102655: [AtCoder]ABC265 F - Manhattan Cafe

Problem Statement

In an $N$-dimensional space, the Manhattan distance $d(x,y)$ between two points $x=(x_1, x_2, \dots, x_N)$ and $y = (y_1, y_2, \dots, y_N)$ is defined by:

A point $x=(x_1, x_2, \dots, x_N)$ is said to be a lattice point if the components $x_1, x_2, \dots, x_N$ are all integers.

You are given lattice points $p=(p_1, p_2, \dots, p_N)$ and $q = (q_1, q_2, \dots, q_N)$ in an $N$-dimensional space.
How many lattice points $r$ satisfy $d(p,r) \leq D$ and $d(q,r) \leq D$? Find the count modulo $998244353$.

Constraints

• $1 \leq N \leq 100$
• $0 \leq D \leq 1000$
• $-1000 \leq p_i, q_i \leq 1000$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $D$ 
$p_1$ $p_2$ $\dots$ $p_N$
$q_1$ $q_2$ $\dots$ $q_N$

Output

Print the answer.

Sample Input 1

1 5
0
3

Sample Output 1

8

When $N=1$, we consider points in a one-dimensional space, that is, on a number line.
$8$ lattice points satisfy the conditions: $-2,-1,0,1,2,3,4,5$.

Sample Input 2

3 10
2 6 5
2 1 2

Sample Output 2

632

Sample Input 3

10 100
3 1 4 1 5 9 2 6 5 3
2 7 1 8 2 8 1 8 2 8

Sample Output 3

145428186

问题描述

在一个N维空间中，两个点$x=(x_1, x_2, \dots, x_N)$和$y = (y_1, y_2, \dots, y_N)$之间的曼哈顿距离$d(x,y)$定义为：

如果点$x=(x_1, x_2, \dots, x_N)$的各个分量$x_1, x_2, \dots, x_N$都是整数，则称该点为格点。

给你一个N维空间中的格点$p=(p_1, p_2, \dots, p_N)$和$q = (q_1, q_2, \dots, q_N)$。
满足$d(p,r) \leq D$和$d(q,r) \leq D$的格点$r$有多少个？ 找到答案对$998244353$取模。

约束条件

• $1 \leq N \leq 100$
• $0 \leq D \leq 1000$
• $-1000 \leq p_i, q_i \leq 1000$
• 输入中的所有值都是整数。

输入

从标准输入以以下格式给出输入：

$N$ $D$ 
$p_1$ $p_2$ $\dots$ $p_N$
$q_1$ $q_2$ $\dots$ $q_N$

输出

打印答案。

样例输入1

1 5
0
3

样例输出1

8

当$N=1$时，我们考虑一维空间中的点，即数轴上的点。
有8个格点满足条件：$-2,-1,0,1,2,3,4,5$。

样例输入2

3 10
2 6 5
2 1 2

样例输出2

632

样例输入3

10 100
3 1 4 1 5 9 2 6 5 3
2 7 1 8 2 8 1 8 2 8

样例输出3

145428186