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Cow Ski Area
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 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, 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. 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 3 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 |
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;
}