Problem Description
Conflicts are everywhere in the world, from the young to the elderly, from families to countries. Conflicts cause quarrels, fights or even wars. How wonderful the world will be if all conflicts can be eliminated.
Edward contributes his lifetime to invent a 'Conflict Resolution Terminal' and he has finally succeeded. This magic item has the ability to eliminate all the conflicts. It works like this:
If any two people have conflict, they should simply put their hands into the 'Conflict Resolution Terminal' (which is simply a plastic tube). Then they play 'Rock, Paper and Scissors' in it. After they have decided what they will play, the tube should be opened and no one will have the chance to change. Finally, the winner have the right to rule and the loser should obey it. Conflict Eliminated!
But the game is not that fair, because people may be following some patterns when they play, and if the pattern is founded by others, the others will win definitely.
Alice and Bob always have conflicts with each other so they use the 'Conflict Resolution Terminal' a lot. Sadly for Bob, Alice found his pattern and can predict how Bob plays precisely. She is very kind that doesn't want to take advantage of that. So she tells Bob about it and they come up with a new way of eliminate the conflict:
They will play the 'Rock, Paper and Scissors' for N round. Bob will set up some restricts on Alice.
But the restrict can only be in the form of "you must play the same (or different) on the ith and jth rounds". If Alice loses in any round or break any of the rules she loses, otherwise she wins.
Will Alice have a chance to win?
Input
The first line contains an integer T(1 <= T <= 50), indicating the number of test cases.
Each test case contains several lines.
The first line contains two integers N,M(1 <= N <= 10000, 1 <= M <= 10000), representing how many round they will play and how many restricts are there for Alice.
The next line contains N integers B1,B2, ...,BN, where Bi represents what item Bob will play in the ith round. 1 represents Rock, 2 represents Paper, 3 represents Scissors.
The following M lines each contains three integers A,B,K(1 <= A,B <= N,K = 0 or 1) represent a restrict for Alice. If K equals 0, Alice must play the same on Ath and Bth round. If K equals 1, she must play different items on Ath and Bthround.
Output
For each test case in the input, print one line: "Case #X: Y", where X is the test case number (starting with 1) and Y is "yes" or "no" represents whether Alice has a chance to win.
Sample Input
2 3 3 1 1 1 1 2 1 1 3 1 2 3 1 5 5 1 2 3 2 1 1 2 1 1 3 1 1 4 1 1 5 1 2 3 0
Sample Output
Case #1: no Case #2: yes
Hint
'Rock, Paper and Scissors' is a game which played by two person. They should play Rock, Paper or Scissors by their hands at the same time. Rock defeats scissors, scissors defeats paper and paper defeats rock. If two people play the same item, the game is tied..
题意：
A和B两个人要用 N局“石头布剪刀”的游戏来决定胜负，现在给出A在每一轮游戏选择的手势（1表示石头，2表示布，3表示剪刀）。另外A给B设置了M个如a b k的限制，当k = 0时，要求B在第a局游戏和第b局游戏出的手势必须一样；当 k = 1时，要求B在第a局游戏和第b局游戏出的手势不能一样。对于B来说，输掉任意一局 或者 违反规则都是他输，反之他赢。这也就意味着在A和B全平局（每局出的手势一样）的情况下，是B赢。问你B有没有机会获胜。
分析:
因为Alice已经知道Bob的所有出拳，所以用a[maxn][2]数组保存那些出拳的结果。然后对于下面的要求作出Alice在第i轮选0还是选1的推断即可。看看是否有方法满足本2-SAT的n个解。
因为Alice一次都不能输，所以根据Bob出的拳，Alice只可以赢或者平局，即每次有两种选择，是2-SAT模型
然后会有一些矛盾对，假设第a次可以出a1，a2， 第b次可以出b1和b2
如果第a次和b次要求相同， 但是a1和b1不同，说明这个矛盾，建立连接 a1—>b2, b1—>a2.(a1与b1不同时最多有4种可能的情况需要考虑)
同理，第a次和b次要求不相同，但是a1和b2相同，说明这个矛盾，建立链接a1—>b1, b2—>a2
……
然后用2-SAT判断即可.
#include <cstdio>
#include <cstring>
#include <queue>
#include <vector>
#include <stack>
#include <algorithm>
#define MAXN 60000+100
#define MAXM 200000+10
#define INF 100000000
using namespace std;
struct Edge
{
int from, to, next;
};
Edge edge[MAXM];
int head[MAXN], edgenum;
int low[MAXN], dfn[MAXN];
int dfs_clock;
int sccno[MAXN], scc_cnt;
stack<int> S;
bool Instack[MAXN];
int N, M;//N局游戏 M个限制
int TT = 1;
void init()
{
edgenum = 0;
memset(head, -1, sizeof(head));
}
void addEdge(int u, int v)
{
Edge E = {u, v, head[u]};
edge[edgenum] = E;
head[u] = edgenum++;
}
void getMap()
{
int a, b, k;
for(int i = 1; i <= N; i++)
{
scanf("%d", &k);
switch(k)
{
//对手出石头
case 1: //在第i局出布 或者 出石头
addEdge(i + 4 * N, i);//不出布 就必须出石头
addEdge(i + 3 * N, i + N);//不出石头 就必须出布
break;
//对手出布
case 2: //在第i局出剪刀 或者 出布
addEdge(i + 5 * N, i + N);//不出剪刀 就必须出布
addEdge(i + 4 * N, i + 2 * N);//不出布 就必须出剪刀
break;
//对手出剪刀
case 3: //在第i局出石头 或者 出剪刀
addEdge(i + 3 * N, i + 2 * N);//不出石头 就必须出剪刀
addEdge(i + 5 * N, i);//不出剪刀 就必须出石头
break;
}
}
//每局只能出一个
for(int i = 1; i <= N; i++)
{
//出了石头就不能出其他的
addEdge(i, i + 4 * N);
addEdge(i, i + 5 * N);
//出了布 就不能出其他的
addEdge(i + N, i + 3 * N);
addEdge(i + N, i + 5 * N);
//出了剪刀 就不能出其他的
addEdge(i + 2 * N, i + 3 * N);
addEdge(i + 2 * N, i + 4 * N);
}
while(M--)
{
scanf("%d%d%d", &a, &b, &k);
if(k == 1)//a局和b局不同
{
addEdge(a, b + 3 * N);//a局出石头 b局一定不能出石头
addEdge(b, a + 3 * N);//b局出石头 a局一定不能出石头
addEdge(a + N, b + 4 * N);//a局出布 b局一定不能出布
addEdge(b + N, a + 4 * N);//a局出布 b局一定不能出布
addEdge(a + 2 * N, b + 5 * N);//a局出剪刀 b局一定不能出剪刀
addEdge(b + 2 * N, a + 5 * N);//b局出剪刀 a局一定不能出剪刀
}
else//a局和b局一样
{
addEdge(a, b);//a局出石头 b局也出石头
addEdge(b, a);//b局出石头 a局也出石头
addEdge(a + N, b + N);//a局出布 b局也出布
addEdge(b + N, a + N);//b局出布 a局也出布
addEdge(a + 2 * N, b + 2 * N);//a局出剪刀 b局也出剪刀
addEdge(b + 2 * N, a + 2 * N);//b局出剪刀 a局也出剪刀
}
}
}
void tarjan(int u, int fa)
{
int v;
low[u] = dfn[u] = ++dfs_clock;
S.push(u);
Instack[u] = true;
for(int i = head[u]; i != -1; i = edge[i].next)
{
v = edge[i].to;
if(!dfn[v])
{
tarjan(v, u);
low[u] = min(low[u], low[v]);
}
else if(Instack[v])
low[u] = min(low[u], dfn[v]);
}
if(low[u] == dfn[u])
{
scc_cnt++;
for(;;)
{
v = S.top(); S.pop();
Instack[v] = false;
sccno[v] = scc_cnt;
if(v == u) break;
}
}
}
void find_cut(int l, int r)
{
memset(low, 0, sizeof(low));
memset(dfn, 0, sizeof(dfn));
memset(sccno, 0, sizeof(sccno));
memset(Instack, false, sizeof(Instack));
dfs_clock = scc_cnt = 0;
for(int i = l; i <= r; i++)
if(!dfn[i]) tarjan(i, -1);
}
void solve()
{
printf("Case #%d: ", TT++);
for(int i = 1; i <= 3 * N; i++)
{
if(sccno[i] == sccno[i + 3 * N])
{
printf("no
");
return ;
}
}
printf("yes
");
}
int main()
{
int t;
scanf("%d", &t);
while(t--)
{
scanf("%d%d", &N, &M);
init();
getMap();
find_cut(1, 6 * N);
solve();
}
return 0;
}