Problem:
Score : 1200 points

Problem Statement

#nck {
width: 30px;
height: auto;
}

Imagine a game played on a line. Initially, the player is located at position 0 with N candies in his possession, and the exit is at position E. There are also N bears in the game. The i-th bear is located at x_i. The maximum moving speed of the player is 1 while the bears do not move at all.

When the player gives a candy to a bear, it will provide a coin after T units of time. More specifically, if the i-th bear is given a candy at time t, it will put a coin at its position at time t+T. The purpose of this game is to give candies to all the bears, pick up all the coins, and go to the exit. Note that the player can only give a candy to a bear if the player is at the exact same position of the bear. Also, each bear will only produce a coin once. If the player visits the position of a coin after or at the exact same time that the coin is put down, the player can pick up the coin. Coins do not disappear until collected by the player.

Shik is an expert of this game. He can give candies to bears and pick up coins instantly. You are given the configuration of the game. Please calculate the minimum time Shik needs to collect all the coins and go to the exit.

Constraints

1 \leq N \leq 100,000

1 \leq T, E \leq 10^9

0 < x_i < E

x_i < x_{i+1} for 1 \leq i < N

All input values are integers.

Partial Scores

In test cases worth 600 points, N \leq 2,000.

Input

The input is given from Standard Input in the following format:

N E T
x_1 x_2 ... x_N

Output

Print an integer denoting the answer.

Sample Input 1

3 9 1
1 3 8

Sample Output 1

12

The optimal strategy is to wait for the coin after treating each bear. The total time spent on waiting is 3 and moving is 9. So the answer is 3 + 9 = 12.

Sample Input 2

3 9 3
1 3 8

Sample Output 2

16

Sample Input 3

2 1000000000 1000000000
1 999999999

Sample Output 3

2999999996

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

using System;
using System.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Number = System.Int64;

namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var xs = new long[n + 1];
            long t;
            {
                var e = sc.Long();
                t = sc.Long();
                for (int i = 0; i < n; i++)
                    xs[i] = sc.Long();
                xs[n] = e;
            }
            var dp = Enumerate(n + 1, x => 1L << 60);
            dp[0] = xs[0];

            var min = 1L << 60;
            var deq = new Deque<KeyValuePair<int, long>>();
            deq.PushBack(new KeyValuePair<int, long>(0, dp[0]));

            for (int i = 1; i <= n; i++)
            {
                while (deq.Count > 0)
                {
                    var p = deq[0].Key;
                    if (2 * (xs[i - 1] - xs[p]) >= t)
                        min = Math.Min(min, 3 * (xs[i - 1] - xs[p]) + deq.PopFront().Value);
                    else break;
                }
                dp[i] = Math.Min(dp[i], min);
                if (deq.Count > 0)
                    dp[i] = Math.Min(dp[i], deq[0].Value + t + xs[i - 1] - xs[deq[0].Key]);
                /*
                for (int j = i + 1; j <= n; j++)
                {
                    var dx = xs[j] - xs[j - 1];
                    dp[j] = Math.Min(dp[j], Math.Max(3 * d, t + d) + dp[i] + dx);
                    d += dx;
                }
                Debug.WriteLine(dp.Items.AsJoinedString());
                */
                var dx = xs[i] - xs[i - 1];
                min += 3 * dx;
                dp[i] += dx;
                //
                while (deq.Count > 0)
                {
                    var p = deq[deq.Count - 1];
                    if (p.Value + xs[i] - xs[p.Key] > dp[i]) deq.PopBack();
                    else break;
                }
                deq.PushBack(new KeyValuePair<int, Number>(i, dp[i]));
            }
            IO.Printer.Out.WriteLine(dp[n]);
        }
        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 public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}
#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public 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(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.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 = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; 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 = 0; 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 (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else 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; }
        private 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 Deque<T>

public class Deque<T>
{
    T[] buf;
    int offset;
    int count;
    int capacity;
    public int Count { get { return count; } }
    public Deque(int cap) { buf = new T[capacity = cap]; }
    public Deque() { buf = new T[capacity = 8]; }
    public T this[int index]
    {
        get { return buf[getIndex(index)]; }
        set { buf[getIndex(index)] = value; }
    }
    private int getIndex(int index)
    {
        if (index >= capacity)
            throw new IndexOutOfRangeException("out of range");
        var ret = index + offset;
        if (ret >= capacity)
            ret -= capacity;
        return ret;
    }
    public void PushFront(T item)
    {
        if (count == capacity) Extend();
        if (--offset < 0) offset += buf.Length;
        buf[offset] = item;
        ++count;
    }
    public T PopFront()
    {
        if (count == 0)
            throw new InvalidOperationException("collection is empty");
        --count;
        var ret = buf[offset++];
        if (offset >= capacity) offset -= capacity;
        return ret;
    }
    public void PushBack(T item)
    {
        if (count == capacity) Extend();
        var id = count++ + offset;
        if (id >= capacity) id -= capacity;
        buf[id] = item;
    }
    public T PopBack()
    {
        if (count == 0)
            throw new InvalidOperationException("collection is empty");
        return buf[getIndex(--count)];
    }
    public void Insert(T item, int index)
    {
        if (index > count) throw new IndexOutOfRangeException();
        this.PushFront(item);
        for (int i = 0; i < index; i++)
            this[i] = this[i + 1];
        this[index] = item;
    }
    public T RemoveAt(int index)
    {
        if (index < 0 || index >= count) throw new IndexOutOfRangeException();
        var ret = this[index];
        for (int i = index; i > 0; i--)
            this[i] = this[i - 1];
        this.PopFront();
        return ret;
    }
    private void Extend()
    {
        T[] newBuffer = new T[capacity << 1];
        if (offset > capacity - count)
        {
            var len = buf.Length - offset;
            Array.Copy(buf, offset, newBuffer, 0, len);
            Array.Copy(buf, 0, newBuffer, len, count - len);
        }
        else Array.Copy(buf, offset, newBuffer, 0, count);
        buf = newBuffer;
        offset = 0;
        capacity <<= 1;
    }
    public T[] Items
    {
        get
        {
            var a = new T[count];
            for (int i = 0; i < count; i++)
                a[i] = this[i];
            return a;
        }
    }
}

#endregion