Problem:
Score : 600 points

Problem Statement

There are N cities in Republic of AtCoder. The size of the i-th city is A_{i}.
Takahashi would like to build N-1 bidirectional roads connecting two cities so that any city can be reached from any other city by using these roads.

Assume that the cost of building a road connecting the i-th city and the j-th city is |i-j| \times D + A_{i} + A_{j}.
For Takahashi, find the minimum possible total cost to achieve the objective.

Constraints

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

1 \leq D \leq 10^9

1 \leq A_{i} \leq 10^9

A_{i} and D are integers.

Input

Input is given from Standard Input in the following format:

N D
A_1 A_2 ... A_N

Output

Print the minimum possible total cost.

Sample Input 1

3 1
1 100 1

Sample Output 1

106

This cost can be achieved by, for example, building roads connecting City 1, 2 and City 1, 3.

Sample Input 2

3 1000
1 100 1

Sample Output 2

2202

Sample Input 3

6 14
25 171 7 1 17 162

Sample Output 3

497

Sample Input 4

12 5
43 94 27 3 69 99 56 25 8 15 46 8

Sample Output 4

658

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

#[allow(unused_imports)]
use std::cmp::{min,max};
#[allow(unused_imports)]
use std::collections::{BTreeMap,BTreeSet};
#[allow(unused_imports)]
use std::ops::*;
#[allow(unused_imports)]
use std::collections::BinaryHeap;

#[allow(unused_macros)]
macro_rules! ite {
    ($c:expr, $t:expr, $f:expr) => {{
        if $c { $t } else { $f }
    }};
}

// ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
// diff: using Parser
#[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")
    };
}

const inf: i64 = 1i64 << 58;

fn main() {
    input! {
        n: usize,
        d: i64,
        xs: [i64; n],
    }
    let mut uf = UnionFind::new(n+1);
    let mut res = 0;
    //for _ in 0..3 {
    while uf.num(0) != n {
        let mut closest: Vec<(i64,usize)> = vec![(inf,n); n];
        // left
        {
            let mut min_1 = inf;
            let mut min_1_id = n;
            let mut min_2 = inf;
            let mut min_2_id = n;
            for i in 0..n {
                min_1 += d;
                min_2 += d;
                if closest[i].0 > min_1 && uf.root(i) != uf.root(min_1_id) {
                    closest[i] = (min_1, min_1_id);
                }
                if closest[i].0 > min_2 && uf.root(i) != uf.root(min_2_id) {
                    closest[i] = (min_2, min_2_id);
                }
                if xs[i] < min_1 && uf.root(i) == uf.root(min_1_id) {
                    min_1 = xs[i];
                    min_1_id = i;
                }
                else if xs[i] < min_1 && uf.root(i) != uf.root(min_1_id) {
                    min_2 = min_1;
                    min_2_id = min_1_id;
                    min_1 = xs[i];
                    min_1_id = i;
                }
                else if xs[i] < min_2 && uf.root(i) == uf.root(min_2_id) {
                    min_2 = xs[i];
                    min_2_id = i;
                }
                else if xs[i] < min_2 && uf.root(i) != uf.root(min_1_id) {
                    min_2 = xs[i];
                    min_2_id = i;
                }
            }
        }
        // right
        {
            let mut min_1 = inf;
            let mut min_1_id = n;
            let mut min_2 = inf;
            let mut min_2_id = n;
            for i in (0..n).rev() {
                min_1 += d;
                min_2 += d;
                if closest[i].0 > min_1 && uf.root(i) != uf.root(min_1_id) {
                    closest[i] = (min_1, min_1_id);
                }
                if closest[i].0 > min_2 && uf.root(i) != uf.root(min_2_id) {
                    closest[i] = (min_2, min_2_id);
                }
                if xs[i] < min_1 && uf.root(i) == uf.root(min_1_id) {
                    min_1 = xs[i];
                    min_1_id = i;
                }
                else if xs[i] < min_1 && (uf.root(i) != uf.root(min_1_id) || min_2_id == n) {
                    min_2 = min_1;
                    min_2_id = min_1_id;
                    min_1 = xs[i];
                    min_1_id = i;
                }
                else if xs[i] < min_2 && uf.root(i) == uf.root(min_2_id) {
                    min_2 = xs[i];
                    min_2_id = i;
                }
                else if xs[i] < min_2 && uf.root(i) != uf.root(min_1_id) {
                    min_2 = xs[i];
                    min_2_id = i;
                }
            }
        }
        let mut zs = vec![];
        for i in 0..n {
            zs.push((closest[i].0 + xs[i], i));
        }
        zs.sort();
        for &(d, i) in &zs {
            if uf.root(i) != uf.root(closest[i].1) && closest[i].1 != n {
                uf.unite(i, closest[i].1);
                res += d;
            }
        }
        for i in 0..n {
        }
    }
    println!("{}", res);
}

use std::io::BufRead;
use std::io;
use std::str;

// ref: tatsuya6502 <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e>
// ref: wariuni <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e#comment-7040a5ae96305e884eb9>
// diff: using std::io::BufRead::fill_buf()
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();
                (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();
            }
        }
    }
}

struct UnionFindNode {
    parent: usize,
    num: usize,
}

impl UnionFindNode {
    fn new(parent: usize) -> UnionFindNode {
        UnionFindNode {
            parent: parent,
            num: 1,
        }
    }
}

pub struct UnionFind(Vec<UnionFindNode>);

impl UnionFind {
    pub fn new(n: usize) -> UnionFind {
        let mut xs = vec![];
        for i in 0..n {
            xs.push(UnionFindNode::new(i));
        }
        UnionFind(xs)
    }
    pub fn root(&mut self, i: usize) -> usize {
        let parent = self.0[i].parent;
        if parent == i {
            i
        }
        else {
            let res = self.root(parent);
            self.0[i].parent = res;
            res
        }
    }
    pub fn unite(&mut self, i: usize, k: usize) {
        let i = self.root(i);
        let k = self.root(k);
        if i == k {
            return;
        }
        let (i,k) = if self.0[i].num < self.0[k].num {
            (i,k)
        }
        else {
            (k,i)
        };
        self.0[i].parent = k;
        self.0[k].num += self.0[i].num;
    }
    pub fn is_same_set(&mut self, i: usize, k: usize) -> bool {
        self.root(i) == self.root(k)
    }
    pub fn num(&mut self, i: usize) -> usize {
        let i = self.root(i);
        self.0[i].num
    }
}

use std::ops::*;

const MOD : u64 = 1_000_000_007;
#[derive(Clone,Copy,PartialEq,Debug)]
pub struct Mod (pub u64);

impl Mod {
    pub fn new(x: u64) -> Mod {
        Mod(x % MOD)
    }
    pub fn pow(self, n: u64) -> Mod {
        if n == 0 {
            return Mod(1);
        }
        let t = self.pow(n/2);
        if n % 2 == 0 {
            t * t
        }
        else {
            t * t * self
        }
    }
}

impl Add for Mod {
    type Output = Self;
    fn add(self, rhs: Self) -> Self::Output {
        Mod((self.0 + rhs.0) % MOD)
    }
}

impl AddAssign for Mod {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl Sub for Mod {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self::Output {
        Mod((self.0 + MOD - rhs.0 % MOD) % MOD)
    }
}

impl SubAssign for Mod {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl Neg for Mod {
    type Output = Mod;
    fn neg(self) -> Mod {
        Mod(MOD - self.0)
    }
}

impl Mul for Mod {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self::Output {
        Mod((self.0 % MOD) * (rhs.0 % MOD) % MOD)
    }
}

impl MulAssign for Mod {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl Div for Mod {
    type Output = Self;
    fn div(self, rhs: Self) -> Self::Output {
        if rhs.0 == 0 {
            loop {}
        }
        assert!(rhs.0 != 0);
        self * rhs.pow(MOD-2)
    }
}

struct Writer {
    s: String,
}

use std::fmt::Display;
impl Writer {
    pub fn new() -> Writer {
        Writer {
            s: String::new(),
        }
    }
    pub fn flush(&mut self) {
        print!("{}", self.s);
        self.s.clear();
    }
    pub fn write<T: Display>(&mut self, x: T) {
        self.s.push_str(&format!("{}", x));
    }
    pub fn writeln<T: Display>(&mut self, x: T) {
        self.s.push_str(&format!("{}", x));
        self.s.push('\n');
    }
    #[allow(dead_code)]
    pub fn write_vec<T: Display>(&mut self, xs: &Vec<T>) {
        if xs.len() == 0 {
            self.writeln("");
            return;
        }
        self.write(&format!("{}", xs[0]));
        for i in 1..xs.len() {
            self.write(&format!(" {}", xs[i]));
        }
        self.writeln("");
    }
}