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: Accepted)

#[derive(Default)]
//NOTE
//declare variables to reduce the number of parameters for dp and dfs etc.
struct Solver {
    n: usize,
    a: Vec<i64>,
    memo: Vec<Vec<Option<i64>>>,
}
const MAX_N: usize = 200010;
impl Solver {
    fn dp(&mut self, pos: usize, skip_cnt: usize) -> i64 {
        if skip_cnt >= 3 {
            return -(1i64 << 50);
        }
        if pos >= self.n {
            if self.n % 2 != 0 {
                if skip_cnt >= 1 {
                    return 0;
                }
                return -(1i64 << 50);
            }
            return 0;
        }
        if let Some(cached) = self.memo[pos][skip_cnt] {
            return cached;
        }
        let mut result = -(1i64 << 50);
        if self.n % 2 == 0 {
            //not skip
            result = std::cmp::max(result, self.dp(pos + 2, skip_cnt) + self.a[pos]);
            //skip
            if skip_cnt == 0 {
                result = std::cmp::max(result, self.dp(pos + 1, 1));
            }
        } else {
            //not skip
            result = std::cmp::max(result, self.dp(pos + 2, skip_cnt) + self.a[pos]);

            //skip if it is ok
            result = std::cmp::max(result, self.dp(pos + 1, skip_cnt + 1));
        }

        self.memo[pos][skip_cnt] = Some(result);
        result
    }
    fn solve(&mut self) {
        let stdin = std::io::stdin();
        let mut scn = Scanner {
            stdin: stdin.lock(),
        };
        let n: usize = scn.read();
        let a: Vec<i64> = scn.vec(n);

        let mut tmp1 = 0;
        let mut tmp2 = 0;
        /*for (idx, x) in a.iter().enumerate() {
            if idx % 2 == 0 {
                tmp1 += x;
            } else {
                tmp2 += x;
            }
        }
        eprintln!("{} {}", tmp1, tmp2);*/
        let memo = vec![vec![None; 3]; MAX_N];
        self.n = n;

        self.a = a;
        self.memo = memo;
        let result1 = self.dp(0, 0);

        let memo = vec![vec![None; 3]; MAX_N];
        self.memo = memo;
        self.a.reverse();
        let result2 = self.dp(0, 0);
        let result = std::cmp::max(result1, result2);
        println!("{}", result);
        //println!("{} {}", result1, result2);
    }
}

fn main() {
    std::thread::Builder::new()
        .stack_size(64 * 1024 * 1024) // 64MB
        .spawn(|| {
            let mut solver = Solver::default();
            solver.solve()
        })
        .unwrap()
        .join()
        .unwrap();
}
pub mod utils {
    const DYX: [(isize, isize); 8] = [
        (0, 1),   //right
        (1, 0),   //down
        (0, -1),  //left
        (-1, 0),  //top
        (1, 1),   //down right
        (-1, 1),  //top right
        (1, -1),  //down left
        (-1, -1), //top left
    ];

    pub fn try_adj(y: usize, x: usize, dir: usize, h: usize, w: usize) -> Option<(usize, usize)> {
        let ny = y as isize + DYX[dir].0;
        let nx = x as isize + DYX[dir].1;
        if ny >= 0 && nx >= 0 {
            let ny = ny as usize;
            let nx = nx as usize;
            if ny < h && nx < w {
                Some((ny, nx))
            } else {
                None
            }
        } else {
            None
        }
    }
}

//snippet from kenkoooo
pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn read_line(&mut self) -> String {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b'\n' || b == b'\r')
            .take_while(|&b| b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }

    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}