Problem:
Score : 300 points

Problem Statement

There are N sightseeing spots on the x-axis, numbered 1, 2, ..., N.
Spot i is at the point with coordinate A_i.
It costs |a - b| yen (the currency of Japan) to travel from a point with coordinate a to another point with coordinate b along the axis.

You planned a trip along the axis.
In this plan, you first depart from the point with coordinate 0, then visit the N spots in the order they are numbered, and finally return to the point with coordinate 0.

However, something came up just before the trip, and you no longer have enough time to visit all the N spots, so you decided to choose some i and cancel the visit to Spot i.
You will visit the remaining spots as planned in the order they are numbered.
You will also depart from and return to the point with coordinate 0 at the beginning and the end, as planned.

For each i = 1, 2, ..., N, find the total cost of travel during the trip when the visit to Spot i is canceled.

Constraints

2 \leq N \leq 10^5

-5000 \leq A_i \leq 5000 (1 \leq i \leq N)

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print N lines.
In the i-th line, print the total cost of travel during the trip when the visit to Spot i is canceled.

Sample Input 1

3
3 5 -1

Sample Output 1

12
8
10

Spot 1, 2 and 3 are at the points with coordinates 3, 5 and -1, respectively.
For each i, the course of the trip and the total cost of travel when the visit to Spot i is canceled, are as follows:

For i = 1, the course of the trip is 0 \rightarrow 5 \rightarrow -1 \rightarrow 0 and the total cost of travel is 5 + 6 + 1 = 12 yen.

For i = 2, the course of the trip is 0 \rightarrow 3 \rightarrow -1 \rightarrow 0 and the total cost of travel is 3 + 4 + 1 = 8 yen.

For i = 3, the course of the trip is 0 \rightarrow 3 \rightarrow 5 \rightarrow 0 and the total cost of travel is 3 + 2 + 5 = 10 yen.

Sample Input 2

5
1 1 1 2 0

Sample Output 2

4
4
4
2
4

Sample Input 3

6
-679 -2409 -3258 3095 -3291 -4462

Sample Output 3

21630
21630
19932
8924
21630
19288

Solution in C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include<bits/stdc++.h>
using namespace std;
#define REP(i,st,ed) for(register int i=st,i##end=ed;i<=i##end;++i)
#define DREP(i,st,ed) for(register int i=st,i##end=ed;i>=i##end;--i)
typedef long long ll;
template<typename T>inline bool chkmin(T &x,T y){return (y<x)?(x=y,1):0;}
template<typename T>inline bool chkmax(T &x,T y){return (y>x)?(x=y,1):0;}
inline int read(){
	int x;
	char c;
	int f=1;
	while((c=getchar())!='-' && (c>'9' || c<'0'));
	if(c=='-') f=-1,c=getchar();
	x=c^'0';
	while((c=getchar())>='0' && c<='9') x=(x<<1)+(x<<3)+(c^'0');
	return x*f;
}
inline ll readll(){
	ll x;
	char c;
	int f=1;
	while((c=getchar())!='-' && (c>'9' || c<'0'));
	if(c=='-') f=-1,c=getchar();
	x=c^'0';
	while((c=getchar())>='0' && c<='9') x=(x<<1ll)+(x<<3ll)+(c^'0');
	return x*f;
}
const int maxn=20,mod=1e9+7;
inline int ksm(int x,int y){
	int res=1;
	while(y){
		if(y&1) res=(ll)res*x%mod;
		y>>=1;
		x=(ll)x*x%mod;
	}
	return res;
}
inline void add(int &x,int y){
	x+=y;
	if(x>=mod) x-=mod;
}
int ans,a[maxn];
int dp[maxn][1<<16];
int fac[1<<16],ifac[1<<16];
inline int C(int n,int m){
	if(n<m) return 0;
	return (ll)fac[n]*ifac[m]%mod*ifac[n-m]%mod;
}
int main(){
#ifndef ONLINE_JUDGE
	freopen("f.in","r",stdin);
	freopen("f.out","w",stdout);
#endif
	int n=read(),m=read();
	REP(i,1,m) a[i]=read();
	dp[m][0]=fac[0]=1;
	sort(a+1,a+m+1);
	int All=(1<<n);
	REP(i,1,All-1) fac[i]=(ll)i*fac[i-1]%mod;
	ifac[All-1]=ksm(fac[All-1],mod-2);
	DREP(i,All-1,1) ifac[i-1]=(ll)i*ifac[i]%mod;
	DREP(i,m,1) REP(j,0,All-1) if(dp[i][j]){
		add(dp[i-1][j],dp[i][j]);
		REP(k,0,n-1) if((j&(1<<k))==0)
			add(dp[i-1][j|(1<<k)],(ll)dp[i][j]*C(All-a[i]-j,(1<<k)-1)%mod*fac[(1<<k)]%mod);
	}
	REP(i,0,(1<<n)-1) ans=(ans+(ll)(__builtin_popcount(i)%2==0?1:(mod-1))*dp[0][i]%mod*fac[iend-i])%mod;
	printf("%lld\n",(ll)ans*ksm(2,n)%mod);
	return 0;
}