Problem:
Score : 600 points

Problem Statement

We have a grid with A horizontal rows and B vertical columns, with the squares painted white. On this grid, we will repeatedly apply the following operation:

Assume that the grid currently has a horizontal rows and b vertical columns. Choose "vertical" or "horizontal".

If we choose "vertical", insert one row at the top of the grid, resulting in an (a+1) \times b grid.

If we choose "horizontal", insert one column at the right end of the grid, resulting in an a \times (b+1) grid.

Then, paint one of the added squares black, and the other squares white.

Assume the grid eventually has C horizontal rows and D vertical columns. Find the number of ways in which the squares can be painted in the end, modulo 998244353.

Constraints

1 \leq A \leq C \leq 3000

1 \leq B \leq D \leq 3000

A, B, C, and D are integers.

Input

Input is given from Standard Input in the following format:

A B C D

Output

Print the number of ways in which the squares can be painted in the end, modulo 998244353.

Sample Input 1

1 1 2 2

Sample Output 1

3

Any two of the three squares other than the bottom-left square can be painted black.

Sample Input 2

2 1 3 4

Sample Output 2

65

Sample Input 3

31 41 59 265

Sample Output 3

387222020

Solution in Java (OpenJDK 11.0.6) (judge verdict: Accepted)



import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.StringTokenizer;

public class Main {
    private static final long MOD = 998_244_353;

    public static void main(final String[] args) {
        final FastScanner scanner = new FastScanner(System.in);
        final int a = scanner.nextInt();
        final int b = scanner.nextInt();
        final int c = scanner.nextInt();
        final int d = scanner.nextInt();

        if (a == c && b == d) {
            System.out.println(1);
            return;
        }

        final long[][] addedCol = new long[c + 1][d + 1];
        addedCol[a][b] = 1;
        for (int i = a + 1; i <= c; i++) {
            addedCol[i][b] = addedCol[i - 1][b] * b % MOD;
        }

        final long[][] addedRow = new long[c + 1][d + 1];
        addedRow[a][b] = 1;
        for (int i = b + 1; i <= d; i++) {
            addedRow[a][i] = addedRow[a][i - 1] * a % MOD;
        }

        for (int i = a + 1; i <= c; i++) {
            for (int j = b + 1; j <= d; j++) {
                addedCol[i][j] = (addedCol[i - 1][j] + addedRow[i - 1][j]) * j % MOD;
                addedRow[i][j] = (addedCol[i][j - 1] + addedRow[i][j - 1] * i) % MOD;
            }
        }

        System.out.println((addedCol[c][d] + addedRow[c][d]) % MOD);
    }

    private static class FastScanner {
        private final BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(final InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
        }

        String next() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (final IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }
    }
}