Problem:
Score : 900 points

Problem Statement

There are N robots and M exits on a number line.
The N + M coordinates of these are all integers and all distinct.
For each i (1 \leq i \leq N), the coordinate of the i-th robot from the left is x_i.
Also, for each j (1 \leq j \leq M), the coordinate of the j-th exit from the left is y_j.

Snuke can repeatedly perform the following two kinds of operations in any order to move all the robots simultaneously:

Increment the coordinates of all the robots on the number line by 1.

Decrement the coordinates of all the robots on the number line by 1.

Each robot will disappear from the number line when its position coincides with that of an exit, going through that exit.
Snuke will continue performing operations until all the robots disappear.

When all the robots disappear, how many combinations of exits can be used by the robots?
Find the count modulo 10^9 + 7.
Here, two combinations of exits are considered different when there is a robot that used different exits in those two combinations.

Constraints

1 \leq N, M \leq 10^5

1 \leq x_1 < x_2 < ... < x_N \leq 10^9

1 \leq y_1 < y_2 < ... < y_M \leq 10^9

All given coordinates are integers.

All given coordinates are distinct.

Input

Input is given from Standard Input in the following format:

N M
x_1 x_2 ... x_N
y_1 y_2 ... y_M

Output

Print the number of the combinations of exits that can be used by the robots when all the robots disappear, modulo 10^9 + 7.

Sample Input 1

2 2
2 3
1 4

Sample Output 1

3

The i-th robot from the left will be called Robot i, and the j-th exit from the left will be called Exit j.
There are three possible combinations of exits (the exit used by Robot 1, the exit used by Robot 2) as follows:

(Exit 1, Exit 1)

(Exit 1, Exit 2)

(Exit 2, Exit 2)

Sample Input 2

3 4
2 5 10
1 3 7 13

Sample Output 2

8

The exit for each robot can be chosen independently, so there are 2^3 = 8 possible combinations of exits.

Sample Input 3

4 1
1 2 4 5
3

Sample Output 3

1

Every robot uses Exit 1.

Sample Input 4

4 5
2 5 7 11
1 3 6 9 13

Sample Output 4

6

Sample Input 5

10 10
4 13 15 18 19 20 21 22 25 27
1 5 11 12 14 16 23 26 29 30

Sample Output 5

22

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

using System;
using System.Text;
using System.Collections.Generic;
using System.Linq;
class Solve{
    public Solve(){}
    StringBuilder sb;
    ReadData re;
    public static int Main(){
        new Solve().Run();
        return 0;
    }
    void Run(){
        sb = new StringBuilder();
        re = new ReadData();
        Calc();
        Console.Write(sb.ToString());
    }
    void Calc(){
        Data[] D;
        {
            int n = re.i();
            int M = re.i();
            int[] x = re.ia();
            int[] y = re.ia();
            List<Data> DL = new List<Data>();
            for(int i=0;i<n;i++){
                int bf = 0;
                int bl = M;
                while(bf != bl){
                    int bc = (bf+bl)/2;
                    if(bc == M || x[i] < y[bc]){
                        bl = bc;
                    }
                    else{
                        bf = bc+1;
                    }
                }
                if(bf != 0 && bf != M){
                    DL.Add(new Data(x[i]-y[bf-1],y[bf]-x[i]));
                }
            }
            D = DL.ToArray();
        }
        int N = D.Length;
        if(N == 0){
            sb.Append("1\n");
            return;
        }
        Array.Sort(D,(x,y)=>(x.L == y.L ? y.R - x.R : x.L - y.L));
        Compression Comp;
        {
            long[] R = new long[N];
            for(int i=0;i<N;i++){
                R[i] = D[i].R;
            }
            Comp = new Compression(R);
        }
        SegTree Seg = new SegTree(Comp.size,this);
        long count = 1;
        for(int i=0;i<N;i++){
            if(i != 0 && D[i].L == D[i-1].L && D[i].R == D[i-1].R){
                continue;
            }
            int p = Comp.ToPoint(D[i].R);
            long C = 1;
            if(p != 0){
                C += Seg.Sum(0,p-1);
            }
            Seg.Add(p,C);
            count += C;
        }
        count %= Define.mod;
        sb.Append(count+"\n");
    }
}
class Compression{
    public int size;
    public long[] A;
    public Compression(long[] a){
        long[] b = new long[a.Length];
        for(int i=0;i<a.Length;i++){
            b[i] = a[i];
        }
        a = b;
        Array.Sort(a);
        List<long> AL = new List<long>();
        if(a.Length != 0){
            AL.Add(a[0]);
        }
        for(int i=1;i<a.Length;i++){
            if(a[i] != a[i-1]){
                AL.Add(a[i]);
            }
        }
        A = AL.ToArray();
        size = A.Length;
    }
    public int ToPoint(long X){
        int bf = 0;
        int bl = size-1;
        while(bf != bl){
            int bc = (bf+bl)/2;
            if(A[bc] >= X){
                bl = bc;
            }
            else{
                bf = bc+1;
            }
        }
        return bf;
    }
    public long Value(int i){
        return A[i];
    }
}

struct SegTree{
    Solve Sol;
    public long[] X;
    //葉の最初の場所
    public int segf;
    //segfのbit
    public int segfb;
    //葉の深さの内浅い方
    public int depth;
    //浅い葉の内座標が最大のもの
    public int borderP;
    //はみ出すものの数
    public int border;
    long C;
    public SegTree(int N,Solve So){
        C = 0;
        Sol = So;
        X = new long[2*N-1];
        for(depth = 1;(1 << depth) <= N;depth++){
            ;
        }
        segf = N-1;
        borderP = (1 << depth) - 2;
        border = 2*N - borderP - 2;
        segfb = border/2;
    }
    public int ToBit(int p){
        if(p >= border){
            return segfb + p - border;
        }
        else{
            return p;
        }
    }
    //葉のある場所
    public int ToLeaf(int p){
        if(p >= border){
            return segf + p - border;
        }
        else{
            return borderP + p + 1;
        }
    }
    //葉の示す頂点
    public int ToPoint(int leaf){
        if(leaf > borderP){
            return leaf - borderP - 1;
        }
        else{
            return leaf - segf + border;
        }
    }
    public void Add(int p,long value){
        //葉の作業
        X[ToLeaf(p)] = (X[ToLeaf(p)] + value) % Define.mod;
        //親の作業
        for(int v=ToLeaf(p);v!=0;v=(v-1)/2){
            int spa = (v-1)/2;
            X[spa] = (X[spa] + value) % Define.mod;
        }
    }
    //vの子の内bより大きいものに(bは子孫,bはToBitしたもの)
    void More(int b,int v,int depth){
        if((b & (1 << depth)) == 0){
            //右の子は全てbより大きい
            if(((b+1) % (1 << depth)) != 0){
                More(b,v*2+1,depth-1);
                Act(v*2+2);
            }
            else{
                Act(v*2+2);
            }
        }
        else{
            //左の子は全てb未満
            More(b,v*2+2,depth-1);
        }
    }
    //vの子の内bより小さなものに(bは子孫,bはToBitしたもの)
    void Less(int b,int v,int depth){
        if((b & (1 << depth)) == 0){
            //右の子は全てbより大きい
            Less(b,v*2+1,depth-1);
        }
        else{
            //左の子は全てb未満
            if(b % (1 << depth) != 0){
                Act(v*2+1);
                Less(b,v*2+2,depth-1);
            }
            else{
                //右の子は全てb以上
                Act(v*2+1);
            }
        }
    }
    void Section(int bl,int br,int v,int dl,int dr){
        if(v >= segf){
            //葉
            Act(v);
        }
        else{
            if((bl & (1 << dl)) == 0 && (br & (1 << dr)) == 0){
                //両方左
                Section(bl,br,v*2+1,dl-1,dr-1);
            }
            else if((bl & (1 << dl)) != 0 && (br & (1 << dr)) != 0){
                //両方右
                Section(bl,br,v*2+2,dl-1,dr-1);
            }
            else{
                //左と右に分かれた
                if(bl % (1 << dl) != 0){
                    More(bl-1,v*2+1,dl-1);
                }
                else{
                    //左は全て含まれる
                    Act(v*2+1);
                }
                if((br + 1) % (1 << dl) != 0){
                    Less(br+1,v*2+2,dr-1);
                }
                else{
                    //右は全て含まれる
                    Act(v*2+2);
                }
            }
        }
    }
    public long Sum(int l0,int r0){
        C = 0;
        int bl = ToBit(l0);
        int br = ToBit(r0);
        int dl = l0 >= border ? depth - 2 : depth - 1;
        int dr = r0 >= border ? depth - 2 : depth - 1;
        Section(bl,br,0,dl,dr);
        return C;
    }
    void Act(int v){
        C += X[v];
    }
}
struct Data{
    public int L;
    public int R;
    public Data(int l0,int r0){
        L = l0;
        R = r0;
    }
}
class ReadData{
    string[] str;
    int counter;
    public ReadData(){
        counter = 0;
    }
    public string s(){
        if(counter == 0){
            str = Console.ReadLine().Split(' ');
            counter = str.Length;
        }
        counter--;
        return str[str.Length-counter-1];
    }
    public int i(){
        return int.Parse(s());
    }
    public long l(){
        return long.Parse(s());
    }
    public double d(){
        return double.Parse(s());
    }
    public int[] ia(int N){
        int[] ans = new int[N];
        for(int j=0;j<N;j++){
            ans[j] = i();
        }
        return ans;
    }
    public int[] ia(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        int[] ans = new int[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = int.Parse(str[j]);
        }
        return ans;
    }
    public long[] la(int N){
        long[] ans = new long[N];
        for(int j=0;j<N;j++){
            ans[j] = l();
        }
        return ans;
    }
    public long[] la(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        long[] ans = new long[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = long.Parse(str[j]);
        }
        return ans;
    }
    public double[] da(int N){
        double[] ans = new double[N];
        for(int j=0;j<N;j++){
            ans[j] = d();
        }
        return ans;
    }
    public double[] da(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        double[] ans = new double[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = double.Parse(str[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int[] f,int[] t){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<f.Length;j++){
            ans[f[j]].Add(t[j]);
            ans[t[j]].Add(f[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int M){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
    public List<int>[] g(){
        int N = i();
        int M = i();
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
}
public static class Define{
    public const long mod = 1000000007;
}
public static class Debug{
    public static void Print(double[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(long[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(int[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
}