HDU 3681 Prison Break（BFS+二分+状态压缩DP）

Problem Description

Rompire is a robot kingdom and a lot of robots live there peacefully. But one day, the king of Rompire was captured by human beings. His thinking circuit was changed by human and thus became a tyrant. All those who are against him were put into jail, including our clever Micheal#1. Now it’s time to escape, but Micheal#1 needs an optimal plan and he contacts you, one of his human friends, for help.
The jail area is a rectangle contains n×m little grids, each grid might be one of the following: 
1) Empty area, represented by a capital letter ‘S’. 
2) The starting position of Micheal#1, represented by a capital letter ‘F’. 
3) Energy pool, represented by a capital letter ‘G’. When entering an energy pool, Micheal#1 can use it to charge his battery ONLY ONCE. After the charging, Micheal#1’s battery will become FULL and the energy pool will become an empty area. Of course, passing an energy pool without using it is allowed.
4) Laser sensor, represented by a capital letter ‘D’. Since it is extremely sensitive, Micheal#1 cannot step into a grid with a laser sensor. 
5) Power switch, represented by a capital letter ‘Y’. Once Micheal#1 steps into a grid with a Power switch, he will certainly turn it off. 

In order to escape from the jail, Micheal#1 need to turn off all the power switches to stop the electric web on the roof—then he can just fly away. Moving to an adjacent grid (directly up, down, left or right) will cost 1 unit of energy and only moving operation costs energy. Of course, Micheal#1 cannot move when his battery contains no energy. 

The larger the battery is, the more energy it can save. But larger battery means more weight and higher probability of being found by the weight sensor. So Micheal#1 needs to make his battery as small as possible, and still large enough to hold all energy he need. Assuming that the size of the battery equals to maximum units of energy that can be saved in the battery, and Micheal#1 is fully charged at the beginning, Please tell him the minimum size of the battery needed for his Prison break.

 

Input

Input contains multiple test cases, ended by 0 0. For each test case, the first line contains two integer numbers n and m showing the size of the jail. Next n lines consist of m capital letters each, which stands for the description of the jail.You can assume that 1<=n,m<=15, and the sum of energy pools and power switches is less than 15.

 

Output

For each test case, output one integer in a line, representing the minimum size of the battery Micheal#1 needs. If Micheal#1 can’t escape, output -1.

 

题目大意：略。太麻烦了不写了。

思路：首先此题显然可以二分答案然后判定，于是题目转化为判定问题，给定能量energy，问这个能量能否走完所有的Y。

注意到有意义的点其实只有F、G、Y，可以事先进行数次BFS，把他们之间的最短路径找出来。

注意到，Y+G不超过15，那么可以用状态压缩，记录每个点是否走过，现在在哪一个点上。用一个数组记录在某个状态下剩余的能量最多是多少。然后随便搞搞就可以水了。

PS：没有Y的时候应该是输出0吧。走到G的时候，就算没有能量了，也能充能。走到最后一个Y的时候，就算没有能量了，也能脱出。

 

代码（250MS）：

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cstring>
#include <queue>
using namespace std;
typedef long long LL;

const int MAXN = 1 << 16;
const int MAXV = 20;

int fx[] = {-1, 0, 1, 0};
int fy[] = {0, 1, 0, -1};

char mat[MAXV][MAXV];
int id[MAXV][MAXV], dis[MAXV][MAXV], Gcnt, Ycnt;
int n, m;

bool isOn(int state, int i) {
    return (state >> i) & 1;
}

int moveState(int state, int i) {
    return state ^ (1 << i);
}

bool atEnd(int state) {
    return state % (1 << Ycnt) == 0;
}

bool isGYF(char c) {
    return c == 'G' || c == 'Y' || c == 'F';
}

int vis[MAXV][MAXV];

void bfsdis(int x, int y) {
    memset(vis, 0x7f, sizeof(vis));
    vis[x][y] = 0;
    queue<pair<int, int> > que;
    que.push(make_pair(x, y));
    int st = id[x][y];
    while(!que.empty()) {
        int x = que.front().first, y = que.front().second; que.pop();
        if(isGYF(mat[x][y])) dis[st][id[x][y]] = vis[x][y];
        for(int i = 0; i < 4; ++i) {
            int tx = x + fx[i], ty = y + fy[i];
            if(mat[tx][ty] != 'D' && vis[x][y] + 1 < vis[tx][ty]) {
                vis[tx][ty] = vis[x][y] + 1;
                que.push(make_pair(tx, ty));
            }
        }
    }
}

void init() {
    Ycnt = 0;
    for(int i = 1; i <= n; ++i) {
        for(int j = 1; j <= m; ++j) {
            if(mat[i][j] != 'Y') continue;
            id[i][j] = Ycnt++;
        }
    }
    Gcnt = Ycnt;
    for(int i = 1; i <= n; ++i) {
        for(int j = 1; j <= m; ++j) {
            if(mat[i][j] != 'G') continue;
            id[i][j] = Gcnt++;
        }
    }
    for(int i = 1; i <= n; ++i) {
        for(int j = 1; j <= m; ++j) {
            if(mat[i][j] != 'F') continue;
            id[i][j] = Gcnt;
        }
    }
    memset(dis, 0x7f, sizeof(dis));
    for(int i = 1; i <= n; ++i) {
        for(int j = 1; j <= m; ++j) {
            if(!isGYF(mat[i][j])) continue;
            bfsdis(i, j);
        }
    }
}

int best[MAXN][MAXV];
bool inque[MAXN][MAXV];

bool check(int energy) {
    memset(inque, 0, sizeof(inque));
    memset(best, 0, sizeof(best));
    best[(1 << Gcnt) - 1][Gcnt] = energy;
    queue<pair<int, int> > que;
    que.push(make_pair((1 << Gcnt) - 1, Gcnt));
    while(!que.empty()) {
        int state = que.front().first, pos = que.front().second; que.pop();
        inque[state][pos] = false;
        for(int i = 0; i < Gcnt; ++i) {
            if(!isOn(state, i)) continue;
            int newState = moveState(state, i);
            if(i >= Ycnt) {
                if(best[state][pos] >= dis[pos][i] && best[newState][i] == 0) {
                    best[newState][i] = energy;
                    inque[newState][i] = true;
                    que.push(make_pair(newState, i));
                }
            } else {
                if(best[state][pos] - dis[pos][i] >= 0 && atEnd(newState)) return true;
                if(best[state][pos] - dis[pos][i] > best[newState][i]) {
                    best[newState][i] = best[state][pos] - dis[pos][i];
                    if(!inque[newState][i]) {
                        inque[newState][i] = true;
                        que.push(make_pair(newState, i));
                    }
                }
            }
        }
    }
    return false;
}

int solve() {
    if(Ycnt == 0) return 0;
    int st = Gcnt;
    for(int i = 0; i < Ycnt; ++i)
        if(dis[i][st] > 2014) return -1;
    int l = 1, r = dis[st][0];
    for(int i = 0; i < Ycnt - 1; ++i) r += dis[i][i + 1];
    while(l < r) {
        int mid = (l + r) >> 1;
        if(!check(mid)) l = mid + 1;
        else r = mid;
    }
    return l;
}

int main() {
    while(scanf("%d%d", &n, &m) != EOF) {
        if(n == 0 && m == 0) break;
        memset(mat, 'D', sizeof(mat));
        for(int i = 1; i <= n; ++i) scanf("%s", &mat[i][1]);
        init();
        printf("%d\n", solve());
    }
}