
#define mp make_pair
#define fi first
#define se second
#define eps 1e-6
#define chu(x) if(DEBUG_Switch) cout<<"["<<#x<<" "<<(x)<<"]"<<endl
#define du3(a,b,c) scanf("%d %d %d",&(a),&(b),&(c))
#define du2(a,b) scanf("%d %d",&(a),&(b))
#define du1(a) scanf("%d",&(a));
using namespace std;
typedef long long ll;
ll gcd(ll a, ll b) {return b ? gcd(b, a % b) : a;}
ll lcm(ll a, ll b) {return a / gcd(a, b) * b;}
ll powmod(ll a, ll b, ll MOD) { if (a == 0ll) {return 0ll;} a %= MOD; ll ans = 1; while (b) {if (b & 1) {ans = ans * a % MOD;} a = a * a % MOD; b >>= 1;} return ans;}
ll poww(ll a, ll b) { if (a == 0ll) {return 0ll;} ll ans = 1; while (b) {if (b & 1) {ans = ans * a ;} a = a * a ; b >>= 1;} return ans;}
void Pv(const vector<int> &V) {int Len = sz(V); for (int i = 0; i < Len; ++i) {printf("%d", V[i] ); if (i != Len - 1) {printf(" ");} else {printf("\n");}}}
void Pvl(const vector<ll> &V) {int Len = sz(V); for (int i = 0; i < Len; ++i) {printf("%lld", V[i] ); if (i != Len - 1) {printf(" ");} else {printf("\n");}}}
inline long long readll() {long long tmp = 0, fh = 1; char c = getchar(); while (c < '0' || c > '9') {if (c == '-') fh = -1; c = getchar();} while (c >= '0' && c <= '9') tmp = tmp * 10 + c - 48, c = getchar(); return tmp * fh;}
inline int readint() {int tmp = 0, fh = 1; char c = getchar(); while (c < '0' || c > '9') {if (c == '-') fh = -1; c = getchar();} while (c >= '0' && c <= '9') tmp = tmp * 10 + c - 48, c = getchar(); return tmp * fh;}
void pvarr_int(int *arr, int n, int strat = 1) {if (strat == 0) {n--;} repd(i, strat, n) {printf("%d%c", arr[i], i == n ? '\n' : ' ');}}
void pvarr_LL(ll *arr, int n, int strat = 1) {if (strat == 0) {n--;} repd(i, strat, n) {printf("%lld%c", arr[i], i == n ? '\n' : ' ');}}
const int maxn = 1e4 + 10;
struct node
{
int f, t, w;
node() {}
node(int ff, int tt, int ww)
{
f = ff; t = tt; w = ww;
}
bool operator < (const node& b) const
{
return w < b.w;
}
};
map<pii, int> vis;
int far[maxn];
int dsu_sz[maxn];
void dsu_init(int n)
{
repd(i, 0, n) {
far[i] = i;
dsu_sz[i] = 1;
}
}
int findpar(int x)
{
if (x == far[x]) {
return x;
} else {
return far[x] = findpar(far[x]);
}
}
void mg(int x, int y)
{
x = findpar(x);
y = findpar(y);
if (x == y) {
return;
}
if (dsu_sz[x] > dsu_sz[y]) {
dsu_sz[x] += dsu_sz[y];
far[y] = x;
} else {
dsu_sz[y] += dsu_sz[x];
far[x] = y;
}
}
vector<node> v, others;
std::vector<pii> son[maxn];
void mst()
{
sort(v.begin(), v.end());
for (auto &now : v)
{
int x = now.f;
int y = now.t;
int w = now.w;
if (findpar(x) == findpar(y))
{
others.push_back(now);
continue;
}
mg(x, y);
son[x].push_back(mp(y, w));
son[y].push_back(mp(x, w));
}
}
int depth[maxn], fa[maxn][21];
ll dist[maxn];
void dfs(int rt, int prev, int dis)
{
depth[rt] = depth[prev] + 1;
dist[rt] = dist[prev] + dis;
fa[rt][0] = prev;
for (int i = 1; i < 20; i++)
{
fa[rt][i] = fa[fa[rt][i - 1]][i - 1];
}
for (int i = 0; i < son[rt].size(); i++)
{
if (son[rt][i].fi == prev)
continue;
dfs(son[rt][i].fi, rt, son[rt][i].se);
}
}
int LCA(int x, int y)
{
if (depth[x] < depth[y])
swap(x, y);
for (int i = 19; i >= 0; i--)
{
if (depth[x] - (1 << i) >= depth[y])
{
x = fa[x][i];
}
}
if (x == y)
{
return x;
}
for (int i = 19; i >= 0; i--)
{
if (fa[x][i] != fa[y][i])
{
x = fa[x][i];
y = fa[y][i];
}
}
return fa[x][0];
}
ll finddist(int a, int b)
{
ll u = LCA(a, b);
ll L = dist[a] + dist[b] - 2 * dist[u];
return L;
}
const ll inf = 1e18;
int main()
{
int t;
scanf("%d", &t);
for (int icase = 1; icase <= t; ++icase)
{
int cnt = 0;
vis.clear();
v.clear();
others.clear();
int m;
scanf("%d", &m);
dsu_init(m * 2);
for (int i = 1; i <= m; ++i)
{
int x1, x2, y1, y2, w;
scanf("%d %d %d %d %d", &x1, &y1, &x2, &y2, &w);
int l, r;
if (vis.count(mp(x1, y1)) == 0)
{
vis[mp(x1, y1)] = ++cnt;
}
if (vis.count(mp(x2, y2)) == 0)
{
vis[mp(x2, y2)] = ++cnt;
}
l = vis[mp(x1, y1)];
r = vis[mp(x2, y2)];
v.push_back(node(l, r, w));
}
mst();
repd(i, 1, cnt)
{
if (!depth[i])
{
dfs(i, 0, 0);
}
}
ll ans = inf;
for (auto now : others)
{
ll num = finddist(now.t, now.f);
num += now.w;
ans = min(ans, num);
}
if (others.size() == 0)
{
ans = 0;
}
printf("Case #%d: %lld\n", icase, ans );
for (int i = 0; i <= cnt; ++i)
{
depth[i]=0;
son[i].clear();
}
}
return 0;
}