POJ2375 Cow Ski Area 【强连通分量】+【DFS】

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Cow Ski Area

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 2323		Accepted: 660

Description

Farmer John’s cousin, Farmer Ron, who lives in the mountains of Colorado, has recently taught his cows to ski. Unfortunately, his cows are somewhat timid and are afraid to ski among crowds of people at the local resorts, so FR has decided to construct his own private ski area behind his farm.

FR’s ski area is a rectangle of width W and length L of ‘land squares’ (1 <= W <= 500; 1 <= L <= 500). Each land square is an integral height H above sea level (0 <= H <= 9,999). Cows can ski horizontally and vertically between any two adjacent land squares, but never diagonally. Cows can ski from a higher square to a lower square but not the other way and they can ski either direction between two adjacent squares of the same height.

FR wants to build his ski area so that his cows can travel between any two squares by a combination of skiing (as described above) and ski lifts. A ski lift can be built between any two squares of the ski area, regardless of height. Ski lifts are bidirectional. Ski lifts can cross over each other since they can be built at varying heights above the ground, and multiple ski lifts can begin or end at the same square. Since ski lifts are expensive to build, FR wants to minimize the number of ski lifts he has to build to allow his cows to travel between all squares of his ski area.

Find the minimum number of ski lifts required to ensure the cows can travel from any square to any other square via a combination of skiing and lifts.

Input

* Line 1: Two space-separated integers: W and L

* Lines 2..L+1: L lines, each with W space-separated integers corresponding to the height of each square of land.

Output

* Line 1: A single integer equal to the minimal number of ski lifts FR needs to build to ensure that his cows can travel from any square to any other square via a combination of skiing and ski lifts

Sample Input

9 3
1 1 1 2 2 2 1 1 1
1 2 1 2 3 2 1 2 1
1 1 1 2 2 2 1 1 1

Sample Output

Hint

This problem has huge input data,use scanf() instead of cin to read data to avoid time limit exceed.

OUTPUT DETAILS:

FR builds the three lifts. Using (1, 1) as the lower-left corner,

the lifts are (3, 1) <-> (8, 2), (7, 3) <-> (5, 2), and (1, 3) <->

(2, 2). All locations are now connected. For example, a cow wishing

to travel from (9, 1) to (2, 2) would ski (9, 1) -> (8, 1) -> (7,

1) -> (7, 2) -> (7, 3), take the lift from (7, 3) -> (5, 2), ski

(5, 2) -> (4, 2) -> (3, 2) -> (3, 3) -> (2, 3) -> (1, 3), and then

take the lift from (1, 3) – > (2, 2). There is no solution using

fewer than three lifts.

Source

USACO 2004 December Gold

题意：本题描述了一片滑雪场，并且规定奶牛从一个点只能向它相邻的并且高度不大于它的点运动，现在想要在某些点对之间加上缆车使得奶牛也可以从较低点到达较高点，问最少需要多少辆这样的缆车就可以使得奶牛可以从任意一个点运动到滑雪场的每个角落。

题解：对于相邻的高度相同的点，实际上路线是双向的，所以它们构成了强连通，然后这道题可以找出所有强连通，于是就构成了一个DAG，然后答案就是max(入度为0的点，出度为0的点)，特别的，当整幅图已经是强连通时答案为0.这题求强连通可以用DFS，省去不少麻烦。

#include <stdio.h>
#include <string.h>
#define maxn 502

int map[maxn][maxn], scc[maxn][maxn];
int sccNum, id, head[maxn * maxn], n, m;
bool in[maxn * maxn], out[maxn * maxn];
struct Node{
    int to, next;
} E[maxn * maxn << 2];

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

void getMap(int n, int m)
{
    int i, j; id = 0;
    for(i = 1; i <= n; ++i)
        for(j = 1; j <= m; ++j)
            scanf("%d", &map[i][j]);
}

void DFS(int x, int y)
{
    if(scc[x][y]) return;
    scc[x][y] = sccNum;    
    if(x + 1 <= n && map[x+1][y] == map[x][y]) DFS(x + 1, y);
    if(x - 1 >= 0 && map[x-1][y] == map[x][y]) DFS(x - 1, y);
    if(y + 1 <= m && map[x][y+1] == map[x][y]) DFS(x, y + 1);
    if(y - 1 >= 0 && map[x][y-1] == map[x][y]) DFS(x, y - 1);
}

void solve(int n, int m)
{
    memset(scc, 0, sizeof(scc));
    memset(in, 0, sizeof(in));
    memset(out, 0, sizeof(out));
    memset(head, -1, sizeof(head));
    sccNum = 0;
    int i, j, ans1 = 0, ans2 = 0;
    for(i = 1; i <= n; ++i)
        for(j = 1; j <= m; ++j)
            if(!scc[i][j]){
                ++sccNum; DFS(i, j);
            }
    for(i = 1; i <= n; ++i)
        for(j = 1; j <= m; ++j){
            if(i + 1 <= n && scc[i][j] != scc[i+1][j]){
                if(map[i][j] > map[i+1][j]){
                    addEdge(scc[i][j], scc[i+1][j]);
                    in[scc[i+1][j]] = out[scc[i][j]] = 1;
                } else{
                    addEdge(scc[i+1][j], scc[i][j]);
                    in[scc[i][j]] = out[scc[i+1][j]] = 1;
                }
            }
            if(j + 1 <= m && scc[i][j] != scc[i][j+1]){
                if(map[i][j] > map[i][j+1]){
                    addEdge(scc[i][j], scc[i][j+1]);
                    in[scc[i][j+1]] = out[scc[i][j]] = 1;
                } else{
                    addEdge(scc[i][j+1], scc[i][j]);
                    in[scc[i][j]] = out[scc[i][j+1]] = 1;
                }
            }
        }
    if(sccNum != 1)
        for(i = 1; i <= sccNum; ++i){
            if(!in[i]) ++ans1;
            if(!out[i]) ++ans2;
        }
    if(ans1 < ans2) ans1 = ans2;
    printf("%d\n", ans1);
}

int main()
{
    while(scanf("%d%d", &m, &n) == 2){
        getMap(n, m);
        solve(n, m);
    }
    return 0;
}

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