HDU 4085 Peach Blossom Spring 記憶化搜索枚舉子集 斯坦納樹

HDU 4085 Peach Blossom Spring 記憶化搜索枚舉子集 斯坦納樹

題目鏈接：點擊打開鏈接

題意:

第一行輸入n個點 m條可修建的無向邊 k個人

下面給出修建的邊和修建該邊的花費。

開始時k個人在1-k的每個點上（一個點各一人）

目標:從m條給定邊中修建部分邊使得花費和最小

讓k個人移動到 [n-k+1, n] 後面的k個點上（每個點放一個人）。

思路:

首先就是一道斯坦納樹，還是先求一個dp數組（求解方法：點擊打開鏈接）

dp[i][j] 表示以i為根 ，j為8個點中是否在 i 的子樹裡 時的最小花費。

現在的問題就是如何求答案。

因為一個人到他的目標點這條路徑可能和別人的不連通，也就是多個最小生成樹，

我們枚舉2*k個點哪些點是在一個連通分量裡的，

則對於狀態x中，表示人的二進制是低k位，表示目標點的是高k位，x中1表示這些點是在一個連通分量裡的，

對於這個x的最小花費就是min( dp[ anypoint regard root ][x])

而x必須保證低k位中1的個數 和高k位中1的個數相同（即人數和目標點個數相同）

然後記憶化搜索即可。

#include 
#include 
#include 
#include 
#include 

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.math.BigInteger;
import java.text.DecimalFormat;
import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.Deque;
import java.util.HashMap;
import java.util.Iterator;
import java.util.LinkedList;
import java.util.Map;
import java.util.PriorityQueue;
import java.util.Scanner;
import java.util.Stack;
import java.util.StringTokenizer;
import java.util.TreeMap;
import java.util.TreeSet;
import java.util.Queue;
import java.io.File;
import java.io.FileInputStream;
import java.io.FileNotFoundException;
import java.io.FileOutputStream;
public class Main {
	int n, m, k, ans;
	int[][] dis = new int[N][N], dp = new int[N][1<<10];
	int[] e = new int[10];
	int[] vis = new int[N], mem = new int[1<<10];
	int check(int x){
		int a = 0, b = 0;
		for (int i = 0; i < k; i++)
		if ((x&(1 << i))>0)a++;
		for (int i = 0; i < k; i++)
		if ((x&(1 << (i + k)))>0)b++;
		if (a != b)return -1;
		int ans = inf;
		for (int i = 0; i < n; i++)
			ans = min(ans, dp[i][x]);
		return ans;
	}
	int dfs(int x){
		if (mem[x] != -1)return mem[x];
		int tmp = check(x);
		if (tmp == -1)return mem[x] = inf;
		int ans = tmp;
		for (int i = (x-1)&x; i > 0; i = (i - 1)&x){
			ans = min(ans, dfs(i) + dfs(x - i));
		}
		return mem[x] = ans;
	}
	void floyd(){
		for(int z = 0; z < n; z++)
			for(int j = 0; j < n; j++)
				for(int i = 0; i < n; i++)
					dis[i][j] = min(dis[i][j], dis[j][z] + dis[z][i]);
	}
	void input() throws Exception{
		for(int i = 0; i < n; i++)for(int j = 0; j < n; j++)dis[i][j] = (i==j)?0:inf;
		while(m-->0){
			int u = Int() - 1, v = Int() - 1, d = Int();
			dis[u][v] = dis[v][u] = min(dis[u][v], d);
		}
	}
	void work() throws Exception{
		int T = Int(); 
		while(T-->0){
			n = Int(); m = Int(); k = Int();
			input();
			floyd();
			for (int i = 0; i < k; i++)e[i] = i; for (int i = 0; i < k; i++)e[i + k] = n - k + i;
			for (int i = 0; i < n; i++)for (int j = 0; j < (1 << (2 * k)); j++)dp[i][j] = inf;
			for (int i = 0; i < n; i++){
				for (int j = 0; j < 2 * k; j++)
					dp[i][1 << j] = dis[i][e[j]];
			}
			for (int i = 1; i < (1 << (2 * k)); i++){
				if (0 == (i&(i - 1)))continue;
				for (int j = 0; j < n; j++){
					for (int sub = i; sub > 0; sub = (sub - 1)&i)
						dp[j][i] = min(dp[j][i], dp[j][sub] + dp[j][i-sub]);
				}
				for (int j = 0; j < n; j++)vis[j] = 0;
				for (int j = 0; j < n; j++){
					int a = inf, pos = 0;
					for (int z = 0; z < n; z++)
					if (vis[z] == 0 && dp[z][i] <= a)
						a = dp[pos = z][i];
					vis[pos] = 1;
					for (int z = 0; z < n; z++)
						dp[pos][i] = min(dp[pos][i], dp[z][i] + dis[z][pos]);
				}
			}
			for(int i = 1; i < (1<<(2*k)); i++)mem[i] = -1;
			mem[0] = 0;
			ans = dfs((1<<(2*k))-1);
			if(ans == inf)out.println("No solution");
			else out.println(ans);
		}
	}

    public static void main(String[] args) throws Exception{
        Main wo = new Main();
    	in = new BufferedReader(new InputStreamReader(System.in));
    	out = new PrintWriter(System.out);
  //  	in = new BufferedReader(new InputStreamReader(new FileInputStream(new File("input.txt"))));
  //  	out = new PrintWriter(new File("output.txt"));
        wo.work();
        out.close();
    }

	static int N = 55;
	static int M = 2005;
	DecimalFormat df=new DecimalFormat("0.0000");
	static int inf = (int)1e8;
	static long inf64 = (long) 1e18*2;
	static double eps = 1e-8;
	static double Pi = Math.PI;
	static int mod = 1000000009 ;
	
	private String Next() throws Exception{
    	while (str == null || !str.hasMoreElements())
    	    str = new StringTokenizer(in.readLine());
    	return str.nextToken();
    }
    private int Int() throws Exception{
    	return Integer.parseInt(Next());
    }
    private long Long() throws Exception{
    	return Long.parseLong(Next());
    }
    private double Double() throws Exception{
    	return Double.parseDouble(Next());
    }
    StringTokenizer str;
    static Scanner cin = new Scanner(System.in);
    static BufferedReader in;
    static PrintWriter out;
  /*  
	class Edge{
		int from, to, dis, nex;
		Edge(){}
		Edge(int from, int to, int dis, int nex){
			this.from = from;
			this.to = to;
			this.dis = dis;
			this.nex = nex;
		}
	}
	Edge[] edge = new Edge[M<<1];
	int[] head = new int[N];
	int edgenum;
	void init_edge(){for(int i = 0; i < N; i++)head[i] = -1; edgenum = 0;}
	void add(int u, int v, int dis){
		edge[edgenum] = new Edge(u, v, dis, head[u]);
		head[u] = edgenum++;
	}/**/
	int upper_bound(int[] A, int l, int r, int val) {// upper_bound(A+l,A+r,val)-A;
		int pos = r;
		r--;
		while (l <= r) {
			int mid = (l + r) >> 1;
			if (A[mid] <= val) {
				l = mid + 1;
			} else {
				pos = mid;
				r = mid - 1;
			}
		}
		return pos;
	}

	int Pow(int x, int y) {
		int ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			y >>= 1;
			x = x * x;
		}
		return ans;
	}
	double Pow(double x, int y) {
		double ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			y >>= 1;
			x = x * x;
		}
		return ans;
	}
	int Pow_Mod(int x, int y, int mod) {
		int ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			ans %= mod;
			y >>= 1;
			x = x * x;
			x %= mod;
		}
		return ans;
	}
	long Pow(long x, long y) {
		long ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			y >>= 1;
			x = x * x;
		}
		return ans;
	}
	long Pow_Mod(long x, long y, long mod) {
		long ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			ans %= mod;
			y >>= 1;
			x = x * x;
			x %= mod;
		}
		return ans;
	}

	int gcd(int x, int y){
		if(x>y){int tmp = x; x = y; y = tmp;}
		while(x>0){
			y %= x;
			int tmp = x; x = y; y = tmp;
		}
		return y;
	}
	int max(int x, int y) {
		return x > y ? x : y;
	}

	int min(int x, int y) {
		return x < y ? x : y;
	}

	double max(double x, double y) {
		return x > y ? x : y;
	}

	double min(double x, double y) {
		return x < y ? x : y;
	}

	long max(long x, long y) {
		return x > y ? x : y;
	}

	long min(long x, long y) {
		return x < y ? x : y;
	}

	int abs(int x) {
		return x > 0 ? x : -x;
	}

	double abs(double x) {
		return x > 0 ? x : -x;
	}

	long abs(long x) {
		return x > 0 ? x : -x;
	}

	boolean zero(double x) {
		return abs(x) < eps;
	}
	double sin(double x){return Math.sin(x);}
	double cos(double x){return Math.cos(x);}
	double tan(double x){return Math.tan(x);}
	double sqrt(double x){return Math.sqrt(x);}
}