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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)


import java.io.*;
import java.math.*;
import java.util.*;

import static java.util.Arrays.*;

public class Main {
	private static final int mod = (int)924844033;

	final Random random = new Random(0);
	final IOFast io = new IOFast();

	/// MAIN CODE
	public void run() throws IOException {
//		int TEST_CASE = Integer.parseInt(new String(io.nextLine()).trim());
		int TEST_CASE = 1;
		while(TEST_CASE-- != 0) {
			int n = io.nextInt();
			long e = io.nextInt();
			long T = io.nextInt();
			long[] x = io.nextLongArray(n);
			long[] d = new long[n];
			for (int i = 0; i < n; i++) {
				d[i] = x[i] - (i > 0 ? x[i-1] : 0);
			}
			
			SegRMQLong seg = new SegRMQLong(n+1);
			long[] ans = new long[n+1];
			for (int i = 0, l = 0; i < n; i++) {
				seg.add(0, l, 3 * d[i]);
				seg.add(l, i, d[i]);
				for (; (x[i] - x[l]) * 2 > T; l++) {
					seg.set(l, ans[l] + d[l] + (x[i] - x[l]) * 3);
				}
				// T - 2*d(j)
				ans[i+1] = ans[i] + d[i] + T;
				ans[i+1] = Math.min(ans[i+1], seg.get(0, i));
//				dump(i, ans[i] + d[i] + T, seg.get(0, 1), seg.get(1, 2));
				seg.set(i, ans[i] + T + d[i]);
			}
//			dump(ans);
			io.out.println(ans[n] + e - x[n-1]);
		}
	}
	
	static
	class SegRMQLong {
		final long INF = 1L<<60;
		
		final int n;
		final long[] seg;
		final long[] add;
		
		public SegRMQLong(final int n) {
			this.n = Integer.highestOneBit(n) << 1;
			seg = new long[this.n << 1];
			add = new long[this.n << 1];
			Arrays.fill(seg, INF);
		}
		
		void propagate(int curL, int curR, int k) {
			if(add[k] != 0) {
				final int curM = (curL + curR) / 2;
				add(curL, curR, 2 * k + 1, add[k], curL, curM);
				add(curL, curR, 2 * k + 2, add[k], curM, curR);
				add[k] = 0;
			}
		}

		long get(int l, int r) { return get(l, r, 0, 0, n); }
		long get(int l, int r, int k, int curL, int curR) {
			if(curR <= l || curL >= r) return INF;
			if(l <= curL && curR <= r) { return seg[k]; }
			propagate(curL, curR, k);
			final int curM = (curL + curR) / 2;
			return Math.min(
					get(l, r, 2 * k + 1, curL, curM),
					get(l, r, 2 * k + 2, curM, curR));
		}

		void add(int l, int r, long v) { add(l, r, 0, v, 0, n); }
		void add(int l, int r, int k, long v, int curL, int curR) {
			if(curR <= l || curL >= r) return;
			if(l <= curL && curR <= r) { seg[k] += v; add[k] += v; return; }
			final int curM = (curL + curR) / 2;
			propagate(curL, curR, k);
			add(l, r, 2 * k + 1, v, curL, curM);
			add(l, r, 2 * k + 2, v, curM, curR);
			seg[k] = Math.min(seg[2*k+1], seg[2*k+2]);
		}
		
		void set(int i, long v) {
			add(i, i + 1, 0);
			
			i += n - 1;
			seg[i] = v;
			add[i] = 0;
			while(i != 0) {
				i = (i - 1) / 2;
				seg[i] = Math.min(seg[2*i+1], seg[2*i+2]) + add[i];
			}
		}
	}
	
	void shuffle(int[] p) {
		for (int i = 1; i < p.length; i++) {
			int idx = random.nextInt(i + 1);
			swap(p, i, idx);
		}
	}
	
	/// TEMPLATE
	static int gcd(int n, int r) { return r == 0 ? n : gcd(r, n%r); }
	static long gcd(long n, long r) { return r == 0 ? n : gcd(r, n%r); }
	
	static <T> void swap(T[] x, int i, int j) { T t = x[i]; x[i] = x[j]; x[j] = t; }
	static void swap(int[] x, int i, int j) { int t = x[i]; x[i] = x[j]; x[j] = t; }

	void printArrayLn(int[] xs) { for(int i = 0; i < xs.length; i++) io.out.print(xs[i] + (i==xs.length-1?"\n":" ")); }
	void printArrayLn(long[] xs) { for(int i = 0; i < xs.length; i++) io.out.print(xs[i] + (i==xs.length-1?"\n":" ")); }
	
	static void dump(Object... o) { System.err.println(Arrays.deepToString(o)); } 
	
	void main() throws IOException {
		//		IOFast.setFileIO("rle-size.in", "rle-size.out");
		try { run(); }
		catch (EndOfFileRuntimeException e) { }
		io.out.flush();
	}
	public static void main(String[] args) throws IOException { new Main().main(); }
	
	static class EndOfFileRuntimeException extends RuntimeException {
		private static final long serialVersionUID = -8565341110209207657L; }

	static
	public class IOFast {
		private BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
		private PrintWriter out = new PrintWriter(System.out);

		void setFileIn(String ins) throws IOException { in.close(); in = new BufferedReader(new FileReader(ins)); }
		void setFileOut(String outs) throws IOException { out.flush(); out.close(); out = new PrintWriter(new FileWriter(outs)); }
		void setFileIO(String ins, String outs) throws IOException { setFileIn(ins); setFileOut(outs); }

		private static int pos, readLen;
		private static final char[] buffer = new char[1024 * 8];
		private static char[] str = new char[500*8*2];
		private static boolean[] isDigit = new boolean[256];
		private static boolean[] isSpace = new boolean[256];
		private static boolean[] isLineSep = new boolean[256];

		static { for(int i = 0; i < 10; i++) { isDigit['0' + i] = true; } isDigit['-'] = true; isSpace[' '] = isSpace['\r'] = isSpace['\n'] = isSpace['\t'] = true; isLineSep['\r'] = isLineSep['\n'] = true; }
		public int read() throws IOException { if(pos >= readLen) { pos = 0; readLen = in.read(buffer); if(readLen <= 0) { throw new EndOfFileRuntimeException(); } } return buffer[pos++]; }
		public int nextInt() throws IOException { int len = 0; str[len++] = nextChar(); len = reads(len, isSpace); int i = 0; int ret = 0; if(str[0] == '-') { i = 1; } for(; i < len; i++) ret = ret * 10 + str[i] - '0'; if(str[0] == '-') { ret = -ret; } return ret; }
		public long nextLong() throws IOException { int len = 0; str[len++] = nextChar(); len = reads(len, isSpace); int i = 0; long ret = 0; if(str[0] == '-') { i = 1; } for(; i < len; i++) ret = ret * 10 + str[i] - '0'; if(str[0] == '-') { ret = -ret; } return ret; }
		public char nextChar() throws IOException { while(true) { final int c = read(); if(!isSpace[c]) { return (char)c; } } }
		int reads(int len, boolean[] accept) throws IOException { try { while(true) { final int c = read(); if(accept[c]) { break; } if(str.length == len) { char[] rep = new char[str.length * 3 / 2]; System.arraycopy(str, 0, rep, 0, str.length); str = rep; } str[len++] = (char)c; } } catch(EndOfFileRuntimeException e) { ; } return len; }
		int reads(char[] cs, int len, boolean[] accept) throws IOException { try { while(true) { final int c = read(); if(accept[c]) { break; } cs[len++] = (char)c; } } catch(EndOfFileRuntimeException e) { ; } return len; }
		public char[] nextLine() throws IOException { int len = 0; str[len++] = nextChar(); len = reads(len, isLineSep); try { if(str[len-1] == '\r') { len--; read(); } } catch(EndOfFileRuntimeException e) { ; } return Arrays.copyOf(str, len); }
		public String nextString() throws IOException { return new String(next()); }
		public char[] next() throws IOException { int len = 0; str[len++] = nextChar(); len = reads(len, isSpace); return Arrays.copyOf(str, len); }
//		public int next(char[] cs) throws IOException { int len = 0; cs[len++] = nextChar(); len = reads(cs, len, isSpace); return len; }
		public double nextDouble() throws IOException { return Double.parseDouble(nextString()); }
		public long[] nextLongArray(final int n) throws IOException { final long[] res = new long[n]; for(int i = 0; i < n; i++) { res[i] = nextLong(); } return res; }
		public int[] nextIntArray(final int n) throws IOException { final int[] res = new int[n]; for(int i = 0; i < n; i++) { res[i] = nextInt(); } return res; }
		public int[][] nextIntArray2D(final int n, final int k) throws IOException { final int[][] res = new int[n][]; for(int i = 0; i < n; i++) { res[i] = nextIntArray(k); } return res; }
		public int[][] nextIntArray2DWithIndex(final int n, final int k) throws IOException { final int[][] res = new int[n][k+1]; for(int i = 0; i < n; i++) { for(int j = 0; j < k; j++) { res[i][j] = nextInt(); } res[i][k] = i; } return res; }
		public double[] nextDoubleArray(final int n) throws IOException { final double[] res = new double[n]; for(int i = 0; i < n; i++) { res[i] = nextDouble(); } return res; }
	}
}