1. 程式人生 > 實用技巧 >POJ1679 The Unique MST

POJ1679 The Unique MST

題目大意是給你一個無向圖,讓你判斷該圖的最小生成樹(MST)是否唯一。
要判斷最小生成樹是否唯一,只需判斷次小生成樹和最小生成樹的大小關係即可。要求次小生成樹,就得先求最小生成樹,在最小生成樹的基礎上進行列舉非MST邊,每加一條非MST邊,就會形成一個環,然後求這個環內的最大邊權,依次列舉求最小值即可,這個環就是從非MST邊的端點到兩端點的LCA,所以要求一下LCA,順便維護一下樹上ST表,進行快速的查詢最大值。

#pragma GCC optimize(2)
#include <algorithm>
#include <iostream>
#include <cstring>
#include <cstdio>
#include <vector>
#include <cmath>
#include <map>
using namespace std;
typedef long long ll;
#define sc(z) scanf("%d", &(z))
#define _ff(i, a, b) for (int i = (a); i <= (b); ++i)
#define _f(i, a, b) for (int i = (a); i < (b); ++i)
#define _rr(i, a, b) for (int i = (a); i >= (b); --i)
#define _r(i, a, b) for (int i = (a); i > (b); --i)
const int inf = 0x3f3f3f3f;

const int maxn = 105;
int mp[maxn][maxn];
vector<int> G[maxn];
struct node{
    int from, to, id, w;
}e[maxn * maxn];
vector<node> C;
int dep[maxn], anc[maxn][25], v[maxn * maxn];
int mx[maxn][25];
int n, m, tot, fa[maxn], f[maxn];

void dfs(int u, int v) {
    _ff(i, 1, 20) {
        anc[u][i] = anc[anc[u][i - 1]][i - 1];
        mx[u][i] = max(mx[u][i - 1], mx[anc[u][i - 1]][i - 1]);
    }
    _f(i, 0, G[u].size()) {
		int k = G[u][i];
        if (k == v) continue;
		anc[k][0] = u;
		dep[k] = dep[u] + 1;
        mx[k][0] = mp[k][u];
		dfs(k, u);
	}
}

int lca(int u, int v) {
    if (dep[u] < dep[v]) swap(u, v);
    _rr(i, 20, 0) {
        if (dep[anc[u][i]] >= dep[v])
        	u = anc[u][i];
    }
    if (u == v) return v;
    _rr(i, 20, 0) {
        if (anc[u][i] != anc[v][i])
            u = anc[u][i], v = anc[v][i];
    }
    return anc[u][0];
}

bool cmp(node a, node b) {
    return a.w < b.w;
}

int find_fa(int x) {
    while (x != fa[x]) x = fa[x] = fa[fa[x]];
    return x;
}

int qmax(int u, int v) {
    if (u == v) return 0;
    int k = log(dep[u] - dep[v] + 1);
    return mx[u][k];
}

void solve() {
    sort(e + 1, e + 1 + tot, cmp);
    _ff(i, 1, n) fa[i] = i;
    int cnt = 0, minn = 0, ans = inf;
    _ff(i, 1, n) fa[i] = i;
    _ff(i, 1, tot) {
        int x = find_fa(e[i].from), y = find_fa(e[i].to);
        if (x == y) {
            C.push_back(e[i]);
            continue;
        }
        fa[x] = y;
        G[e[i].from].push_back(e[i].to);
        G[e[i].to].push_back(e[i].from);
        minn += e[i].w;
        v[e[i].id>>1] = 1;
        if (++cnt == n - 1) break;
    }
    _ff(i, cnt + 1, tot) C.push_back(e[i]);
    dep[1] = 0, dep[0] = -1;
    dfs(1, 0);
    _f(i, 0, C.size()) {
        if (v[C[i].id>>1]) continue;
        int fr = C[i].from, to = C[i].to;
        int ca = lca(fr, to);
        int tmp = max(qmax(fr, ca), qmax(to, ca));
        ans = min(ans, minn - tmp + C[i].w);
        v[C[i].id>>1] = 1;
    }
    if (ans == minn) cout << "Not Unique!" << endl;
    else printf("%d\n", minn);
}

signed main() {
    int kase; scanf("%d", &kase);
    while (kase--) {
        tot = 0;
        int x, y, z; scanf("%d%d", &n, &m);
        _ff(i, 1, m) {
            scanf("%d%d%d", &x, &y, &z);
            ++tot;
            e[tot] = {x, y, tot + 1, z};
            ++tot;
            e[tot] = {y, x, tot + 1, z};
            mp[x][y] = mp[y][x] = z;
        }
        solve();
        C.clear();
        _ff(i, 1, n) G[i].clear();
        memset(v, 0, sizeof(v));
        memset(mx, 0, sizeof(mx));
    }
    return 0;
}