Problem:
Score : 500 points

Problem Statement

Takahashi has come to a party as a special guest.
There are N ordinary guests at the party. The i-th ordinary guest has a power of A_i.

Takahashi has decided to perform M handshakes to increase the happiness of the party (let the current happiness be 0).
A handshake will be performed as follows:

Takahashi chooses one (ordinary) guest x for his left hand and another guest y for his right hand (x and y can be the same).

Then, he shakes the left hand of Guest x and the right hand of Guest y simultaneously to increase the happiness by A_x+A_y.

However, Takahashi should not perform the same handshake more than once. Formally, the following condition must hold:

Assume that, in the k-th handshake, Takahashi shakes the left hand of Guest x_k and the right hand of Guest y_k. Then, there is no pair p, q (1 \leq p < q \leq M) such that (x_p,y_p)=(x_q,y_q).

What is the maximum possible happiness after M handshakes?

Constraints

1 \leq N \leq 10^5

1 \leq M \leq N^2

1 \leq A_i \leq 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M
A_1 A_2 ... A_N

Output

Print the maximum possible happiness after M handshakes.

Sample Input 1

5 3
10 14 19 34 33

Sample Output 1

202

Let us say that Takahashi performs the following handshakes:

In the first handshake, Takahashi shakes the left hand of Guest 4 and the right hand of Guest 4.

In the second handshake, Takahashi shakes the left hand of Guest 4 and the right hand of Guest 5.

In the third handshake, Takahashi shakes the left hand of Guest 5 and the right hand of Guest 4.

Then, we will have the happiness of (34+34)+(34+33)+(33+34)=202.

We cannot achieve the happiness of 203 or greater, so the answer is 202.

Sample Input 2

9 14
1 3 5 110 24 21 34 5 3

Sample Output 2

1837

Sample Input 3

9 73
67597 52981 5828 66249 75177 64141 40773 79105 16076

Sample Output 3

8128170

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[macro_export]
macro_rules ! chmax { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: max ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! chmin { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: min ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! max { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: max ( $ x , max ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! min { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: min ( $ x , min ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! dvec { ( $ t : expr ; $ len : expr ) => { vec ! [ $ t ; $ len ] } ; ( $ t : expr ; $ len : expr , $ ( $ rest : expr ) ,* ) => { vec ! [ dvec ! ( $ t ; $ ( $ rest ) ,* ) ; $ len ] } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[macro_export]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut parser = Parser :: from_str ( $ s ) ; input_inner ! { parser , $ ( $ r ) * } } ; ( parser = $ parser : ident , $ ( $ r : tt ) * ) => { input_inner ! { $ parser , $ ( $ r ) * } } ; ( new_stdin_parser = $ parser : ident , $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let reader = std :: io :: BufReader :: new ( stdin . lock ( ) ) ; let mut $ parser = Parser :: new ( reader ) ; input_inner ! { $ parser , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { input ! { new_stdin_parser = parser , $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! input_inner { ( $ parser : ident ) => { } ; ( $ parser : ident , ) => { } ; ( $ parser : ident , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ parser , $ t ) ; input_inner ! { $ parser $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! read_value { ( $ parser : ident , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ parser , $ t ) ) ,* ) } ; ( $ parser : ident , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ parser , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ parser : ident , chars ) => { read_value ! ( $ parser , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ parser : ident , usize1 ) => { read_value ! ( $ parser , usize ) - 1 } ; ( $ parser : ident , $ t : ty ) => { $ parser . next ::<$ t > ( ) . expect ( "Parse error" ) } ; }
use std::io;
use std::io::BufRead;
use std::str;
pub struct Parser<R> {
    reader: R,
    buf: Vec<u8>,
    pos: usize,
}
impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}
impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                if len == 0 {
                    break;
                }
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return str::from_utf8(&self.buf[begin..end]).unwrap().parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
}
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
fn solve() {
    let out = stdout();
    let mut out = BufWriter::new(out.lock());
    input!{
        n:usize,m:usize,
        a:[usize;n],
    }

    let mut amax = 0;
    for i in 0..n {
        chmax!(amax,a[i]);
    }

    let mut N = 1;
    while N < (amax+1)*2 + 1 {
        N *= 2;
    }

    let mut tmp = vec![0; N];
    for i in 0..n {
        tmp[a[i]] += 1;
    }
    let mut x = vec![Complex::new(0.,0.); N];
    for i in 0..N {
        if tmp[i] >= 0 {
            x[i] = Complex::new(tmp[i] as f64, 0.);
        }
    }

    let X = fast_fourier_transform(&x, false);
    let mut Z = vec![Complex::new(0.,0.); N];
    for i in 0..N {
        Z[i] = X[i] * X[i];
    }
    let z = fast_fourier_transform(&Z, true);
    let mut tot = 0;
    let mut remaining = m as i64;
    for i in (0..N).rev() {
        let v = (z[i].x + 0.5) as i64;
        let take = min(v, remaining);
        tot += take * i as i64;
        remaining -= take;
    }
    println!("{}", tot);
}
mod complex {
    #[derive(Clone, Copy, Debug)]
    pub struct Complex {
        pub x: f64,
        pub y: f64,
    }
    
    impl Complex {
        pub fn new(x: f64, y: f64) -> Self { Complex { x: x, y: y } }
        pub fn polar(r: f64, theta: f64) -> Self { Complex::new(r * theta.cos(), r * theta.sin()) }
        pub fn conj(&self) -> Self { Complex::new(self.x, -self.y) }
        pub fn abs(&self) -> f64 { (self.x * self.x + self.y * self.y).sqrt() }
        pub fn arg(&self) -> f64 { self.y.atan2(self.x) }
    }

    use std::ops::*;
    
    impl Add for Complex {
        type Output = Self;
        fn add(self, rhs: Self) -> Self { Complex::new(self.x + rhs.x, self.y + rhs.y) }
    }
    
    impl Sub for Complex {
        type Output = Self;
        fn sub(self, rhs: Self) -> Self { Complex::new(self.x - rhs.x, self.y - rhs.y) }
    }
    
    impl Mul for Complex {
        type Output = Self;
        fn mul(self, rhs: Self) -> Self { Complex::new(self.x * rhs.x - self.y * rhs.y, self.x * rhs.y + self.y * rhs.x) }
    }
    
    impl Div for Complex {
        type Output = Self;
        fn div(self, rhs: Self) -> Self {
            let z = self * rhs.conj();
            let a = rhs.x * rhs.x + rhs.y * rhs.y;
            Complex::new(z.x / a, z.y / a)
        }
    }
    
    impl AddAssign for Complex { fn add_assign(&mut self, rhs: Self) { *self = *self + rhs } }
    impl SubAssign for Complex { fn sub_assign(&mut self, rhs: Self) { *self = *self - rhs } }
    impl MulAssign for Complex { fn mul_assign(&mut self, rhs: Self) { *self = *self * rhs } }
    impl DivAssign for Complex { fn div_assign(&mut self, rhs: Self) { *self = *self / rhs } }
}
pub type Complex = complex::Complex;
pub fn fast_fourier_transform(arr: &[Complex], inv: bool) -> Vec<Complex> {
    let n = arr.len();
    assert!(n.count_ones() == 1, "the length of array is not square");
    let mut a: Vec<_> = arr.to_vec();
    let mut tmp: Vec<_> = (0..n).map(|_| Complex::new(0., 0.)).collect();
    let mut ai: Vec<_> = (0..n).map(|i| i).collect();
    let mut ti: Vec<_> = (0..n).map(|_| 0).collect();
    let bit = n.trailing_zeros();
    let f = if inv { -1.0 } else { 1.0 };
    for si in (0..bit).rev() {
        let s = 1 << si;
        std::mem::swap(&mut a, &mut tmp);
        std::mem::swap(&mut ai, &mut ti);
        let zeta = Complex::polar(1.0, std::f64::consts::PI * 2.0 * f / (s << 1) as f64);
        let mut z_i = Complex::new(1.0, 0.0);
        let mut ev = 0;
        let mut od = 1;
        for i in 0..n {
            if (i & s) != 0 {
                a[i] = (tmp[i - s] - tmp[i]) * z_i;
                ai[i] = ti[od];
                od += 2;
                z_i *= zeta;
            }
            else {
                a[i] = tmp[i] + tmp[i + s];
                ai[i] = ti[ev];
                ev += 2;
                z_i = Complex::new(1.0, 0.0);
            }
        }
    }
 
    std::mem::swap(&mut a, &mut tmp);
    let inv_n = if inv { n as f64 } else { 1.0 };
    for i in 0..n { a[ai[i]] = Complex::new(tmp[i].x / inv_n, tmp[i].y / inv_n); }
    a
}