【CODEFORCES】 C. Riding in a Lift

C. Riding in a Lift

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

Imagine that you are in a building that has exactly n floors. You can move between the floors in a lift. Let's number the floors from bottom to top with integers from 1 to n. Now you're on the floor number a. You are very bored, so you want to take the lift. Floor numberb has a secret lab, the entry is forbidden. However, you already are in the mood and decide to make k consecutive trips in the lift.

Let us suppose that at the moment you are on the floor number x (initially, you were on floor a). For another trip between floors you choose some floor with number y (y ≠ x) and the lift travels to this floor. As you cannot visit floor b with the secret lab, you decided that the distance from the current floor x to the chosen y must be strictly less than the distance from the current floor x to floor b with the secret lab. Formally, it means that the following inequation must fulfill: |x - y| < |x - b|. After the lift successfully transports you to floor y, you write down number y in your notepad.

Your task is to find the number of distinct number sequences that you could have written in the notebook as the result of k trips in the lift. As the sought number of trips can be rather large, find the remainder after dividing the number by 1000000007 (109 + 7).

Input

The first line of the input contains four space-separated integers n, a, b, k (2 ≤ n ≤ 5000, 1 ≤ k ≤ 5000, 1 ≤ a, b ≤ n, a ≠ b).

Output

Print a single integer — the remainder after dividing the sought number of sequences by 1000000007 (109 + 7).

Sample test(s)

input

5 2 4 1

output

2

input

5 2 4 2

output

2

input

5 3 4 1

output

0

Note

Two sequences p1, p2, ..., pk and q1, q2, ..., qk are distinct, if there is such integer j (1 ≤ j ≤ k), that pj ≠ qj.

Notes to the samples:

1. In the first sample after the first trip you are either on floor 1, or on floor 3, because |1 - 2| < |2 - 4| and |3 - 2| < |2 - 4|.
2. In the second sample there are two possible sequences: (1, 2); (1, 3). You cannot choose floor 3 for the first trip because in this case no floor can be the floor for the second trip.
3. In the third sample there are no sought sequences, because you cannot choose the floor for the first trip.

题解：这是一道动归加优化的题目，动归方程很好写。D[i][j]=sigma(d[i-1][k])  (abs(k-j)<abs(k-b)) 但是很容易发现这样会超时因为时间复杂度是10^9，经过简单的分析以后可以发现每次都会重复的取某一段区间的和，所以这里采用前缀和进行优化（如果a>b就是后缀和，可以写一个函数省得麻烦）优化过后就可以过了。

时间给了两秒，非常充裕。

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cstdlib>
const long long mod=1000000007;

using namespace std;

long long n,a,b,k,d[5005][5005],ans,sum[5005];

void solve(int p)
{
    if (a<b) for (int i=1;i<=b;i++) sum[i]=(sum[i-1]+d[p][i])%mod;
    else for (int i=n;i>=b;i--) sum[i]=(sum[i+1]+d[p][i])%mod;
}

int main()
{
    scanf("%I64d%I64d%I64d%I64d",&n,&a,&b,&k);
    memset(d,0,sizeof(d));
    d[0][a]=1;
    for (int i=1;i<=k;i++)
    {
        solve(i-1);
        for (int j=1;j<=n;j++) if ((b-a)*(b-j)>0 && a<b) d[i][j]=(sum[(j+b-1)/2]%mod-d[i-1][j]%mod+mod)%mod;
        else if ((b-a)*(b-j)>0 && a>b) d[i][j]=(sum[(j+b)/2+1]%mod-d[i-1][j]%mod+mod)%mod;
    }
    for (int j=1;j<=n;j++)
        ans=(ans+d[k][j])%mod;
    printf("%I64d",ans);
    return 0;
}