Problem:
Score : 600 points

Problem Statement

There are N robots numbered 1 to N placed on a number line. Robot i is placed at coordinate X_i. When activated, it will travel the distance of D_i in the positive direction, and then it will be removed from the number line. All the robots move at the same speed, and their sizes are ignorable.

Takahashi, who is a mischievous boy, can do the following operation any number of times (possibly zero) as long as there is a robot remaining on the number line.

Choose a robot and activate it. This operation cannot be done when there is a robot moving.

While Robot i is moving, if it touches another robot j that is remaining in the range [X_i, X_i + D_i) on the number line, Robot j also gets activated and starts moving. This process is repeated recursively.

How many possible sets of robots remaining on the number line are there after Takahashi does the operation some number of times? Compute this count modulo 998244353, since it can be enormous.

Constraints

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

-10^9 \leq X_i \leq 10^9

1 \leq D_i \leq 10^9

X_i \neq X_j (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 D_1
:
X_N D_N

Output

Print the number of possible sets of robots remaining on the number line, modulo 998244353.

Sample Input 1

2
1 5
3 3

Sample Output 1

3

There are three possible sets of robots remaining on the number line: \{1, 2\}, \{1\}, and \{\}.

These can be achieved as follows:

If Takahashi activates nothing, the robots \{1, 2\} will remain.

If Takahashi activates Robot 1, it will activate Robot 2 while moving, after which there will be no robots on the number line. This state can also be reached by activating Robot 2 and then Robot 1.

If Takahashi activates Robot 2 and finishes doing the operation, the robot \{1\} will remain.

Sample Input 2

3
6 5
-1 10
3 3

Sample Output 2

5

There are five possible sets of robots remaining on the number line: \{1, 2, 3\}, \{1, 2\}, \{2\}, \{2, 3\}, and \{\}.

Sample Input 3

4
7 10
-10 3
4 3
-4 3

Sample Output 3

16

None of the robots influences others.

Sample Input 4

20
-8 1
26 4
0 5
9 1
19 4
22 20
28 27
11 8
-3 20
-25 17
10 4
-18 27
24 28
-11 19
2 27
-2 18
-1 12
-24 29
31 29
29 7

Sample Output 4

110

Remember to print the count modulo 998244353.

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using Yorukatsu037.Questions;
using Yorukatsu037.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace Yorukatsu037.Questions
{
    /// <summary>
    /// ABC158 F
    /// </summary>
    public class QuestionF : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var n = inputStream.ReadInt();
            var robots = new Robot[n];

            for (int i = 0; i < n; i++)
            {
                var xd = inputStream.ReadIntArray();
                robots[i] = new Robot(xd[0], xd[1]);
            }

            Array.Sort(robots);
            Array.Reverse(robots);

            var unitedRobots = new UnionFindTree(robots.Select(r => r.Destination));
            var destinations = new SegmentTree<int>(robots.Select(r => r.Destination).ToArray(), (d1, d2) => Math.Max(d1, d2), int.MinValue);

            const int mod = 998244353;
            var counts = new Modular[robots.Length + 1];
            for (int i = 0; i < counts.Length; i++)
            {
                counts[i] = new Modular(0, mod);
            }
            counts[0] = new Modular(1, mod);

            for (int i = 0; i < robots.Length; i++)
            {
                var currentRobot = robots[i];
                var inRangeRobotIndex = BoundaryBinarySearch(mid => robots[mid].Coordinate < currentRobot.Destination, -1, i);

                while (!unitedRobots.IsInSameGroup(inRangeRobotIndex, i))
                {
                    var hasntUnited = BoundaryBinarySearch(mid => !unitedRobots.IsInSameGroup(mid, i), i, inRangeRobotIndex);
                    unitedRobots.Unite(hasntUnited, i);
                }

                var outOfRangeRobotIndex = BoundaryBinarySearch(mid => !unitedRobots.IsInSameGroup(mid, i), i, -1);

                counts[i + 1] += counts[i] + counts[outOfRangeRobotIndex + 1];
            }

            yield return counts[robots.Length].Value;
        }

        struct Robot : IComparable<Robot>
        {
            public int Coordinate { get; }
            public int Length { get; }
            public int Destination { get; }

            public Robot(int coordinate, int length)
            {
                Coordinate = coordinate;
                Length = length;
                Destination = coordinate + length;
            }

            public int CompareTo(Robot other) => Coordinate.CompareTo(other.Coordinate);

            public override string ToString() => $"X:{Coordinate}, D:{Length}";
        }

        private static int BoundaryBinarySearch(Predicate<int> predicate, int ng, int ok)
        {
            // めぐる式二分探索
            // Span.BinarySearchだとできそうでできない（lower_boundがダメ）
            while (Math.Abs(ok - ng) > 1)
            {
                int mid = (ok + ng) / 2;

                if (predicate(mid))
                {
                    ok = mid;
                }
                else
                {
                    ng = mid;
                }
            }
            return ok;
        }

        public class SegmentTree<T> : IEnumerable<T>
        {
            private readonly T[] _data;
            private readonly T _identityElement;
            private readonly Func<T, T, T> _queryOperation;

            private readonly int _leafOffset;   // n - 1
            private readonly int _leafLength;   // n (= 2^k)

            public int Length { get; }          // 実データ長

            public SegmentTree(ICollection<T> data, Func<T, T, T> queryOperation, T identityElement)
            {
                Length = data.Count;
                _leafLength = GetMinimumPow2(data.Count);
                _leafOffset = _leafLength - 1;
                _data = new T[_leafOffset + _leafLength];
                _queryOperation = queryOperation;
                _identityElement = identityElement;

                data.CopyTo(_data, _leafOffset);
                BuildTree();
            }

            public T this[int index]
            {
                set
                {
                    if (index < 0 || index >= Length)
                    {
                        throw new IndexOutOfRangeException($"{nameof(index)}がデータの範囲外です。");
                    }
                    index += _leafOffset;
                    _data[index] = value;
                    while (index > 0)
                    {
                        // 一つ上の親の更新
                        index = (index - 1) / 2;
                        _data[index] = _queryOperation(_data[index * 2 + 1], _data[index * 2 + 2]);
                    }
                }
            }

            public T Query(int begin, int end)
            {
                if (begin < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(begin), $"{nameof(begin)}は0以上の数でなければなりません。");
                }
                if (end > Length)
                {
                    throw new ArgumentOutOfRangeException(nameof(end), $"{nameof(end)}は{nameof(Length)}以下でなければなりません。");
                }
                if (begin >= end)
                {
                    throw new ArgumentException($"{nameof(begin)},{nameof(end)}", $"{nameof(end)}は{nameof(begin)}より大きい数でなければなりません。");
                }
                return Query(begin, end, 0, 0, _leafLength);
            }

            private T Query(int begin, int end, int index, int left, int right)
            {
                if (right <= begin || end <= left)      // 範囲外
                {
                    return _identityElement;
                }
                else if (begin <= left && right <= end) // 全部含まれる
                {
                    return _data[index];
                }
                else    // 一部だけ含まれる
                {
                    var leftValue = Query(begin, end, index * 2 + 1, left, (left + right) / 2);     // 左の子
                    var rightValue = Query(begin, end, index * 2 + 2, (left + right) / 2, right);   // 右の子
                    return _queryOperation(leftValue, rightValue);
                }
            }

            private void BuildTree()
            {
                for (int i = _leafLength + Length; i < _data.Length; i++)
                {
                    _data[i] = _identityElement;
                }

                for (int i = _leafLength - 2; i >= 0; i--)  // 葉の親から順番に一つずつ上がっていく
                {
                    _data[i] = _queryOperation(_data[2 * i + 1], _data[2 * i + 2]); // f(left, right)
                }
            }

            private int GetMinimumPow2(int n)
            {
                var p = 1;
                while (p < n)
                {
                    p *= 2;
                }
                return p;
            }

            public IEnumerator<T> GetEnumerator()
            {
                var upperIndex = _leafOffset + Length;
                for (int i = _leafOffset; i < upperIndex; i++)
                {
                    yield return _data[i];
                }
            }

            System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator();
        }


        // See https://kumikomiya.com/competitive-programming-with-c-sharp/
        public class UnionFindTree
        {
            private UnionFindNode[] _nodes;
            public int Count => _nodes.Length;
            public int Groups { get; private set; }

            public UnionFindTree(IEnumerable<int> destinations)
            {
                _nodes = destinations.Select(i => new UnionFindNode(i)).ToArray();
                Groups = _nodes.Length;
            }

            public void Unite(int index1, int index2)
            {
                var united = _nodes[index1].Unite(_nodes[index2]);
                if (united)
                {
                    Groups--;
                }
            }

            public bool IsInSameGroup(int index1, int index2) => _nodes[index1].IsInSameGroup(_nodes[index2]);
            public int GetGroupSizeOf(int index) => _nodes[index].GetGroupSize();
            public int this[int index] => _nodes[index].Destination;

            private class UnionFindNode
            {
                private int _height;        // rootのときのみ有効
                private int _groupSize;     // 同上
                private UnionFindNode _parent;
                public int Destination { get; private set; }

                public UnionFindNode(int id)
                {
                    _height = 0;
                    _groupSize = 1;
                    _parent = this;
                    Destination = id;
                }

                public UnionFindNode FindRoot()
                {
                    if (_parent != this) // not ref equals
                    {
                        var root = _parent.FindRoot();
                        _parent = root;
                    }

                    return _parent;
                }

                public int GetGroupSize() => FindRoot()._groupSize;

                public bool Unite(UnionFindNode other)
                {
                    var thisRoot = this.FindRoot();
                    var otherRoot = other.FindRoot();

                    var max = Math.Max(thisRoot.Destination, otherRoot.Destination);
                    thisRoot.Destination = max;
                    otherRoot.Destination = max;

                    if (thisRoot == otherRoot)
                    {
                        return false;
                    }

                    if (thisRoot._height < otherRoot._height)
                    {
                        thisRoot._parent = otherRoot;
                        otherRoot._groupSize += thisRoot._groupSize;
                        otherRoot._height = Math.Max(thisRoot._height + 1, otherRoot._height);
                        return true;
                    }
                    else
                    {
                        otherRoot._parent = thisRoot;
                        thisRoot._groupSize += otherRoot._groupSize;
                        thisRoot._height = Math.Max(otherRoot._height + 1, thisRoot._height);
                        return true;
                    }
                }

                public bool IsInSameGroup(UnionFindNode other) => this.FindRoot() == other.FindRoot();

                public override string ToString() => $"{Destination} root:{FindRoot().Destination}";
            }
        }


        [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.LayoutKind.Auto)]
        public struct Modular : IEquatable<Modular>, IComparable<Modular>
        {
            private const int _defaultMod = 1000000007;
            private readonly long _value;    // 0 <= value < Mod の値をとる
            public int Mod { get; }
            public int Value => (int)_value;

            public Modular(long value, int mod = _defaultMod)
            {
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }
                Mod = mod;

                if (value >= 0 && value < mod)
                {
                    _value = value;
                }
                else
                {
                    _value = value % mod;
                    if (Value < 0)
                    {
                        _value += mod;
                    }
                }
            }

            private static void CheckModEquals(Modular a, Modular b)
            {
                if (a.Mod != b.Mod)
                {
                    throw new ArgumentException($"{nameof(a)}, {nameof(b)}", $"両者の法（{nameof(Mod)}）は等しくなければなりません。");
                }
            }

            public static Modular operator +(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value + b._value;
                if (result > a.Mod)
                {
                    result -= a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator -(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value - b._value;
                if (result < 0)
                {
                    result += a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator *(Modular a, Modular b)
            {
                CheckModEquals(a, b);
                return new Modular(a._value * b._value, a.Mod);
            }

            public static Modular operator /(Modular a, Modular b)  // こいつは値渡しの方が速いっぽい
            {
                CheckModEquals(a, b);
                return a * Pow(b, a.Mod - 2);
            }

            // 需要は不明だけど一応
            public static bool operator ==(Modular left, Modular right) => left.Equals(right);
            public static bool operator !=(Modular left, Modular right) => !(left == right);
            public static bool operator <(Modular left, Modular right) => left.CompareTo(right) < 0;
            public static bool operator <=(Modular left, Modular right) => left.CompareTo(right) <= 0;
            public static bool operator >(Modular left, Modular right) => left.CompareTo(right) > 0;
            public static bool operator >=(Modular left, Modular right) => left.CompareTo(right) >= 0;

            public static explicit operator int(Modular a) => a.Value;
            public static explicit operator long(Modular a) => a._value;

            public static Modular Pow(Modular a, int n)
            {
                switch (n)
                {
                    case 0:
                        return new Modular(1, a.Mod);
                    case 1:
                        return a;
                    default:
                        var p = Pow(a, n / 2);
                        return p * p * Pow(a, n % 2);
                }
            }

            private static Dictionary<int, List<int>> FactorialCache { get; } = new Dictionary<int, List<int>>();
            private static Dictionary<int, int[]> FactorialInverseCache { get; } = new Dictionary<int, int[]>();
            private static Dictionary<int, List<int>> PermutationCache { get; } = new Dictionary<int, List<int>>();
            const int maxFactorial = 1000000;

            public static Modular Factorial(int n, int mod = _defaultMod)
            {
                if (n < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
                }
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }

                if (!FactorialCache.ContainsKey(mod))
                {
                    FactorialCache.Add(mod, new List<int>() { 1 });
                }

                var cache = FactorialCache[mod];
                for (int i = cache.Count; i <= n; i++)  // Countが1（0!までキャッシュ済み）のとき1!～n!まで計算
                {
                    cache.Add((int)((long)cache[i - 1] * i % mod));
                }
                return new Modular(cache[n], mod);
            }

            public static Modular Permutation(int n, int r, int mod = _defaultMod)
            {
                CheckNR(n, r);

                if (!PermutationCache.ContainsKey(mod ^ n))
                {
                    PermutationCache.Add(mod ^ n, new List<int>() { 1 });
                }

                var cache = PermutationCache[mod ^ n];
                for (int i = cache.Count; i <= r; i++)  // Countが1（nP0までキャッシュ済み）のときnP1～nPrまで計算
                {
                    cache.Add((int)((long)cache[i - 1] * (n - (i - 1)) % mod));
                }
                return new Modular(cache[r], mod);
            }

            public static Modular Combination(int n, int r, int mod = _defaultMod)
            {
                if (!FactorialInverseCache.ContainsKey(mod))
                {
                    InitializeCombinationTable(maxFactorial, mod);
                }
                CheckNR(n, r);
                r = Math.Min(r, n - r);
                return n == r ? new Modular(1, mod) : new Modular(FactorialCache[mod][n], mod) * new Modular(FactorialInverseCache[mod][r], mod) * new Modular(FactorialInverseCache[mod][n - r], mod);
            }

            private static void InitializeCombinationTable(int max, int mod)
            {
                Factorial(max);
                FactorialInverseCache.Add(mod, new int[max + 1]);

                long fInv = (new Modular(1, mod) / Factorial(max, mod)).Value;
                FactorialInverseCache[mod][max] = (int)fInv;
                for (int i = max - 1; i >= 0; i--)
                {
                    fInv = (fInv * (i + 1)) % mod;
                    FactorialInverseCache[mod][i] = (int)fInv;
                }
            }

            public static Modular CombinationWithRepetition(int n, int r, int mod = _defaultMod) => Combination(n + r - 1, r, mod);

            private static void CheckNR(int n, int r)
            {
                if (n < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません");
                }
                if (r < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
                }
                if (n < r)
                {
                    throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
                }
            }

            public override string ToString() => $"{_value} (mod {Mod})";

            public bool Equals(Modular other) => _value == other._value && Mod == other.Mod;

            public int CompareTo(Modular other) => Mod == other.Mod ? _value.CompareTo(other._value) : 0;

            public override int GetHashCode() => Value ^ Mod;
        }


    }
}


namespace Yorukatsu037
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionF();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace Yorukatsu037.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace Yorukatsu037.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion