Problem:
Score : 1000 points

Problem Statement

There are M chairs arranged in a line. The coordinate of the i-th chair (1 ≤ i ≤ M) is i.

N people of the Takahashi clan played too much games, and they are all suffering from backaches. They need to sit in chairs and rest, but they are particular about which chairs they sit in. Specifically, the i-th person wishes to sit in a chair whose coordinate is not greater than L_i, or not less than R_i. Naturally, only one person can sit in the same chair.

It may not be possible for all of them to sit in their favorite chairs, if nothing is done.
Aoki, who cares for the health of the people of the Takahashi clan, decides to provide additional chairs so that all of them can sit in chairs at their favorite positions.

Additional chairs can be placed at arbitrary real coordinates. Find the minimum required number of additional chairs.

Constraints

1 ≤ N,M ≤ 2 × 10^5

0 ≤ L_i < R_i ≤ M + 1(1 ≤ i ≤ N)

All input values are integers.

Input

Input is given from Standard Input in the following format:

N M
L_1 R_1
:
L_N R_N

Output

Print the minimum required number of additional chairs.

Sample Input 1

4 4
0 3
2 3
1 3
3 4

Sample Output 1

0

The four people can sit in chairs at the coordinates 3, 2, 1 and 4, respectively, and no more chair is needed.

Sample Input 2

7 6
0 7
1 5
3 6
2 7
1 6
2 6
3 7

Sample Output 2

2

If we place additional chairs at the coordinates 0 and 2.5, the seven people can sit at coordinates 0, 5, 3, 2, 6, 1 and 2.5, respectively.

Sample Input 3

3 1
1 2
1 2
1 2

Sample Output 3

2

Sample Input 4

6 6
1 6
1 6
1 5
1 5
2 6
2 6

Sample Output 4

2

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

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
//using System.Numerics;
using Number = System.Int64;
using static System.Math;
//using static MathEx;
//using P = System.Collections.Generic.KeyValuePair<int, int>;

using C = System.Int32;
using V = System.Int32;

namespace Program
{
    public class Solver
    {


        public void Solve()
        {
            var rand = new Random(0);
            for (int _ = 0; _ < 1; _++)
            {

                var n = ri;
                var m = ri;
                //var n = rand.Next(50, 100);
                //var m = rand.Next(50, 100);
                var A = Enumerate(m + 5, x => new List<int>());
                var stary = new StarySkyTree(m);
                stary.Push(0, m, 500000);
                for (int i = 0; i < m; i++)
                    stary.Push(i, m, -1);
                var L = new int[n];
                var R = new int[n];
                for (int i = 0; i < n; i++)
                {
                    var l = ri;
                    var r = ri;
                    //var l = rand.Next(0, 40);
                    //var r = rand.Next(m - 40, m + 2);
                    L[i] = l;
                    R[i] = r;
                }
                for (int i = 0; i < n; i++)
                {

                    var l = L[i];
                    var r = R[i];
                    A[l].Add(r);
                    stary.Push(m - r, m, 1);
                }
                var cnt = 0;
                var pqL = new PriorityQueue<int>();
                long min = 1000000000;
                var dame = 0;
                for (int i = 0; i <= m; i++, cnt++)
                {
                    //Debug.WriteLine(stary.Items.Select(x => x - 500000).AsJoinedString());
                    foreach (var x in A[i])
                    {
                        pqL.Enqueue(x);
                        stary.Push(m - x, m, -1);
                    }
                    while (pqL.Count > cnt)
                    {
                        var p = pqL.Dequeue();
                        if (p == m + 1)
                            dame++;
                        else stary.Push(m - p, m, 1);
                    }
                    var max = Max(0, stary.Query(0, m) - 500000) + dame;
                    min = Min(min, max);

                }
                if (min < n - m) min = n - m;
                IO.Printer.Out.WriteLine(min);
                //var res = debug(L, R, m);
                //Debug.WriteLine(res);
                //if (min != res)
                {

                }
            }
        }
        public int debug(int[] L, int[] R, int m)
        {
            var n = L.Length;
            var src = n + m + 8;
            var sink = n + m + 9;
            var G = new MinCostFlow(n + m + 10);
            for (int i = 1; i <= m; i++)
                G.AddDirectedEdge(n + i, sink, 1, 0);
            for (int i = 0; i < n; i++)
            {
                G.AddDirectedEdge(src, i, 1, 0);
                for (int j = 1; j <= L[i]; j++)
                    G.AddDirectedEdge(i, n + j, 1, 0);
                for (int j = R[i]; j <= m; j++)
                    G.AddDirectedEdge(i, n + j, 1, 0);
                G.AddDirectedEdge(i, sink, 1, 1);

            }
            var cost = G.Execute(src, sink);

            return cost.Value;
        }
        //*
        int ri => sc.Integer();
        long rl => sc.Long();
        double rd => sc.Double();
        string rs => sc.Scan();
        char rc => sc.Char();

        [System.Diagnostics.Conditional("DEBUG")]
        void put(params object[] a) => Debug.WriteLine(string.Join(" ", a));

        //*/
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

        static T[] Enumerate<T>(int n, Func<int, T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f(i);
            return a;
        }
        static void Swap<T>(ref T a, ref T b)
        {
            var tmp = a;
            a = b;
            b = tmp;
        }
    }
}

#region main

static class Ex
{
    public static string AsString(this IEnumerable<char> ie)
    {
        return new string(ie.ToArray());
    }

    public static string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ")
    {
        return string.Join(st, ie);
    }

    public static void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}

#endregion
#region Ex

namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;

    public class Printer: StreamWriter
    {
        static Printer()
        {
            Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false };
        }

        public static Printer Out { get; set; }

        public override IFormatProvider FormatProvider
        {
            get { return CultureInfo.InvariantCulture; }
        }

        public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true))
        {
        }

        public Printer(Stream stream, Encoding encoding) : base(stream, encoding)
        {
        }

        public void Write<T>(string format, T[] source)
        {
            base.Write(format, source.OfType<object>().ToArray());
        }

        public void WriteLine<T>(string format, T[] source)
        {
            base.WriteLine(format, source.OfType<object>().ToArray());
        }
    }

    public class StreamScanner
    {
        public StreamScanner(Stream stream)
        {
            str = stream;
        }

        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof;

        public bool IsEndOfStream
        {
            get { return isEof; }
        }

        private byte read()
        {
            if (isEof) return 0;
            if (ptr < len) return buf[ptr++];
            ptr = 0;
            if ((len = str.Read(buf, 0, 1024)) > 0) return buf[ptr++];
            isEof = true;
            return 0;
        }

        public char Char()
        {
            byte b;
            do b = read(); while ((b < 33 || 126 < b) && !isEof);
            return (char)b;
        }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }

        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }

        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0;
            byte b;
            var ng = false;
            do b = read(); while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-')
            {
                ng = true;
                b = read();
            }
            for (; ; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                ret = ret * 10 + b - '0';
            }
        }

        public int Integer()
        {
            return (isEof) ? int.MinValue : (int)Long();
        }

        public double Double()
        {
            var s = Scan();
            return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN;
        }

        static T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n)
        {
            return enumerate(n, Char);
        }

        public string[] Scan(int n)
        {
            return enumerate(n, Scan);
        }

        public double[] Double(int n)
        {
            return enumerate(n, Double);
        }

        public int[] Integer(int n)
        {
            return enumerate(n, Integer);
        }

        public long[] Long(int n)
        {
            return enumerate(n, Long);
        }
    }
}

#endregion



#region BinaryHeap
public class PriorityQueue<T>
{
    readonly List<T> heap = new List<T>();
    readonly IComparer<T> cmp;
    public PriorityQueue() { cmp = Comparer<T>.Default; }
    public PriorityQueue(Comparison<T> f) { cmp = Comparer<T>.Create(f); }
    public PriorityQueue(IComparer<T> c) { cmp = c; }
    public void Enqueue(T item)
    {

        var i = heap.Count;
        heap.Add(item);
        while (i > 0)
        {
            var p = (i - 1) / 2;
            if (cmp.Compare(heap[p], item) <= 0)
                break;
            heap[i] = heap[p];
            i = p;
        }
        heap[i] = item;

    }
    public T Dequeue()
    {
        var ret = heap[0];
        var i = 0;
        var x = heap[heap.Count - 1];

        while ((i * 2) + 1 < heap.Count - 1)
        {
            var a = i * 2 + 1;
            var b = i * 2 + 2;
            if (b < heap.Count - 1 && cmp.Compare(heap[b], heap[a]) < 0) a = b;
            if (cmp.Compare(heap[a], x) >= 0)
                break;
            heap[i] = heap[a];
            i = a;
        }
        heap[i] = x;
        heap.RemoveAt(heap.Count - 1);
        return ret;

    }
    public T Peek() { return heap[0]; }
    public int Count { get { return heap.Count; } }
    public bool Any() { return heap.Count > 0; }
    public T[] Items
    {
        get
        {
            var ret = heap.ToArray();
            Array.Sort(ret, cmp);
            return ret;
        }
    }
}
#endregion
#region Lazy Propagation
public interface ILazyData<T1, T2>
{
    T1 Merge(T1 l, T1 r);
    T1 Identity { get; }
    T1 Apply(int l, int r, T1 val, T2 x);
    T2 Push(T2 val, T2 x);
    T2 Zero { get; }
}
#endregion
#region LazyPropagation Segment Tree
public class LazyPropagationSegmentTree<T1, T2, U>
    where U : struct, ILazyData<T1, T2>
{
    int sz;
    int n;
    T1[] data;
    T2[] lazy;
    bool[] flag;
    U op;
    public LazyPropagationSegmentTree(int size)
    {
        sz = size;
        n = 1;
        while (n < sz)
            n <<= 1;
        data = new T1[n * 2];
        lazy = new T2[n * 2];
        flag = new bool[n * 2];
        for (int i = 1; i < n * 2; i++)
        {
            data[i] = op.Identity;
            lazy[i] = op.Zero;
        }
    }
    public LazyPropagationSegmentTree(T1[] a)
    {

        sz = a.Length;
        n = 1;
        while (n < sz)
            n <<= 1;
        data = new T1[n * 2];
        lazy = new T2[n * 2];
        flag = new bool[n * 2];
        for (int i = 0; i < n; i++)
        {
            if (i < sz)
                data[i + n] = a[i];
            else data[i + n] = op.Identity;
            lazy[i + n] = op.Zero;
        }
        for (int i = n - 1; i >= 1; i--)
        {
            data[i] = op.Merge(data[i * 2], data[i * 2 + 1]);
            lazy[i] = op.Zero;

        }
    }
    private void lazyEval(int k, int a, int b)
    {
        if (!flag[k])
            return;
        push(a, (a + b) / 2, lazy[k], k * 2, a, (a + b) / 2);
        push((a + b) / 2, b, lazy[k], k * 2 + 1, (a + b) / 2, b);
        lazy[k] = op.Zero;
        flag[k] = false;
    }
    private void eval(int k)
    {
        data[k] = op.Merge(data[k * 2], data[k * 2 + 1]);
    }

    public void Push(int a, int b, T2 v)
    {
        push(a, b, v, 1, 0, n);
    }
    private void push(int a, int b, T2 x, int k, int l, int r)
    {
        if (r <= a || b <= l)
            return;
        else if (a <= l && r <= b)
        {
            flag[k] = true;
            lazy[k] = op.Push(lazy[k], x);
            data[k] = op.Apply(l, r, data[k], x);
        }
        else
        {
            lazyEval(k, l, r);
            push(a, b, x, k << 1, l, (l + r) >> 1);
            push(a, b, x, (k << 1) + 1, (l + r) >> 1, r);
            eval(k);
        }
    }
    public T1 Query(int a, int b) { return query(a, b, 1, 0, n); }
    private T1 query(int a, int b, int k, int l, int r)
    {
        if (r <= a || b <= l)
            return op.Identity;
        if (a <= l && r <= b)
            return data[k];
        else
        {
            lazyEval(k, l, r);
            var vl = query(a, b, k << 1, l, (l + r) >> 1);
            var vr = query(a, b, (k << 1) + 1, (l + r) >> 1, r);
            eval(k);
            return op.Merge(vl, vr);
        }
    }
    public T1[] Items
    {
        get
        {
            var ret = new T1[sz];
            for (int i = 0; i < ret.Length; i++)
                ret[i] = Query(i, i + 1);
            return ret;
        }
    }
}
#endregion

#region StarySkyTree
public class StarySkyTree: LazyPropagationSegmentTree<long, long, StarySky>
{
    public StarySkyTree(int size) : base(size) { }
}
public struct StarySky: ILazyData<long, long>
{
    public long Identity { get { return 0; } }
    public long Zero { get { return 0; } }
    public long Apply(int l, int r, long val, long x) { return val + x; }
    public long Merge(long l, long r) { return Math.Max(l, r); }
    public long Push(long val, long x) { return val + x; }
}
#endregion

/// <summary>
/// 辺の追加を行い，最小費用流を求めることが可能なグラフです．
/// </summary>
public class MinCostFlow
{
    bool hasng;
    readonly List<Edge>[] G;
    /// <summary>
    /// 頂点数 <paramref name="V"/> で辺が存在しないグラフを構築します．
    /// </summary>
    /// <param name="V">頂点の数</param>
    /// <remarks>この操作は O(<paramref name="V"/>) で実行されます．</remarks>
    public MinCostFlow(int V)
    {
        G = new List<Edge>[V];
        for (int i = 0; i < V; i++)
            G[i] = new List<Edge>();

    }
    /// <summary>
    /// 頂点 <paramref name="from"/> から頂点 <paramref name="to"/> への容量 <paramref name="cap"/> ，コスト <paramref name="cost"/> の有向辺を追加します．
    /// </summary>
    /// <param name="from">始点となる頂点の 0-indexed での番号</param>
    /// <param name="to">終点となる頂点の 0-indexed での番号</param>
    /// <param name="cap">辺の容量</param>
    /// <param name="cost">辺のコスト</param>
    /// <remarks>この操作は O(1) で実行されます．</remarks>
    public void AddDirectedEdge(int from, int to, C cap, V cost)
    {
        addEdge(from, to, cap, 0, cost);
    }

    /// <summary>
    /// 頂点 <paramref name="src"/> から頂点 <paramref name="sink"/> へ最大で流量 <paramref name="f"/> だけ最小費用フローを流します．
    /// </summary>
    /// <param name="src">ソースとなる頂点の 0-indexed での番号</param>
    /// <param name="sink">シンクとなる頂点の 0-indexed での番号</param>
    /// <param name="f">ソースからシンクへと流す最大の容量．-1を指定した場合，最小費用最大流を求める．</param>
    /// <param name="inf">ソースからシンクまでの距離の最大値</param>
    /// <remarks>この操作はPrimal Dual法に基づき，最悪計算量 O(VE + FE log V) で実行されます．Vは頂点の数，Eは辺の数，Fは流量を表します．</remarks>
    public KeyValuePair<C, V> Execute(int src, int sink, C f = C.MaxValue, V inf = 1000)
    {
        var n = G.Length;
        var dist = new V[n];
        var prev = new int[n];
        var prevEdge = new Edge[n];
        var potential = new V[n];


        C flow = 0;
        V cost = 0;
        var first = hasng;
        while (f > 0)
        {
            for (int i = 0; i < G.Length; i++)
                dist[i] = inf;

            //shortest path
            if (first)//spfa
            {
                var q = new Queue<int>();
                q.Enqueue(src); dist[src] = 0;
                var inQ = new bool[n];
                while (q.Count > 0)
                {
                    var p = q.Dequeue();
                    inQ[p] = false;
                    foreach (var e in G[p])
                    {
                        var t = e.To;
                        var d = dist[p] + e.Cost;
                        if (e.Cap > 0 && d < dist[t])
                        {
                            if (!inQ[t])
                            {
                                inQ[t] = true;
                                q.Enqueue(t);
                            }
                            dist[t] = d; prev[t] = p; prevEdge[t] = e;
                        }
                    }
                }
                first = false;
            }
            else//dijkstra
            {
                var vis = new bool[n];
                var pq = new PriorityQueue<KeyValuePair<int, V>>((l, r) => l.Value.CompareTo(r.Value));
                pq.Enqueue(new KeyValuePair<int, V>(src, 0));
                dist[src] = 0;
                while (pq.Count > 0)
                {
                    var p = pq.Dequeue().Key;
                    if (vis[p]) continue;
                    vis[p] = true;
                    foreach (var e in G[p])
                    {
                        if (e.Cap <= 0) continue;
                        var t = e.To;
                        if (vis[t]) continue;
                        var d = dist[p] + e.Cost + potential[p] - potential[t];
                        if (dist[t] > d)
                        {
                            dist[t] = d; prev[t] = p; prevEdge[t] = e;
                            pq.Enqueue(new KeyValuePair<int, V>(t, d));
                        }
                    }
                }


            }


            //update
            {
                if (dist[sink] == inf) break;
                for (int i = 0; i < n; i++)
                    potential[i] += dist[i];
                C df = f;
                V d = 0;
                for (var v = sink; v != src; v = prev[v])
                {
                    var e = prevEdge[v];
                    df = Math.Min(df, e.Cap); d += e.Cost;
                }
                f -= df; cost += df * d; flow += df;
                for (var v = sink; v != src; v = prev[v])
                {
                    var e = prevEdge[v];
                    e.Cap -= df; e.Rev.Cap += df;
                }
            }

        }
        return new KeyValuePair<C, V>(flow, cost);
    }
    void addEdge(int f, int t, C c1, C c2, V e)
    {
        if (e < 0) hasng = true;
        var a = new Edge(t, c1, e);
        var b = new Edge(f, c2, -e);
        Edge.Link(a, b);
        G[f].Add(a);
        G[t].Add(b);
    }
    class Edge
    {
        public static void Link(Edge e1, Edge e2)
        {
            e1.Rev = e2; e2.Rev = e1;
        }
        public int To { get; private set; }
        public Edge Rev { get; private set; }
        public V Cost { get; private set; }
        public C Cap { get; set; }
        public Edge(int t, C c, V e)
        {
            To = t;
            Cap = c;
            Cost = e;
        }
        public override string ToString()
        {
            return string.Format("to: {0}, cap: {1}", To, Cap);
        }
    }
}