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: Wrong Answer)

#region いつもの
using System;
using System.Collections.Generic;
using System.Linq;
using IEnumerable = System.Collections.IEnumerable;
using IEnumerator = System.Collections.IEnumerator;
using Unsafe = System.Runtime.CompilerServices.Unsafe;
using BigInteger = System.Numerics.BigInteger;
using TextReader = System.IO.TextReader;
using StringBuilder = System.Text.StringBuilder;
using static Global;
using static NumGlobal;

static class Global
{
    public static T[] NewArray<T>(int len0, T value) => new T[len0].Fill(value);
    public static T[] NewArray<T>(int len0, Func<T> factory)
    {
        var arr = new T[len0];
        for (int i = 0; i < arr.Length; i++) arr[i] = factory();
        return arr;
    }
    public static T[][] NewArray<T>(int len0, int len1, T value) where T : struct
    {
        var arr = new T[len0][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, value);
        return arr;
    }
    public static T[][] NewArray<T>(int len0, int len1, Func<T> factory)
    {
        var arr = new T[len0][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, factory);
        return arr;
    }
    public static T[][][] NewArray<T>(int len0, int len1, int len2, T value) where T : struct
    {
        var arr = new T[len0][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, value);
        return arr;
    }
    public static T[][][] NewArray<T>(int len0, int len1, int len2, Func<T> factory)
    {
        var arr = new T[len0][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, factory);
        return arr;
    }
    public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, T value) where T : struct
    {
        var arr = new T[len0][][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, value);
        return arr;
    }
    public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, Func<T> factory)
    {
        var arr = new T[len0][][][];
        for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, factory);
        return arr;
    }
    private class ComparerReverseImpl<T> : Comparer<T> where T : IComparable<T> { public override int Compare(T y, T x) => x.CompareTo(y); public override bool Equals(object obj) => obj != null && GetType() == obj.GetType(); public override int GetHashCode() => GetType().GetHashCode(); }
    public static IComparer<T> ComparerReverse<T>() where T : IComparable<T> => new ComparerReverseImpl<T>();
    public static string AllLines<T>(IEnumerable<T> source) => string.Join("\n", source);
    public static string AllJoin<T>(IEnumerable<T> source) => string.Join(" ", source);
}
static class NumGlobal
{
    public static int Pow(int x, int y)
    {
        int res = 1;
        for (; y > 0; y >>= 1)
        {
            if ((y & 1) == 1) res *= x;
            x *= x;
        }
        return res;
    }
    public static BigInteger ParseBigInteger(ReadOnlySpan<char> s)
    {
        // 自前実装の方が速い
        BigInteger res;
        if (s.Length % 9 == 0)
            res = 0;
        else
        {
            res = new BigInteger(int.Parse(s.Slice(0, s.Length % 9)));
            s = s.Slice(s.Length % 9);
        }

        while (s.Length > 0)
        {
            var sp = s.Slice(0, 9);
            res *= 1000_000_000;
            res += int.Parse(sp);
            s = s.Slice(9);
        }
        return res;
    }
    public static int BitCount(int x) { x -= (x >> 1) & 0x55555555; x = (x & 0x33333333) + ((x >> 2) & 0x33333333); x = (x + (x >> 4)) & 0x0f0f0f0f; x += x >> 8; x += x >> 16; return x & 0x3f; }
    public static int BitCount(long x) { x -= (x >> 1) & 0x5555555555555555; x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333); x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0f; x += x >> 8; x += x >> 16; x += x >> 32; return (int)(x & 0x0000007f); }
    public static int MSB(int x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; return BitCount(x) - 1; }
    public static int MSB(long x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; x |= x >> 32; return BitCount(x) - 1; }
    public static int LSB(int x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; return 32 - BitCount(x); }
    public static int LSB(long x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; x |= x << 32; return 64 - BitCount(x); }
}
static class Ext
{
    public static T[] Fill<T>(this T[] arr, T value)
    {
        Array.Fill(arr, value);
        return arr;
    }
    public static T[] Sort<T>(this T[] arr) { Array.Sort(arr); return arr; }
    public static string[] Sort(this string[] arr) => Sort(arr, StringComparer.OrdinalIgnoreCase);
    public static T[] Sort<T, U>(this T[] arr, Func<T, U> selector) where U : IComparable<U> => Sort(arr, (a, b) => selector(a).CompareTo(selector(b)));
    public static T[] Sort<T>(this T[] arr, Comparison<T> comparison) { Array.Sort(arr, comparison); return arr; }
    public static T[] Sort<T>(this T[] arr, IComparer<T> comparer) { Array.Sort(arr, comparer); return arr; }
    public static T[] Reverse<T>(this T[] arr) { Array.Reverse(arr); return arr; }
    public static (int index, T max) MaxBy<T>(this T[] arr) where T : IComparable<T>
    {
        T max = arr[0];
        int maxIndex = 0;
        for (int i = 0; i < arr.Length; i++)
        {
            if (max.CompareTo(arr[i]) < 0)
            {
                max = arr[i];
                maxIndex = i;
            }
        }
        return (maxIndex, max);
    }
    public static (TSource item, TMax max) MaxBy<TSource, TMax>
        (this IEnumerable<TSource> source, Func<TSource, TMax> maxBySelector)
        where TMax : IComparable<TMax>
    {
        TMax max;
        TSource maxByItem;
        var e = source.GetEnumerator();
        e.MoveNext();
        maxByItem = e.Current;
        max = maxBySelector(maxByItem);
        while (e.MoveNext())
        {
            var item = e.Current;
            var next = maxBySelector(item);
            if (max.CompareTo(next) < 0)
            {
                max = next;
                maxByItem = item;
            }
        }
        return (maxByItem, max);
    }
    public static (int index, T min) MinBy<T>(this T[] arr) where T : IComparable<T>
    {
        T min = arr[0];
        int minIndex = 0;
        for (int i = 0; i < arr.Length; i++)
        {
            if (min.CompareTo(arr[i]) > 0)
            {
                min = arr[i];
                minIndex = i;
            }
        }
        return (minIndex, min);
    }
    public static (TSource item, TMin min) MinBy<TSource, TMin>
        (this IEnumerable<TSource> source, Func<TSource, TMin> minBySelector)
        where TMin : IComparable<TMin>
    {
        TMin min;
        TSource minByItem;

        var e = source.GetEnumerator();
        e.MoveNext();
        minByItem = e.Current;
        min = minBySelector(minByItem);
        while (e.MoveNext())
        {
            var item = e.Current;
            var next = minBySelector(item);
            if (min.CompareTo(next) > 0)
            {
                min = next;
                minByItem = item;
            }
        }
        return (minByItem, min);
    }
    public static IComparer<T> Reverse<T>(this IComparer<T> comparer) => Comparer<T>.Create((x, y) => comparer.Compare(y, x));
    public static Dictionary<TKey, int> GroupCount<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector) => source.GroupBy(keySelector).ToDictionary(g => g.Key, g => g.Count());
    public static Dictionary<TKey, int> GroupCount<TKey>(this IEnumerable<TKey> source) => source.GroupCount(i => i);
    public static TValue Get<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key)
    {
        dic.TryGetValue(key, out var v);
        return v;
    }
    public static TValue GetOrInit<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key, TValue value)
    {
        if (dic.TryGetValue(key, out var v))
            return v;
        return dic[key] = value;
    }
}
class ΔDebugView<T> { private IEnumerable<T> collection; public ΔDebugView(IEnumerable<T> collection) { this.collection = collection ?? throw new ArgumentNullException(nameof(collection)); }[System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.RootHidden)] public T[] Items => collection.ToArray(); }
#pragma warning disable CA1819
[System.Diagnostics.DebuggerNonUserCode]
public class ConsoleReader { private string[] ReadLineSplit() => textReader.ReadLine().Split(Array.Empty<char>(), StringSplitOptions.RemoveEmptyEntries); private string[] line = Array.Empty<string>(); private int linePosition; private TextReader textReader; public ConsoleReader(TextReader tr) { textReader = tr; } public int Int => int.Parse(String); public int Int0 => Int - 1; public long Long => long.Parse(String); public double Double => double.Parse(String); public string String { get { if (linePosition >= line.Length) { linePosition = 0; line = ReadLineSplit(); } return line[linePosition++]; } } public class SplitLine { public SplitLine(ConsoleReader cr) { String = cr.ReadLineSplit(); cr.line = Array.Empty<string>(); } public int[] Int => String.Select(x => int.Parse(x)).ToArray(); public int[] Int0 => String.Select(x => int.Parse(x) - 1).ToArray(); public long[] Long => String.Select(x => long.Parse(x)).ToArray(); public double[] Double => String.Select(x => double.Parse(x)).ToArray(); public string[] String { get; } } public SplitLine Split => new SplitLine(this); public class RepeatReader { ConsoleReader cr; int count; public RepeatReader(ConsoleReader cr, int count) { this.cr = cr; this.count = count; } public T[] Select<T>(Func<ConsoleReader, T> factory) { var arr = new T[count]; for (int i = 0; i < count; i++) arr[i] = factory(cr); return arr; } public string[] String => this.Select(cr => cr.String); public int[] Int => this.Select(cr => cr.Int); public int[] Int0 => this.Select(cr => cr.Int - 1); public long[] Long => this.Select(cr => cr.Long); public double[] Double => this.Select(cr => cr.Double); } public RepeatReader Repeat(int count) => new RepeatReader(this, count); }
public class Program
{
    [System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] private ConsoleReader cr;

    public Program(ConsoleReader consoleReader) { this.cr = consoleReader; }
    static void Main(string[] args) => Console.WriteLine(new Program(new ConsoleReader(args.Length > 0 ? new System.IO.StreamReader(args[0]) : Console.In)).Calc());

    #endregion
    public object Calc()
    {
        var N = cr.Int;
        arr = cr.Split.Long;
        var sums = new long[arr.Length];
        sums[0] = arr[0];
        sums[1] = arr[1];
        for (int i = 2; i < sums.Length; i++)
        {
            sums[i] = arr[i] + sums[i - 2];
        }

        long res;
        if (sums.Length % 2 == 0)
        {
            // 片寄せ
            res = Math.Max(sums[^1], sums[^2]);
            // 2つ開ける
            for (int b = 1; b < sums.Length - 1; b += 2)
            {
                var tmp = sums[^1] - sums[b] + sums[b - 1];
                res = Math.Max(res, tmp);
            }

            return res;
        }

        // 片寄せ
        res = Math.Max(sums[^3], sums[^1] - sums[0]);
        res = Math.Max(res, sums[^2]);
        
        // 1つ開ける
        for (int b = 1; b < sums.Length - 3; b++)
        {
            var tmp1 = sums[^1] - sums[b + 1] + sums[b - 1];
            var tmp2 = sums[^2] - sums[b] + sums[b - 1];
            res = Math.Max(res, tmp1);
            res = Math.Max(res, tmp2);
        }


        var dp1 = (long[])sums.Clone(); // スキップ1回
        dp1[0] = -INF;
        dp1[2] = arr[2];
        for (int i = 3; i < dp1.Length; i += 2)
        {
            dp1[i] = Math.Max(dp1[i - 2] + arr[i], sums[i - 3] + arr[i]);
        }
        res = Math.Max(res, dp1[^2]);

        if (sums.Length < 4)
            return res;

        var dp2 = new long[N]; // スキップ2回
        dp2[0] = dp2[1] = dp2[2] = dp2[3] = -INF;
        for (int i = 3; i < dp1.Length; i++)
        {
            dp2[i] = Math.Max(dp2[i - 2] + arr[i], dp1[i - 3] + arr[i]);
        }
        return Math.Max(res, dp2[^1]);
    }
    long[] arr;
    const long INF = long.MaxValue >> 2;
}