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 Rust (1.42.0) (judge verdict: Time Limit Exceeded)

#![allow(unused_imports)]
use std::cmp::*;
use std::collections::*;
use itertools::Itertools;
use proconio::{input, marker::*};
use superslice::*;

type Map = HashMap<(usize, usize), Option<i64>>;
fn main() {
    input! {
        n: usize,
        mut ai: [i64; n],
    }
    let m = n/2;

    let mut memo = Map::new();
    let ans = dfs(&ai, 0, false,  m, &mut memo).unwrap();
    println!("{}", ans);
}

fn dfs(ai: &[i64], idx: usize, prev_selected: bool, count: usize, memo: &mut Map) -> Option<i64> {
    let key = (idx, count);
    if let Some(v) = memo.get(&key) {
        return *v;
    }

    if count == 0 {
        memo.insert(key, Some(0));
        return Some(0);
    }

    if idx == ai.len() {
        memo.insert(key, None);
        return None;
    }

    let not_select = dfs(&ai, idx+1, false, count, memo);
    if prev_selected {
        return not_select;
    }

    let select = dfs(&ai, idx+1, true, count-1, memo);
    match (not_select, select) {
        (None, None) => None,
        (None, Some(s)) => {
            memo.insert(key, Some(s + ai[idx]));
            Some(s + ai[idx])
        }
        (Some(s), None) => {
            memo.insert(key, Some(s));
            Some(s)
        }
        (Some(s1), Some(s2)) => {
            memo.insert(key, Some(max(s1, s2 + ai[idx])));
            Some(max(s1, s2 + ai[idx]))
        }
    }
}