[hdu 6166 Senior Pan]Dijkstra+概率随机

分类：Dijkstra probability data structure

1. 题目链接

[hdu 6166Senior Pan]

2. 题意描述

有一个n个顶点m点边的有向带权图。已知从其中选出的K个点。在这K个点中选出两个点，要使得他们的最短路最小。
数据范围：1≤数据组数≤5, 1≤n,m≤100000,2≤K≤n，1≤边权≤100000。

3. 解题思路

方法一：假设最终的答案对应的两个端点为s,t, 将K个点分成两半，那么，有四种情况:
1. s,t在前一半;
2. s,t在后一半;
3. s在第一半t在第二半;
4. t在第一半s在第二半。
如果是Solution 3或者Solution 4， 只需要将一个超级源点连接所有前一半的点，边权为0，跑一遍Dijkstra。就可以求出前一半点集到后一半点集的最短路，这样就可以得到答案了。
如果是Solution 1或者Solution 2， 按照上面的做法是得不到答案的。
这样，一次Dijkstra跑出答案的概率为12，那么重复x次，成功的概率为1−12k ，当x=20，可以依概率接近1求出答案。
这题，随机技巧性很强。
方法二：只需要一次Dijkstra就可以了。选一个超级源点，并连接所有K个点，权值为0。Dijkstra算法中，每次枚举边的同时，顺便更新答案就好了。感觉这个挺容易写错的。

Tips：将超级源点连接到点集V,并不一定非得新建一个顶点，只需要将点集V中所有点一起放入队列，并将到该点的最短路径置为0就好了。

4. 实现代码

/**方法一**/#include <set>#include <map>#include <queue>#include <stack>#include <ctime>#include <cmath>#include <cctype>#include <cstdio>#include <string>#include <cstring>#include <cassert>#include <cstdlib>#include <iomanip>#include <iostream>#include <algorithm>using namespace std;typedef long long LL;typedef long double LB;typedef unsigned int uint;typedef unsigned long long ULL;typedef pair<int, int> PII;typedef pair<LL, LL> PLL;typedef pair<LB, LB> PLB;typedef vector<int> VI;const int INF = 0x3f3f3f3f;const LL INFL = 0x3f3f3f3f3f3f3f3fLL;const long double PI = acos(-1.0);const long double eps = 1e-4;void debug() { cout << endl; }template<typename T, typename ...R> void debug (T f, R ...r) { cout << "[" << f << "]"; debug (r...); }template<typename T> inline void umax(T &a, T b) { a = max(a, b); }template<typename T> inline void umin(T &a, T b) { a = min(a, b); }template <typename T> inline bool scan_d (T &ret) {    char c; int sgn;    if (c = getchar(), c == EOF) return 0; //EOF    while (c != '-' && (c < '0' || c > '9') ) if((c = getchar()) == EOF) return 0;    sgn = (c == '-') ? -1 : 1;    ret = (c == '-') ? 0 : (c - '0');    while (c = getchar(), c >= '0' && c <= '9') ret = ret * 10 + (c - '0');    ret *= sgn;    return 1;}template<typename T> void print(T x) {    static char s[33], *s1; s1 = s;    if (!x) *s1++ = '0';    if (x < 0) putchar('-'), x = -x;    while(x) *s1++ = (x % 10 + '0'), x /= 10;    while(s1-- != s) putchar(*s1);}template<typename T> void println(T x) { print(x); putchar('\n'); }template<typename T> T randIntv(T a, T b) { return rand() % (b - a + 1) + a; } /*[a, b]*/const int MAXN = 100005;const int MAXE = 100005;int T, n, m, K;int a[MAXN]; LL ans;template<class T>struct Dijkstra {    struct Edge {        T w;        int v, nxt;    } E[MAXE << 1];    typedef pair<T, int> QNode;    int Head[MAXN], erear;    T d[MAXN], INF;    void init() {        erear = 0;        memset(Head, -1, sizeof(Head));    }    void add_edge(int u, int v, T w) {        E[erear].v = v;        E[erear].w = w;        E[erear].nxt = Head[u];        Head[u] = erear++;    }    T run() {        memset(d, 0x3f, sizeof(d));        INF = d[0];        priority_queue<QNode, vector<QNode>, greater<QNode> >Q;        for(int i = 1; i <= K / 2; ++i) Q.push(QNode(0, a[i])); d[a[i]] = 0;        while(!Q.empty()) {            QNode ftp = Q.top(); Q.pop();            int u = ftp.second;            if(ftp.first != d[u]) continue;            for(int i = Head[u]; ~i; i = E[i].nxt) {                int v = E[i].v; T w = E[i].w;                if(d[u] + w < d[v]) {                    d[v] = d[u] + w;                    Q.push(QNode(d[v], v));                }            }        }        T ret = INF;        for(int i = K / 2 + 1; i <= K; ++i) umin(ret, d[a[i]]);        return ret;    }};Dijkstra<LL> dij;int main() {#ifdef ___LOCAL_WONZY___    freopen ("input.txt", "r", stdin);#endif // ___LOCAL_WONZY___    int cas = 0, u, v, w;    scan_d(T);    while(T --) {        scan_d(n), scan_d(m);        dij.init();        for(int i = 1; i <= m; ++i) {            scan_d(u), scan_d(v), scan_d(w);            dij.add_edge(u, v, w);        }        scan_d(K);        for(int i = 1; i <= K; ++i) {            scan_d(a[i]);        }        ans = INFL;        int times = 20;        while(times --) {            random_shuffle(a + 1, a + K + 1);//            for_each(a + 1, a + K + 1, [](int item) { cout << item << " "; }); puts("");            umin(ans, dij.run());        }        printf("Case #%d: %lld\n", ++cas, ans);    }#ifdef ___LOCAL_WONZY___    cout << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC * 1000 << " ms." << endl;#endif // ___LOCAL_WONZY___    return 0;}

/**方法二**/#include <set>#include <map>#include <queue>#include <stack>#include <ctime>#include <cmath>#include <cctype>#include <cstdio>#include <string>#include <cstring>#include <cassert>#include <cstdlib>#include <iomanip>#include <iostream>#include <algorithm>using namespace std;typedef long long LL;typedef long double LB;typedef unsigned int uint;typedef unsigned long long ULL;typedef pair<int, int> PII;typedef pair<LL, LL> PLL;typedef pair<LB, LB> PLB;typedef vector<int> VI;const int INF = 0x3f3f3f3f;const LL INFL = 0x3f3f3f3f3f3f3f3fLL;const long double PI = acos(-1.0);const long double eps = 1e-4;void debug() { cout << endl; }template<typename T, typename ...R> void debug (T f, R ...r) { cout << "[" << f << "]"; debug (r...); }template<typename T> inline void umax(T &a, T b) { a = max(a, b); }template<typename T> inline void umin(T &a, T b) { a = min(a, b); }template <typename T> inline bool scan_d (T &ret) {    char c; int sgn;    if (c = getchar(), c == EOF) return 0; //EOF    while (c != '-' && (c < '0' || c > '9') ) if((c = getchar()) == EOF) return 0;    sgn = (c == '-') ? -1 : 1;    ret = (c == '-') ? 0 : (c - '0');    while (c = getchar(), c >= '0' && c <= '9') ret = ret * 10 + (c - '0');    ret *= sgn;    return 1;}template<typename T> void print(T x) {    static char s[33], *s1; s1 = s;    if (!x) *s1++ = '0';    if (x < 0) putchar('-'), x = -x;    while(x) *s1++ = (x % 10 + '0'), x /= 10;    while(s1-- != s) putchar(*s1);}template<typename T> void println(T x) { print(x); putchar('\n'); }template<typename T> T randIntv(T a, T b) { return rand() % (b - a + 1) + a; } /*[a, b]*/const int MAXN = 100005;const int MAXE = 100005;int T, n, m, K;int a[MAXN]; LL ans;bool flag[MAXN];template<class T>struct Dijkstra {    struct Edge {        T w;        int v, nxt;    } E[MAXE << 1];    typedef tuple<T, int, int> QNode;    int Head[MAXN], erear;    T d[MAXN], INF;    void init() {        erear = 0;        memset(Head, -1, sizeof(Head));    }    void add_edge(int u, int v, T w) {        E[erear].v = v;        E[erear].w = w;        E[erear].nxt = Head[u];        Head[u] = erear++;    }    T run() {        memset(d, 0x3f, sizeof(d));        INF = d[0];        priority_queue<QNode, vector<QNode>, greater<QNode> >Q;        for(int i = 1; i <= K; ++i) Q.push(QNode(0, a[i], a[i])), d[a[i]] = 0;        T ret = INF;        T cost; int u, fu;        while(!Q.empty()) {            tie(cost, u, fu) = Q.top(); Q.pop();            if(cost != d[u]) continue;            for(int i = Head[u]; ~i; i = E[i].nxt) {                int v = E[i].v; T w = E[i].w;                if(flag[v] && v != fu) umin(ret, d[u] + w); /**注意v!=fu的条件**/                if(d[u] + w < d[v]) {                    d[v] = d[u] + w;                    Q.push(QNode(d[v], v, flag[v] ? v : fu));                }            }        }        return ret;    }};Dijkstra<LL> dij;int main() {#ifdef ___LOCAL_WONZY___    freopen ("input.txt", "r", stdin);#endif // ___LOCAL_WONZY___    int cas = 0, u, v, w;    scan_d(T);    while(T --) {        scan_d(n), scan_d(m);        dij.init();        for(int i = 1; i <= m; ++i) {            scan_d(u), scan_d(v), scan_d(w);            dij.add_edge(u, v, w);        }        scan_d(K);        memset(flag, false, sizeof(flag));        for(int i = 1; i <= K; ++i) {            scan_d(a[i]);            flag[a[i]] = true;        }        ans = dij.run();        printf("Case #%d: %lld\n", ++cas, ans);    }#ifdef ___LOCAL_WONZY___    cout << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC * 1000 << " ms." << endl;#endif // ___LOCAL_WONZY___    return 0;}