Problem:
Score : 600 points

Problem Statement

Given is an integer sequence A_1, ..., A_N of length N.

We will choose exactly \left\lfloor \frac{N}{2} \right\rfloor elements from this sequence so that no two adjacent elements are chosen.

Find the maximum possible sum of the chosen elements.

Here \lfloor x \rfloor denotes the greatest integer not greater than x.

Constraints

2 \leq N \leq 2\times 10^5

|A_i|\leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N

Output

Print the maximum possible sum of the chosen elements.

Sample Input 1

6
1 2 3 4 5 6

Sample Output 1

12

Choosing 2, 4, and 6 makes the sum 12, which is the maximum possible value.

Sample Input 2

5
-1000 -100 -10 0 10

Sample Output 2

0

Choosing -10 and 10 makes the sum 0, which is the maximum possible value.

Sample Input 3

10
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000

Sample Output 3

5000000000

Watch out for overflow.

Sample Input 4

27
18 -28 18 28 -45 90 -45 23 -53 60 28 -74 -71 35 -26 -62 49 -77 57 24 -70 -93 69 -99 59 57 -49

Sample Output 4

295

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using CompLib.Util;

public class Program
{
    private int N;
    private int[] A;

    public void Solve()
    {
        var sc = new Scanner();
        N = sc.NextInt();
        A = sc.IntArray();

        //

        // i番目 選んだか? 選んだ - 選んでない + 2
        var dp = new long[N, 2, 5];
        for (int i = 0; i < N; i++)
        {
            for (int j = 0; j < 2; j++)
            {
                for (int k = 0; k < 5; k++)
                {
                    dp[i, j, k] = long.MinValue;
                }
            }
        }

        for (int i = 0; i < N; i++)
        {
            if (i == 0)
            {
                dp[0, 1, 3] = A[0];
                dp[0, 0, 1] = 0;
                continue;
            }

            // A[i]を選ぶ
            for (int j = 1; j <= 4; j++)
            {
                if (dp[i - 1, 0, j - 1] == long.MinValue) continue;
                dp[i, 1, j] = Math.Max(dp[i, 1, j], dp[i - 1, 0, j - 1] + A[i]);
            }

            // 選ばない
            for (int j = 0; j < 4; j++)
            {
                if (dp[i - 1, 0, j + 1] != long.MinValue) dp[i, 0, j] = Math.Max(dp[i, 0, j], dp[i - 1, 0, j + 1]);
                if (dp[i - 1, 1, j + 1] != long.MinValue) dp[i, 0, j] = Math.Max(dp[i, 0, j], dp[i - 1, 1, j + 1]);
            }
        }

        long ans = N % 2 == 0
            ? Math.Max(dp[N - 1, 0, 2], dp[N - 1, 1, 2])
            : Math.Max(dp[N - 1, 0, 1], dp[N - 1, 1, 1]);

        Console.WriteLine(ans);
    }

    public static void Main(string[] args) => new Program().Solve();
}

namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}