Problem:
Score : 400 points

Problem Statement

There are N cells arranged in a row, numbered 1, 2, \ldots, N from left to right.

Tak lives in these cells and is currently on Cell 1. He is trying to reach Cell N by using the procedure described below.

You are given an integer K that is less than or equal to 10, and K non-intersecting segments [L_1, R_1], [L_2, R_2], \ldots, [L_K, R_K].
Let S be the union of these K segments.
Here, the segment [l, r] denotes the set consisting of all integers i that satisfy l \leq i \leq r.

When you are on Cell i, pick an integer d from S and move to Cell i + d. You cannot move out of the cells.

To help Tak, find the number of ways to go to Cell N, modulo 998244353.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq K \leq \min(N, 10)

1 \leq L_i \leq R_i \leq N

[L_i, R_i] and [L_j, R_j] do not intersect (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
L_1 R_1
L_2 R_2
:
L_K R_K

Output

Print the number of ways for Tak to go from Cell 1 to Cell N, modulo 998244353.

Sample Input 1

5 2
1 1
3 4

Sample Output 1

4

The set S is the union of the segment [1, 1] and the segment [3, 4], therefore S = \{ 1, 3, 4 \} holds.

There are 4 possible ways to get to Cell 5:

1 \to 2 \to 3 \to 4 \to 5,

1 \to 2 \to 5,

1 \to 4 \to 5 and

1 \to 5.

Sample Input 2

5 2
3 3
5 5

Sample Output 2

0

Because S = \{ 3, 5 \} holds, you cannot reach to Cell 5.
Print 0.

Sample Input 3

5 1
1 2

Sample Output 3

5

Sample Input 4

60 3
5 8
1 3
10 15

Sample Output 4

221823067

Note that you have to print the answer modulo 998244353.

Solution in Java (OpenJDK 11.0.6) (judge verdict: Time Limit Exceeded)


import java.util.HashMap;
import java.util.Scanner;
import java.util.TreeSet;

public class Main {

  private static int goal;
  private static long count = 0;
  private static Integer[] sortedNumbers;
  private static HashMap<Integer, Long> countByPosition = new HashMap<>();

  public static void main(String[] args) {
    final Scanner scanner = new Scanner(System.in);
    goal = scanner.nextInt();
    final int K = scanner.nextInt();

    TreeSet<Integer> S = new TreeSet<>();
    for (int i = 0; i < K; i++) {
      int left = scanner.nextInt();
      int right = scanner.nextInt();

      while (left <= right) {
        S.add(left);
        left++;
      }
    }

    sortedNumbers = S.toArray(S.toArray(new Integer[0]));

    for (int i = 0; i < sortedNumbers.length; i++) {
      searchWay(1, sortedNumbers[i]);
    }

    System.out.println(count % 998244353);
  }

  private static void searchWay(int currentPosition, int offset) {
    int newPosition = currentPosition + offset;
    if (newPosition == goal) {
      count++;
      if (count >= 998244353) {
        count = count % 998244353;
      }

      return;
    }

    if (newPosition > goal) {
      return;
    }

    Long memorizedCount = countByPosition.get(newPosition);
    if (memorizedCount != null) {
      count += memorizedCount;
      if (count >= 998244353) {
        count = count % 998244353;
      }
      return;
    }

    long countBefore = count;
    for (Integer newOffset : sortedNumbers) {
      if (newPosition + newOffset > goal) {
        break;
      }
      searchWay(newPosition, newOffset);
    }
    countByPosition.put(newPosition, count - countBefore);

    return;
  }
}