POJ2195 Going Home 【最小费用流】+【二分图最佳匹配】

Description

On a grid map there are n little men and n houses. In each unit time, every little man can move one unit step, either horizontally, or vertically, to an adjacent point. For each little man, you need to pay a $1 travel fee for every step he moves, until he enters a house. The task is complicated with the restriction that each house can accommodate only one little man. 

Your task is to compute the minimum amount of money you need to pay in order to send these n little men into those n different houses. The input is a map of the scenario, a ‘.‘ means an empty space, an ‘H‘ represents a house on that point, and am ‘m‘ indicates there is a little man on that point. 

You can think of each point on the grid map as a quite large square, so it can hold n little men at the same time; also, it is okay if a little man steps on a grid with a house without entering that house.

Input

Output

Sample Input

2 2
.m
H.
5 5
HH..m
.....
.....
.....
mm..H
7 8
...H....
...H....
...H....
mmmHmmmm
...H....
...H....
...H....
0 0

Sample Output

2
10
28

Source

题意：给定一张N*M的图，其中‘.’为空地，小人可以走，‘m’为小人，‘H’为房子，小人可以路过，小人一次只能沿着上下左右走一个格子。现在要求每个小人都进入一个不同的房子，求小人走的步数最小数。

题解：这题可以看成最小费用流来解，其中小人到房子的距离为费用，容量为1，设置源点到每个小人的容量为1，费用为0，每个房子到汇点费用为0，容量为1，剩下的就是求最小费用流了。

版本一：最小费用流:125ms

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <queue>

const int maxn = 205;
const int inf = 0x3f3f3f3f;
char str[maxn];
int head[maxn], n, m; // n rows, m columns
struct Node {
    int x, y;
} A[maxn], B[maxn];
int id1, id2, id, source, sink;
struct node {
    int f, c, u, v, next;
} E[maxn * maxn];
bool vis[maxn];
int pre[maxn], dist[maxn];

void addEdge(int u, int v, int c, int f) {
    E[id].u = u; E[id].v = v;
    E[id].c = c; E[id].f = f;
    E[id].next = head[u]; head[u] = id++;

    E[id].u = v; E[id].v = u;
    E[id].c = 0; E[id].f = -f;
    E[id].next = head[v]; head[v] = id++;
}

void getMap() {
    int i, j, dis; Node e; 
    id = id1 = id2 = 0;
    for(i = 0; i < n; ++i) {
        scanf("%s", str);
        for(j = 0; str[j] != '\0'; ++j) {
            if(str[j] == '.') continue;
            e.x = i; e.y = j;
            if(str[j] == 'm') A[id1++] = e;
            else B[id2++] = e;
        }
    }

    memset(head, -1, sizeof(head));
    source = id1 + id2; sink = source + 1;
    for(i = 0; i < id1; ++i) {
        for(j = 0; j < id2; ++j) {
            dis = abs(A[i].x - B[j].x) + abs(A[i].y - B[j].y);
            addEdge(i, id1 + j, 1, dis); // uvcf
        }
        addEdge(source, i, 1, 0);
    }
    for(j = 0; j < id2; ++j)
        addEdge(id1 + j, sink, 1, 0);
}

bool SPFA(int start, int end) {
    std::queue<int> Q; int i, u, v;
    memset(vis, 0, sizeof(vis));
    memset(pre, -1, sizeof(pre));
    memset(dist, 0x3f, sizeof(pre));
    Q.push(start); vis[start] = 1; dist[start] = 0;
    while(!Q.empty()) {
        u = Q.front(); Q.pop();
        vis[u] = 0;
        for(i = head[u]; i != -1; i = E[i].next) {
            v = E[i].v;
            if(E[i].c && dist[v] > dist[u] + E[i].f) {
                dist[v] = dist[u] + E[i].f;
                pre[v] = i;
                if(!vis[v]) {
                    Q.push(v); vis[v] = 1;
                }
            }
        }
    }
    return dist[end] != inf;
}

int Min_Cost_Flow(int start, int end) {
    int ans_cost = 0, u, minCut;
    while(SPFA(start, end)) {
        minCut = inf;
        for(u = pre[end]; u != -1; u = pre[E[u].u]) {
            if(minCut > E[u].c) minCut = E[u].c;
        }
        for(u = pre[end]; u != -1; u = pre[E[u].u]) {
            E[u].c -= minCut; E[u^1].c += minCut;
        }
        ans_cost += minCut * dist[end];
    }
    return ans_cost;
}

void solve() {
    printf("%d\n", Min_Cost_Flow(source, sink));
}

int main() {
    // freopen("stdin.txt", "r", stdin);
    while(scanf("%d%d", &n, &m), n | m) {
        getMap();
        solve();
    }
    return 0;
}

版本二：KM：0ms

#include <stdio.h>
#include <string.h>
#include <stdlib.h>

const int maxn = 105;
const int largeNum = 210;
const int inf = 0x3f3f3f3f;
int n, m; // n rows, m columns
char str[maxn];
struct Node {
    int x, y;
} A[maxn], B[maxn];
int id1, id2;
int G[maxn][maxn];
int Lx[maxn], Ly[maxn];
int match[maxn];
bool visx[maxn], visy[maxn];
int slack[maxn];

void getMap() {
    int i, j, dis; Node e;
    id1 = id2 = 0;
    for(i = 0; i < n; ++i) {
        scanf("%s", str);
        for(j = 0; str[j] != '\0'; ++j) {
            if(str[j] == '.') continue;
            e.x = i; e.y = j;
            if(str[j] == 'm') A[id1++] = e;
            else B[id2++] = e;
        }
    }
    memset(G, 0, sizeof(G));
    for(i = 0; i < id1; ++i) {
        for(j = 0; j < id2; ++j) {
            G[i][j] = largeNum - (abs(A[i].x - B[j].x) + abs(A[i].y - B[j].y));
        }
    }
}

bool DFS(int cur) {
    int t, y;
    visx[cur] = true;
    for(y = 0; y < id2; ++y) {
        if(visy[y]) continue;
        t = Lx[cur] + Ly[y] - G[cur][y];
        if(t == 0) {
            visy[y] = true;
            if(match[y] == -1 || DFS(match[y])) {
                match[y] = cur; return true;
            }
        } else if(slack[y] > t) slack[y] = t; 
    }
    return false;
}

int KM() {
    int i, j, x, d, ret;
    memset(match, -1, sizeof(match));
    memset(Ly, 0, sizeof(Ly));
    for(i = 0; i < id1; ++i) {
        Lx[i] = -inf;
        for(j = 0; j < id2; ++j)
            if(G[i][j] > Lx[i]) Lx[i] = G[i][j];
    }
    for(x = 0; x < id1; ++x) {
        memset(slack, 0x3f, sizeof(slack));
        while(true) {
            memset(visx, 0, sizeof(visx));
            memset(visy, 0, sizeof(visy));
            if(DFS(x)) break;
            d = inf;
            for(i = 0; i < id2; ++i)
                if(!visy[i] && d > slack[i]) 
                    d = slack[i];
            for(i = 0; i < id1; ++i)
                if(visx[i]) Lx[i] -= d;
            for(i = 0; i < id2; ++i)
                if(visy[i]) Ly[i] += d;
                else slack[i] -= d;
        }
    }
    ret = 0;
    for(i = 0; i < id1; ++i)
        if(match[i] > -1) ret += G[match[i]][i];
    return ret;
}

void solve() {
    printf("%d\n", largeNum * id1 - KM());
}

int main() {
    // freopen("stdin.txt", "r", stdin);
    while(scanf("%d%d", &n, &m), n | m) {
        getMap();
        solve();
    }
    return 0;
}

POJ2195 Going Home 【最小费用流】+【二分图最佳匹配】

原文：http://blog.csdn.net/chang_mu/article/details/39995041