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// RNG
// 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);
}
int main()
{
if (DEBUG)
freopen("in.txt","r",stdin);
int TT,tt;
int N,M,K,T,R;
LD P;
int i,j,k,b;
LD d,v;
int S[20];
static int up[100000],down[100000],dist[100000],dir[20][20];
static vector<int> con[100000];
static LD exRN[100001],exNR[100001],exP[100001],exT[100001];
static LD dyn[1<<20][21];
queue<int> Q;
Read(TT);
Fox1(tt,TT)
{
//input
Read(N),Read(M),Read(K),Read(T),Read(R);
Fox(i,N)
con[i].clear();
cin >> P;
Fox(i,K)
Read(S[i]),S[i]--;
while (M--)
{
Read(i),Read(j);
con[i-1].pb(j-1);
}
Fox(i,K)
dyn[(1<<K)-1][i]=0;
//dist from start to each item
Fill(up,-1);
Q.push(0),up[0]=0;
while (!Q.empty())
{
i=Q.front(),Q.pop();
Fox(j,Sz(con[i]))
if (up[k=con[i][j]]<0)
Q.push(k),up[k]=up[i]+1;
}
//dist from each item to other items and to nearest leaf
Fill(down,-1);
Fox(b,K)
{
Fill(dist,-1);
Q.push(S[b]),dist[S[b]]=0;
while (!Q.empty())
{
i=Q.front(),Q.pop();
if ((!Sz(con[i])) && (down[b]<0))
down[b]=dist[i];
Fox(j,Sz(con[i]))
if (dist[k=con[i][j]]<0)
Q.push(k),dist[k]=dist[i]+1;
}
Fox(i,K)
dir[b][i]=dist[S[i]];
}
//expected times for root -> node
exRN[0]=0;
Fox1(i,N)
{
v=(P*T + (1-P)*(R+exRN[i-1])) / P;
exRN[i]=exRN[i-1]+v;
}
//expected times for node -> root
exNR[0]=R;
Fox1(i,N)
exNR[i]=P*(T + exNR[i-1]) + (1-P)*R;
//conditional expected times for node -> node
exP[0]=1;
exT[0]=0;
Fox1(i,N)
{
exP[i]=exP[i-1]*P;
exT[i]=exT[i-1] + exP[i-1]*(1-P)*((i-1)*T + R);
}
//bitmask DP
FoxR(b,(1<<K)-1)
{
d=-1;
Fox(i,K)
if (!(b&(1<<i)))
{
v=dyn[b][i]=dyn[b|(1<<i)][i];
v+=exRN[up[S[i]]];
if ((d<0) || (d>v))
d=v;
}
dyn[b][K]=d;
Fox(i,K)
if (b&(1<<i))
{
d=exNR[down[i]]+dyn[b][K];
Fox(j,K)
if (!(b&(1<<j)))
if (dir[i][j]>=0)
{
k=dir[i][j];
v=exP[k]*(k*T + dyn[b][j]) + exT[k] + (1-exP[k])*dyn[b][K];
Min(d,v);
}
dyn[b][i]=d;
}
}
//output
printf("Case #%d: ",tt);
cout << fixed << setprecision(9) << dyn[0][K] << endl;
}
return(0);
} |