Problem:
E: LISum

問題

長さ $N$ の数列 $A$ が与えられる。
数列 $A$ の最長増加部分列のひとつを $B$ とするとき、$\sum B_i$ の最大値を求めよ。

数列 $A$ の最長増加部分列とは、すべての $i < j$ で $A_i < A_j$ を満たす部分列のうち、最長なものを示す。

制約

入力値は全て整数である。

$1 \leq N \leq 10^5$

$0 \leq A_i \leq 10^5$

入力形式

入力は以下の形式で与えられる。

$N$

$A_1 \dots A_N$

出力

数列 $A$ の最長増加部分列のひとつを $B$ とするとき、$\sum B_i$ の最大値を出力せよ。また、末尾に改行も出力せよ。

サンプル

サンプル入力 1

4
6 4 7 8

サンプル出力 1

21

最長増加部分列は $ (6, 7, 8)$ と $(4, 7, 8)$ である。よって最大値は $21$ である。

サンプル入力 2

3
1000 2 3

サンプル出力 2

5

最長増加部分列は $(2,3)$ のみである。

サンプル入力 3

7
17 17 13 4 20 12 15

サンプル出力 3

31

サンプル入力 4

7
19 16 14 9 4 20 2

サンプル出力 4

39

Solution in C# (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Linq.Expressions;
using System.IO;
using System.Text;
using System.Diagnostics;

using static util;
using P = pair<int, int>;

using Binary = System.Func<System.Linq.Expressions.ParameterExpression,
                           System.Linq.Expressions.ParameterExpression,
                           System.Linq.Expressions.BinaryExpression>;
using Unary = System.Func<System.Linq.Expressions.ParameterExpression,
                          System.Linq.Expressions.UnaryExpression>;

class Program {
    static StreamWriter sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
    static Scan sc = new Scan();
    const int M = 1000000007;
    const int M2 = 998244353;
    const long LM = 1L << 60;
    const double eps = 1e-11;
    static readonly int[] dd = { 0, 1, 0, -1, 0 };
    const string dstring = "RDLU";
    static void Main(string[] args)
    {
        int n = sc.Int;
        var a = sc.IntArr;
        var dp = new int[n];
        for (int i = 0; i < n; i++)
        {
            dp[i] = M;
        }
        var p = new int[n];
        for (int i = 0; i < n; i++)
        {
            p[i] = dp.UpperBound(a[i] - 1);
            dp[dp.UpperBound(a[i] - 1)] = a[i];
        }
        int k = n;
        for (int i = 0; i < n; i++)
        {
            if (dp[i] == M) {
                k = i;
                break;
            }
        }
        for (int i = 0; i < n; i++)
        {
            if (p[i] == k - 1) {
                var b = new long[k];
                b[k - 1] = a[i];
                for (int j = i - 1; j >= 0 ; j--)
                {
                    if (a[j] < b[p[j] + 1]) {
                        b[p[j]] = a[j];
                    }
                }
                DBG(b.Sum());
                return;
            }
        }

        sw.Flush();
    }

    static void DBG(string a) => Console.WriteLine(a);
    static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    static void DBG(params object[] a) => DBG(string.Join(" ", a));
    static void Prt(string a) => sw.WriteLine(a);
    static void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    static void Prt(params object[] a) => Prt(string.Join(" ", a));
}
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; }
    public static bool operator>(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) > 0;
    public static bool operator<(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) < 0;
    public static bool operator>=(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) >= 0;
    public static bool operator<=(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) <= 0;
}
static class util {
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static T sq<T>(T a) => Operator<T>.Multiply(a, a);
    public static T Max<T>(params T[] a) => a.Max();
    public static T Min<T>(params T[] a) => a.Min();
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < 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 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;
    }
}
static class Operator<T> {
    static readonly ParameterExpression x = Expression.Parameter(typeof(T), "x");
    static readonly ParameterExpression y = Expression.Parameter(typeof(T), "y");
    public static readonly Func<T, T, T> Add = Lambda(Expression.Add);
    public static readonly Func<T, T, T> Subtract = Lambda(Expression.Subtract);
    public static readonly Func<T, T, T> Multiply = Lambda(Expression.Multiply);
    public static readonly Func<T, T, T> Divide = Lambda(Expression.Divide);
    public static readonly Func<T, T> Plus = Lambda(Expression.UnaryPlus);
    public static readonly Func<T, T> Negate = Lambda(Expression.Negate);
    public static Func<T, T, T> Lambda(Binary op) => Expression.Lambda<Func<T, T, T>>(op(x, y), x, y).Compile();
    public static Func<T, T> Lambda(Unary op) => Expression.Lambda<Func<T, T>>(op(x), x).Compile();
}

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]); }
}

static class Search
{
    // x より大きいの項の index のうち最小のものを返す
    // x 以下の項の index のうち最大のもの + 1 を返す
    // そこに挿入
    public static int UpperBound<T>(this IList<T> a, T x) where T : IComparable<T> {
        int le = -1, u = a.Count;
        while (le < u - 1) {
            int m = (le + u) >> 1;
            if (a[m].CompareTo(x) > 0) u = m;
            else le = m;
        }
        return le + 1;
    }
    public static int UpperBound<T>(this IList<T> a, T x, Func<T, T, int> compare) {
        int le = -1, u = a.Count;
        while (le < u - 1) {
            int m = (le + u) >> 1;
            if (compare(a[m], x) > 0) u = m;
            else le = m;
        }
        return le + 1;
    }

    // n 未満の項の index のうち最大のもの + 1 を返す
    // n 以上の項の index のうち最小のものを返す
    // そこに挿入
    public static int LowerBound<T>(this IList<T> a, T x) where T : IComparable<T> {
        int l = -1, ue = a.Count;
        while (l < ue - 1) {
            int m = (l + ue) >> 1;
            if (a[m].CompareTo(x) < 0) l = m;
            else ue = m;
        }
        return ue;
    }
    public static int LowerBound<T>(this IList<T> a, T x, Func<T, T, int> compare) {
        int l = -1, ue = a.Count;
        while (l < ue - 1) {
            int m = (l + ue) >> 1;
            if (compare(a[m], x) < 0) l = m;
            else ue = m;
        }
        return ue;
    }
}