Problem:
Score : 700 points

Problem Statement

You are given two sequences a and b, both of length 2N. The i-th elements in a and b are a_i and b_i, respectively.
Using these sequences, Snuke is doing the job of calculating the beauty of pairs of balanced sequences of parentheses (defined below) of length 2N.
The beauty of a pair (s,t) is calculated as follows:

Let X=0.

For each i between 1 and 2N (inclusive), increment X by a_i if s_i = t_i, and increment X by b_i otherwise.

The beauty of (s,t) is the final value of X.

You will be given Q queries. Process them in order.
In the i-th query, update the value of a_{p_i} to x_i, and the value of b_{p_i} to y_i. Then, find the maximum possible beauty of a pair of balanced sequences of parentheses.

In this problem, only the sequences below are defined to be balanced sequences of parentheses.

An empty string

The concatenation of (, s, ) in this order, where s is a balanced sequence of parentheses

The concatenation of s, t in this order, where s and t are balanced sequences of parentheses

Constraints

1 \leq N,Q \leq 10^{5}

-10^{9} \leq a_i,b_i,x_i,y_i \leq 10^{9}

1 \leq p_i \leq 2N

All input values are integers.

Partial Scores

In the test set worth 200 points, N \leq 5 and Q \leq 10.

In the test set worth 300 points, Q = 1.

Input

Input is given from Standard Input in the following format:

N Q
a_1 a_2 ... a_{2N}
b_1 b_2 ... b_{2N}
p_1 x_1 y_1
:
p_Q x_Q y_Q

Output

Print Q lines. The i-th line should contain the response to the i-th query.

Sample Input 1

2 2
1 1 7 3
4 2 3 3
2 4 6
3 2 5

Sample Output 1

15
15

The first query updates a and b to a=(1,4,7,3),b=(4,6,3,3). The maximum possible beauty is 15 for (s,t) =(()(),()()).

The second query updates a and b to a=(1,4,2,3),b=(4,6,5,3). The maximum possible beauty is 15 for (s,t) =(()(),(())).

Sample Input 2

7 7
34 -20 -27 42 44 29 9 11 20 44 27 19 -31 -29
46 -50 -11 20 28 46 12 13 33 -22 -48 -27 35 -17
7 27 34
12 -2 22
4 -50 -12
3 -32 15
8 -7 23
3 -30 11
4 -2 23

Sample Output 2

311
312
260
286
296
292
327

Solution in C# (Mono-csc 3.5.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Text;
using System.Text.RegularExpressions;
using System.Diagnostics;
using static util;
using P = pair<int, int>;

class Program {
    static void Main(string[] args) {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        var solver = new Solver(sw);
        // var t = new Thread(solver.solve, 1 << 26); // 64 MB
        // t.Start();
        // t.Join();
        solver.solve();
        sw.Flush();
    }
}

class Solver {
    StreamWriter sw;
    Scan sc;
    void Prt(string a) => sw.WriteLine(a);
    void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    void Prt(params object[] a) => Prt(string.Join(" ", a));
    public Solver(StreamWriter sw) {
        this.sw = sw;
        this.sc = new Scan();
    }

    public void solve() {
        int n, q;
        sc.Multi(out n, out q);
        n *= 2;
        var a = sc.IntArr;
        var b = sc.IntArr;
        long sum = 0;
        foreach (var item in a)
        {
            sum += item;
        }
        var pls = new SortedSet<P>();
        var mns = new SortedSet<P>();
        for (int i = 1; i < n - 1; i++)
        {
            int d = b[i] - a[i];
            if (d >= 0) {
                pls.Add(new P(d, i));
                sum += d;
            }
            else {
                mns.Add(new P(-d, i));
            }
        }
        for (int _ = 0; _ < q; _++)
        {
            int i, x, y;
            sc.Multi(out i, out x, out y);
            --i;
            if (i > 0 && i < n - 1) {
                {
                    int d = b[i] - a[i];
                    if (d >= 0) {
                        pls.Remove(new P(d, i));
                        sum -= d;
                    }
                    else {
                        mns.Remove(new P(-d, i));
                    }
                }
                {
                    int d = y - x;
                    if (d >= 0) {
                        pls.Add(new P(d, i));
                        sum += d;
                    }
                    else {
                        mns.Add(new P(-d, i));
                    }
                }
            }
            sum += x - a[i];
            a[i] = x;
            b[i] = y;
            if (pls.Count % 2 == 0) {
                Prt(sum);
            }
            else {
                Prt(sum - Math.Min(pls.Min.v1, mns.Min.v1));
            }
        }

    }
}

class pair<T, U> : IComparable<pair<T, U>> {
    public T v1;
    public U v2;
    public pair() : this(default(T), default(U)) {}
    public pair(T v1, U v2) { this.v1 = v1; this.v2 = v2; }
    public int CompareTo(pair<T, U> a) {
        int c = Comparer<T>.Default.Compare(v1, a.v1);
        return c != 0 ? c : Comparer<U>.Default.Compare(v2, a.v2);
    }
    public override string ToString() => v1 + " " + v2;
    public void Deconstruct(out T a, out U b) { a = v1; b = v2; }
}
static class util {
    public static readonly int M = 1000000007;
    // public static readonly int M = 998244353;
    public static readonly long LM = 1L << 60;
    public static readonly double eps = 1e-11;
    public static void DBG(string a) => Console.Error.WriteLine(a);
    public static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    public static void DBG(params object[] a) => DBG(string.Join(" ", a));
    public static void Assert(params bool[] conds) {
        if (conds.Any(x => !x)) throw new Exception();
    }
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static int CompareList<T>(IList<T> a, IList<T> b) where T : IComparable<T> {
        for (int i = 0; i < a.Count && i < b.Count; i++)
            if (a[i].CompareTo(b[i]) != 0) return a[i].CompareTo(b[i]);
        return a.Count.CompareTo(b.Count);
    }
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < w;
    static readonly int[] dd = { 0, 1, 0, -1 };
    // static readonly string dstring = "RDLU";
    public static P[] adjacents(int i, int j)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1])).ToArray();
    public static P[] adjacents(int i, int j, int h, int w)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1]))
                                         .Where(p => inside(p.v1, p.v2, h, w)).ToArray();
    public static P[] adjacents(this P p) => adjacents(p.v1, p.v2);
    public static P[] adjacents(this P p, int h, int w) => adjacents(p.v1, p.v2, h, w);
    public static Dictionary<T, int> compress<T>(this IEnumerable<T> a)
        => a.Distinct().OrderBy(v => v).Select((v, i) => new { v, i }).ToDictionary(p => p.v, p => p.i);
    public static Dictionary<T, int> compress<T>(params IEnumerable<T>[] a) => compress(a.Aggregate(Enumerable.Union));
    public static T[] inv<T>(this Dictionary<T, int> dic) {
        var res = new T[dic.Count];
        foreach (var item in dic) res[item.Value] = item.Key;
        return res;
    }
    public static void swap<T>(ref T a, ref T b) where T : struct { var t = a; a = b; b = t; }
    public static void swap<T>(this IList<T> a, int i, int j) where T : struct { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a) {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
    public static void Deconstruct<T>(this IList<T> v, out T a) {
        a = v[0];
    }
    public static void Deconstruct<T>(this IList<T> v, out T a, out T b) {
        a = v[0]; b = v[1];
    }
    public static void Deconstruct<T>(this IList<T> v, out T a, out T b, out T c) {
        a = v[0]; b = v[1]; c = v[2];
    }
    public static void Deconstruct<T>(this IList<T> v, out T a, out T b, out T c, out T d) {
        a = v[0]; b = v[1]; c = v[2]; d = v[3];
    }
    public static void Deconstruct<T>(this IList<T> v, out T a, out T b, out T c, out T d, out T e) {
        a = v[0]; b = v[1]; c = v[2]; d = v[3]; e = v[4];
    }
}

class Scan {
    StreamReader sr;
    public Scan() { sr = new StreamReader(Console.OpenStandardInput()); }
    public Scan(string path) { sr = new StreamReader(path); }
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public string Str => sr.ReadLine().Trim();
    public pair<T, U> Pair<T, U>() {
        T a; U b;
        Multi(out a, out b);
        return new pair<T, U>(a, b);
    }
    public P P => Pair<int, int>();
    public int[] IntArr => StrArr.Select(int.Parse).ToArray();
    public long[] LongArr => StrArr.Select(long.Parse).ToArray();
    public double[] DoubleArr => StrArr.Select(double.Parse).ToArray();
    public string[] StrArr => Str.Split(new[]{' '}, StringSplitOptions.RemoveEmptyEntries);
    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
                       : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>()   ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    public void Multi<T>(out T a) => a = cv<T>(Str);
    public void Multi<T, U>(out T a, out U b) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]);
    }
    public void Multi<T, U, V>(out T a, out U b, out V c) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]);
    }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]);
    }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]);
    }
}