2016/finals/grundy_graph.cpp

// GRUNDY GRAPH
// Official Solution by Jacob Plachta

#define DEBUG 0

#include <algorithm>
#include <functional>
#include <numeric>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <complex>
#include <cstdlib>
#include <ctime>
#include <cstring>
#include <cassert>
#include <string>
#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <queue>
#include <stack>
#include <bitset>
#include <sstream>
using namespace std;

#define LL long long
#define LD long double
#define PR pair<int,int>

#define Fox(i,n) for (i=0; i<n; i++)
#define Fox1(i,n) for (i=1; i<=n; i++)
#define FoxI(i,a,b) for (i=a; i<=b; i++)
#define FoxR(i,n) for (i=(n)-1; i>=0; i--)
#define FoxR1(i,n) for (i=n; i>0; i--)
#define FoxRI(i,a,b) for (i=b; i>=a; i--)
#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++)
#define Min(a,b) a=min(a,b)
#define Max(a,b) a=max(a,b)
#define Sz(s) int((s).size())
#define All(s) (s).begin(),(s).end()
#define Fill(s,v) memset(s,v,sizeof(s))
#define pb push_back
#define mp make_pair
#define x first
#define y second

template<typename T> T Abs(T x) { return(x<0 ? -x : x); }
template<typename T> T Sqr(T x) { return(x*x); }
string plural(string s) { return(Sz(s) && s[Sz(s)-1]=='x' ? s+"en" : s+"s"); }

const int INF = (int)1e9;
const LD EPS = 1e-9;
const LD PI = acos(-1.0);

//#if DEBUG
#define GETCHAR getchar
/*#else
#define GETCHAR getchar_unlocked
#endif*/

bool Read(int &x)
{
	char c,r=0,n=0;
	x=0;
		for(;;)
		{
			c=GETCHAR();
				if ((c<0) && (!r))
					return(0);
				if ((c=='-') && (!r))
					n=1;
				else
				if ((c>='0') && (c<='9'))
					x=x*10+c-'0',r=1;
				else
				if (r)
					break;
		}
		if (n)
			x=-x;
	return(1);
}

#define LIM 2000002

struct edge{int e, nxt;};
int V, E;
edge e[LIM], er[LIM];
int sp[LIM], spr[LIM];
int group_cnt, comp[LIM];
bool v[LIM];
int stk[LIM];
void fill_forward(int x)
{
  int i;
  v[x]=true;
  for(i=sp[x];i;i=e[i].nxt) if(!v[e[i].e]) fill_forward(e[i].e);
  stk[++stk[0]]=x;
}
void fill_backward(int x)
{
  int i;
  v[x]=false;
  comp[x]=group_cnt;
  for(i=spr[x];i;i=er[i].nxt) if(v[er[i].e]) fill_backward(er[i].e);
}
void add_edge(int v1, int v2) //add edge v1->v2
{
  e [++E].e=v2; e [E].nxt=sp [v1]; sp [v1]=E;
  er[  E].e=v1; er[E].nxt=spr[v2]; spr[v2]=E;
}
void SCC()
{
  int i;
  stk[0]=0;
  memset(v, false, sizeof(v));
  for(i=1;i<=V;i++) if(!v[i]) fill_forward(i);
  group_cnt=0;
  for(i=stk[0];i>=1;i--) if(v[stk[i]]){group_cnt++; fill_backward(stk[i]);}
}

int val[LIM];
bool visA[LIM],visB[LIM];
vector<int> con[LIM];

bool rec(int s,int i)
{
		if ((i!=s) && ((i-1)/2%2))
			return(1);
		if (visB[i])
			return(0);
	visB[i]=1;
	int j;
		Fox(j,Sz(con[i]))
			if (rec(s,con[i][j]))
				return(1);
	return(0);
}

int main()
{
		if (DEBUG)
			freopen("in.txt","r",stdin);
	int T,t;
	int N,M;
	int i,j,a,b;
	Read(T);
		Fox1(t,T)
		{
			printf("Case #%d: ",t);
			Read(N),Read(M);
			N<<=1;
				Fox1(i,N)
					con[i].clear();
			V=N;
			E=0;
			Fill(sp,0);
			Fill(spr,0);
				while (M--)
				{
					Read(i),Read(j);
					con[i].pb(j);
					add_edge(i,j);
						if (j%2)
							j++;
						else
							j--;
						if (i%2)
							i++;
						else
							i--;
					con[j].pb(i);
					add_edge(j,i);
				}
			SCC();
			Fill(visA,0);
			Fill(visB,0);
				Fox1(i,N)
				{
					//Alice
					a=comp[i],b=comp[i+1];
						if (a==b)
							goto Bad;
					visA[a]=visA[b]=1;
					i+=2;
					//Bob
					a=comp[i],b=comp[i+1];
						if ((visA[a]) || (visA[b]) || (rec(i,i)) || (rec(i+1,i+1)))
							goto Bad;
					i++;
				}
			printf("Alice\n");
			continue;
Bad:;
			printf("Bob\n");
		}
	return(0);
}