\(\text{A*}\)

一种启发式搜索
和暴搜的差别是多了一个估值函数，每次取出一个估算最优的状态以期更高效完成任务
重点在于估值函数 \(\text{h*(n)}\) 的设计，若实际代价为 \(\text{h(n)}\)，则
若 \(\text{h*(n)=h(n)}\)，设计得非常好
若 \(\text{h*(n)<h(n)}\)，跑得更快，只是可能搜不出来正确答案
若 \(\text{h*(n)>h(n)}\)，慢是慢些，但正确性还有保证

例如 \(k\) 短路
暴搜的话就是找出所有路径，取第 \(k\) 短
但实际上若精准预估当前状态到终止状态的代价，每次取出最小代价状态，
那么 \(k\) 次到达终止状态后，前 \(k\) 短路就已经被搜出来了
显然，估值函数就定为当前状态到终止状态的最短路
这个可以反向建图 \(\text{dijkstra}\) 预处理

\(\text{sample}\)

P2483 【模板】k 短路 / [SDOI2010] 魔法猪学院
\(\text{code}\)

#include <cstdio>
#include <queue>
#define RE register
#define IN inline
using namespace std;

const int N = 5005, M = 2e5 + 5, INF = 1e18;
int n, m, cnt[N];
double e;

struct node{
	int id; double f, v;
	bool operator < (const node &c) const{return f > c.f;}
};
priority_queue<node> Q;

struct graph{
	struct edge{int to, nxt; double w;}e[M];
	int tot, h[N], vis[N]; double dis[N];
	IN void add(int u, int v, double w){e[++tot] = edge{v, h[u], w}, h[u] = tot;}
	IN void getdis()
	{
		for(RE int i = 1; i < n; i++) dis[i] = INF, vis[i] = 0;
		vis[n] = 0, Q.push(node{n, 0});
		while (!Q.empty())
		{
			node x = Q.top(); Q.pop();
			if (vis[x.id]) continue;
			vis[x.id] = 1;
			for(RE int i = h[x.id]; i; i = e[i].nxt)
			{
				int v = e[i].to;
				if (dis[v] > dis[x.id] + e[i].w)
					Q.push(node{v, dis[v] = dis[x.id] + e[i].w});
			}
		}
	}
}e1, e2;

int main()
{
	scanf("%d%d%lf", &n, &m, &e);
	int u, v; double w;
	for(RE int i = 1; i <= m; i++) scanf("%d%d%lf", &u, &v, &w), e1.add(u, v, w), e2.add(v, u, w);
	e2.getdis(), Q.push(node{1, 0});
	int k = (int)e / e2.dis[1], ans = 0;
	while (!Q.empty())
	{
		node x = Q.top(); Q.pop();
		++cnt[x.id];
		if (x.id == n)
		{
			e -= x.v;
			if (e <= 0){printf("%d\n", ans); return 0;}
			++ans;
		}
		for(RE int i = e1.h[x.id]; i; i = e1.e[i].nxt)
		{
			int v = e1.e[i].to;
			if (cnt[v] < k && x.v + e1.e[i].w + e2.dis[v] <= e)
				Q.push(node{v, x.v + e1.e[i].w + e2.dis[v], x.v + e1.e[i].w});
		}
	}
	printf("%d\n", ans);
}

P2901 [USACO08MAR]Cow Jogging G
\(\text{code}\)

#include <cstdio>
#include <queue>
#define RE register
#define IN inline
typedef long long LL;
using namespace std;

const int N = 1005, M = 2e5 + 5, INF = 1e9;
int n, m, k, cnt[N];

struct node{
	int id; LL f, v;
	bool operator < (const node &c) const{return f > c.f;}
};
priority_queue<node> Q;

struct graph{
	struct edge{int to, nxt, w;}e[M];
	int tot, h[N], vis[N]; LL dis[N];
	IN void add(int u, int v, int w){e[++tot] = edge{v, h[u], w}, h[u] = tot;}
	IN void getdis()
	{
		for(RE int i = 2; i <= n; i++) dis[i] = INF, vis[i] = 0;
		vis[1] = 0, Q.push(node{1, 0});
		while (!Q.empty())
		{
			node x = Q.top(); Q.pop();
			if (vis[x.id]) continue;
			vis[x.id] = 1;
			for(RE int i = h[x.id]; i; i = e[i].nxt)
			{
				int v = e[i].to;
				if (dis[v] > dis[x.id] + e[i].w)
					Q.push(node{v, dis[v] = dis[x.id] + e[i].w});
			}
		}
	}
}e1, e2;

int main()
{
	scanf("%d%d%d", &n, &m, &k);
	for(RE int i = 1, u, v, w; i <= m; i++) scanf("%d%d%d", &u, &v, &w), e1.add(u, v, w), e2.add(v, u, w);
	e2.getdis(), Q.push(node{n, 0});
	int sum = 0;
	while (!Q.empty())
	{
		node x = Q.top(); Q.pop();
		++cnt[x.id];
		if (x.id == 1)
		{
			printf("%d\n", x.v), ++sum;
			if (sum == k) return 0;
		}
		for(RE int i = e1.h[x.id]; i; i = e1.e[i].nxt)
		{
			int v = e1.e[i].to;
			if (cnt[v] < k)
				Q.push(node{v, x.v + e1.e[i].w + e2.dis[v], x.v + e1.e[i].w});
		}
	}
	for(RE int i = sum; i < k; i++) printf("-1\n");
}