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: Accepted)

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.*;
import java.util.function.IntUnaryOperator;
import java.util.function.LongUnaryOperator;
import java.util.function.Predicate;
import java.util.function.Supplier;

public class Main {
    static In in = new In();
    static Out out = new Out();
    static final long inf = 0x1fffffffffffffffL;
    static final int iinf = 0x3fffffff;
    static final double eps = 1e-9;
    static long mod = 998244353;

    void solve(boolean naive) {
        int n = in.nextInt();
        int k = in.nextInt();
        int[] l = new int[k];
        int[] r = new int[k];
        for (int i = 0; i < k; i++) {
            l[i] = in.nextInt();
            r[i] = in.nextInt() + 1;
        }
        long[] dp = new long[n + 2];
        dp[1] = 1;
        for (int i = 2; i <= n + 1; i++) {
            dp[i] = dp[i - 1];
            for (int j = 0; j < k; j++) {
                dp[i] = (dp[i] + (dp[Math.max(0, i - l[j])] + mod - dp[Math.max(0, i - r[j])]) % mod) % mod;
            }
        }
        out.println((dp[n] + mod - dp[n - 1]) % mod);
    }

    public static void main(String[] args) {
        new Main().solve(args.length == 1 && args[0].equals("naive"));
        out.flush();
    }
}

interface Monoid<T> {
    Monoid<Integer> SUM_INTEGER = Integer::sum;
    Monoid<Integer> PRODUCT_INTEGER = (a, b) -> a * b;
    Monoid<Integer> XOR_INTEGER = (a, b) -> a ^ b;
    Monoid<Integer> GCD = (a, b) -> {
        while (a % b != 0) {
            int t = a % b;
            a = b;
            b = t;
        }
        return b;
    };
    Monoid<Integer> MIN_INTEGER = Math::min;
    Monoid<Integer> MAX_INTEGER = Math::max;
    Monoid<Long> SUM_LONG = Long::sum;
    Monoid<Long> PRODUCT_LONG = (a, b) -> a * b;
    Monoid<Long> XOR_LONG = (a, b) -> a ^ b;
    Monoid<Long> MIN_LONG = Math::min;
    Monoid<Long> MAX_LONG = Math::max;
    Monoid<Long> SUM_MOD = (a, b) -> (a + b) % Main.mod;

    T mergeFunction(T x, T y);

    default T merge(T x, T y) {
        T identity = identity();
        if (Objects.equals(x, identity)) {
            if (Objects.equals(y, identity)) {
                return identity;
            }
            return y;
        } else if (Objects.equals(y, identity)) {
            return x;
        } else {
            return mergeFunction(x, y);
        }
    }

    default T identity() {
        return null;
    }

    default boolean equalsIdentity(T obj) {
        return obj == null;
    }
}

class LazySegmentTree<T, U> {
    private List<T> node;
    private List<U> lazy;
    private int n;
    private int m;
    private int height;
    private final Monoid<T> monoidNode;
    private final Monoid<U> monoidLazy;
    private final Merge<T, U> mergeFunction;

    LazySegmentTree(Monoid<T> monoidNode, Monoid<U> monoidLazy, Merge<T, U> mergeFunction) {
        this.monoidNode = monoidNode;
        this.monoidLazy = monoidLazy;
        this.mergeFunction = mergeFunction;
    }

    void build(int m) {
        this.m = m;
        this.n = Integer.highestOneBit(m) * 2;
        this.height = Integer.bitCount(n - 1);
        node = new ArrayList<>(n * 2);
        lazy = new ArrayList<>(n * 2);
        for (int i = 0; i < n; i++) {
            node.add(null);
        }
        for (int i = 0; i < n; i++) {
            node.add(monoidNode.identity());
        }
        for (int i = n - 1; i > 0; i--) {
            node.set(i, monoidNode.merge(node.get(i * 2), node.get(i * 2 + 1)));
        }
        for (int i = 0; i < n * 2; i++) {
            lazy.add(monoidLazy.identity());
        }
    }

    void build(int m, T init) {
        this.m = m;
        this.n = Integer.highestOneBit(m) * 2;
        this.height = Integer.bitCount(n - 1);
        node = new ArrayList<>(n * 2);
        lazy = new ArrayList<>(n * 2);
        for (int i = 0; i < n; i++) {
            node.add(null);
        }
        for (int i = 0; i < n; i++) {
            node.add(i < m ? init : monoidNode.identity());
        }
        for (int i = n - 1; i > 0; i--) {
            node.set(i, monoidNode.merge(node.get(i * 2), node.get(i * 2 + 1)));
        }
        for (int i = 0; i < n * 2; i++) {
            lazy.add(monoidLazy.identity());
        }
    }

    void build(int m, Supplier<T> initSupplier) {
        this.m = m;
        this.n = Integer.highestOneBit(m) * 2;
        this.height = Integer.bitCount(n - 1);
        node = new ArrayList<>(n * 2);
        lazy = new ArrayList<>(n * 2);
        for (int i = 0; i < n; i++) {
            node.add(null);
        }
        for (int i = 0; i < n; i++) {
            node.add(i < m ? initSupplier.get() : monoidNode.identity());
        }
        for (int i = n - 1; i > 0; i--) {
            node.set(i, monoidNode.merge(node.get(i * 2), node.get(i * 2 + 1)));
        }
        for (int i = 0; i < n * 2; i++) {
            lazy.add(monoidLazy.identity());
        }
    }

    void build(List<T> data) {
        this.m = data.size();
        this.n = Integer.highestOneBit(m) * 2;
        this.height = Integer.bitCount(n - 1);
        node = new ArrayList<>(n * 2);
        lazy = new ArrayList<>(n * 2);
        for (int i = 0; i < n; i++) {
            node.add(null);
        }
        for (int i = 0; i < n; i++) {
            node.add(i < m ? data.get(i) : monoidNode.identity());
        }
        for (int i = n - 1; i > 0; i--) {
            node.set(i, monoidNode.merge(node.get(i * 2), node.get(i * 2 + 1)));
        }
        for (int i = 0; i < n * 2; i++) {
            lazy.add(monoidLazy.identity());
        }
    }

    void update(int i, U x) {
        update(i, i + 1, x);
    }

    void update(int l, int r, U x) {
        for (int i = height; i > 0; i--) {
            propagate((l + n) >> i);
            propagate((r + n - 1) >> i);
        }
        for (int ll = l + n, rr = r + n; ll < rr; ll /= 2, rr /= 2) {
            if (ll % 2 == 1) {
                lazy.set(ll, monoidLazy.merge(lazy.get(ll), x));
                propagate(ll);
                ll++;
            }
            if (rr % 2 == 1) {
                rr--;
                lazy.set(rr, monoidLazy.merge(lazy.get(rr), x));
                propagate(rr);
            }
        }
        for (int ll = (l + n) / 2, rr = (r + n - 1) / 2; ll > 0; ll /= 2, rr /= 2) {
            if (monoidLazy.equalsIdentity(lazy.get(ll))) {
                node.set(ll, monoidNode.merge(merge(ll * 2), merge(ll * 2 + 1)));
            }
            if (monoidLazy.equalsIdentity(lazy.get(rr))) {
                node.set(rr, monoidNode.merge(merge(rr * 2), merge(rr * 2 + 1)));
            }
        }
    }

    T get(int i) {
        return get(i, i + 1);
    }

    T get(int l, int r) {
        for (int i = height; i > 0; i--) {
            propagate((l + n) >> i);
            propagate((r + n - 1) >> i);
        }
        T vl = monoidNode.identity();
        T vr = monoidNode.identity();
        for (int ll = l + n, rr = r + n; ll < rr; ll /= 2, rr /= 2) {
            if (ll % 2 == 1) {
                vl = monoidNode.merge(vl, merge(ll));
                ll++;
            }
            if (rr % 2 == 1) {
                rr--;
                vr = monoidNode.merge(merge(rr), vr);
            }
        }
        return monoidNode.merge(vl, vr);
    }

    List<T> getAll() {
        List<T> all = new ArrayList<>();
        for (int i = 0; i < m; i++) {
            all.add(get(i));
        }
        return all;
    }

    int binarySearch(Predicate<T> predicate) {
        return binarySearch(0, m, predicate);
    }

    int binarySearch(int l, int r, Predicate<T> predicate) {
        if (l < 0 || m < r || l >= r) {
            return -1;
        }
        for (int i = height; i > 0; i--) {
            propagate((l + n) >> i);
            propagate((r + n - 1) >> i);
        }
        T res = monoidNode.identity();
        int offset = l;
        for (int i = l + n; i < n * 2 && offset < r; ) {
            T t = monoidNode.merge(res, merge(i));
            if (rangeRight(i) <= r && !predicate.test(t)) {
                res = t;
                offset = rangeRight(i++);
                if (i % 2 == 0) {
                    i /= 2;
                }
            } else {
                i *= 2;
            }
        }
        return offset;
    }

    private void propagate(int k) {
        if (monoidLazy.equalsIdentity(lazy.get(k))) {
            return;
        }
        node.set(k, merge(k));
        if (k < n) {
            lazy.set(k * 2, monoidLazy.merge(lazy.get(k * 2), lazy.get(k)));
            lazy.set(k * 2 + 1, monoidLazy.merge(lazy.get(k * 2 + 1), lazy.get(k)));
        }
        lazy.set(k, monoidLazy.identity());
    }

    private T merge(int i) {
        if (monoidLazy.equalsIdentity(lazy.get(i))) {
            return node.get(i);
        } else {
            return mergeFunction.merge(node.get(i), lazy.get(i), range(i));
        }
    }

    private int rangeRight(int k) {
        return ((k ^ Integer.highestOneBit(k)) + 1) * range(k);
    }

    private int range(int k) {
        return 1 << (height - Integer.bitCount(Integer.highestOneBit(k) - 1));
    }

    @Override
    public String toString() {
        return getAll().toString();
    }

    interface Merge<T, U> {
        T merge(T t, U u, int length);
    }
}

class In {
    private BufferedReader reader = new BufferedReader(new InputStreamReader(System.in), 0x10000);
    private StringTokenizer tokenizer;

    String next() {
        try {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                tokenizer = new StringTokenizer(reader.readLine());
            }
        } catch (IOException ignored) {
        }
        return tokenizer.nextToken();
    }

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

    long nextLong() {
        return Long.parseLong(next());
    }

    double nextDouble() {
        return Double.parseDouble(next());
    }

    char[] nextCharArray() {
        return next().toCharArray();
    }

    char[][] nextCharGrid(int n, int m) {
        char[][] a = new char[n][m];
        for (int i = 0; i < n; i++) {
            a[i] = next().toCharArray();
        }
        return a;
    }

    int[] nextIntArray(int n) {
        int[] a = new int[n];
        for (int i = 0; i < n; i++) {
            a[i] = nextInt();
        }
        return a;
    }

    int[] nextIntArray(int n, IntUnaryOperator op) {
        int[] a = new int[n];
        for (int i = 0; i < n; i++) {
            a[i] = op.applyAsInt(nextInt());
        }
        return a;
    }

    long[] nextLongArray(int n) {
        long[] a = new long[n];
        for (int i = 0; i < n; i++) {
            a[i] = nextLong();
        }
        return a;
    }

    long[] nextLongArray(int n, LongUnaryOperator op) {
        long[] a = new long[n];
        for (int i = 0; i < n; i++) {
            a[i] = op.applyAsLong(nextLong());
        }
        return a;
    }

    List<List<Integer>> nextEdges(int n, int m) {
        List<List<Integer>> res = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            res.add(new ArrayList<>());
        }
        for (int i = 0; i < m; i++) {
            int u = nextInt() - 1;
            int v = nextInt() - 1;
            res.get(u).add(v);
            res.get(v).add(u);
        }
        return res;
    }
}

class Out {
    private PrintWriter out = new PrintWriter(System.out);
    boolean autoFlush = false;

    void println(Object... a) {
        StringJoiner joiner = new StringJoiner(" ");
        for (Object obj : a) {
            joiner.add(String.valueOf(obj));
        }
        out.println(joiner);
        if (autoFlush) {
            out.flush();
        }
    }

    void println(char[] s) {
        out.println(String.valueOf(s));
        if (autoFlush) {
            out.flush();
        }
    }

    void println(int[] a) {
        StringJoiner joiner = new StringJoiner(" ");
        for (int i : a) {
            joiner.add(Integer.toString(i));
        }
        out.println(joiner);
        if (autoFlush) {
            out.flush();
        }
    }

    void println(long[] a) {
        StringJoiner joiner = new StringJoiner(" ");
        for (long i : a) {
            joiner.add(Long.toString(i));
        }
        out.println(joiner);
        if (autoFlush) {
            out.flush();
        }
    }

    void flush() {
        out.flush();
    }
}