use cht_integer::ConvexHullTrick; #[allow(unused_imports)] #[cfg(feature = "dbg")] use dbg::lg; use ngtio::with_stdin; use std::iter::once; use crate::cht_integer::Concave; fn main() { let mut buf = with_stdin(); let n = buf.usize(); let a = buf.i64(); let b = buf.i64(); let w = buf.i64(); let d = buf.vec::(n); let mut cht = ConvexHullTrick::::new(); cht.insert(0, 2 * w); for (i, d) in once(0).chain(d.iter().copied()).enumerate() { let x0 = i as i64; let y0 = (cht.eval(x0) + b * x0 * x0) / 2 + d; let tilt = -2 * a + (-2 * x0 - 1) * b; let intercept = 2 * y0 + 2 * (x0 + 1) * a + b * x0 * (x0 + 1); cht.insert(tilt, intercept); } let end = n as i64 + 1; let ans = (cht.eval(end) + b * end * end) / 2; println!("{}", ans); } // cht_integer {{{ #[allow(dead_code)] mod cht_integer { use std::{ borrow::Borrow, collections::BTreeSet, fmt::Debug, i64::{MAX, MIN}, marker::PhantomData, }; pub trait ConvexOrConcave {} pub enum Convex {} pub enum Concave {} impl ConvexOrConcave for Convex {} impl ConvexOrConcave for Concave {} #[derive(Clone, Debug, Default, Hash, PartialEq)] pub struct ConvexHullTrick { base: ConvexHullTrickBase, __marker: PhantomData C>, } impl ConvexHullTrick { pub fn new() -> Self { Self { base: Default::default(), __marker: PhantomData, } } pub fn multieval(&self, xs: impl Iterator) -> Vec { xs.map(|x| self.eval(x)).collect() } pub fn collect_lines(&self) -> Vec<(i64, i64)> { self.base .set .iter() .map(|&seg| (seg.line.p, -seg.line.q)) .collect() } pub fn eval(&self, x: i64) -> i64 { self.base.eval(x) } pub fn lave(&self, p: i64) -> Option { self.base.lave(p) } pub fn insert(&mut self, tilt: i64, intercept: i64) -> bool { self.base.insert(tilt, intercept) } } impl ConvexHullTrick { pub fn new() -> Self { Self { base: Default::default(), __marker: PhantomData, } } pub fn multieval(&self, xs: impl Iterator) -> Vec { xs.map(|x| self.eval(x)).collect() } pub fn collect_lines(&self) -> Vec<(i64, i64)> { self.base .set .iter() .map(|&seg| (-seg.line.p, seg.line.q)) .collect() } pub fn eval(&self, x: i64) -> i64 { -self.base.eval(x) } pub fn lave(&self, p: i64) -> Option { assert_ne!(p, MIN); self.base.lave(-p).map(|q| -q) } pub fn insert(&mut self, tilt: i64, intercept: i64) -> bool { assert_ne!(tilt, MIN); assert_ne!(intercept, MIN); self.base.insert(-tilt, -intercept) } } #[derive(Clone, Debug, Default, Hash, PartialEq)] pub struct ConvexHullTrickBase { set: BTreeSet, } impl ConvexHullTrickBase { pub fn new() -> Self { Self::default() } pub fn eval(&self, x: i64) -> i64 { assert!( !self.set.is_empty(), "empty maximum is the negative infinity" ); self.set.range(Max(x)..).next().unwrap().line.eval(x) } pub fn lave(&self, p: i64) -> Option { assert!( !self.set.is_empty(), "empty maximum is the negative infinity" ); let &Segment { line: Line { p: p1, q: q1 }, min: Min(min), max: _, } = self.set.range(p..).next()?; if min == MIN { if p1 == p { Some(q1) } else { None } } else { Some(q1 - (p1 - p) * min) } } pub fn insert(&mut self, tilt: i64, intercept: i64) -> bool { let q = -intercept; let p = tilt; if !self.set.is_empty() && self.lave(p).map_or(false, |c| c <= q) { return false; } self.set.take(&p); let line = Line { p, q }; while let Some(&seg1) = self.set.range(..p).next_back() { if seg1.min.0 == MIN || line.eval(seg1.min.0) < seg1.line.eval(seg1.min.0) { break; } self.set.remove(&seg1); } while let Some(&seg1) = self.set.range(p..).next() { if seg1.max.0 == MAX || line.eval(seg1.max.0) < seg1.line.eval(seg1.max.0) { break; } self.set.remove(&seg1); } if let Some(&seg1) = self.set.range(..p).next_back() { self.set.remove(&seg1); match seg1.line.brace(line) { Err(x) => { debug_assert!(seg1.min.0 < x); self.set.insert(Segment { max: Max(x), ..seg1 }); } Ok(brace) => { if seg1.min.0 < brace.min.0 { self.set.insert(Segment { max: Max(brace.min.0), ..seg1 }); } self.set.insert(brace); } } } if let Some(&seg1) = self.set.range(p..).next() { self.set.remove(&seg1); match line.brace(seg1.line) { Err(x) => { debug_assert!(x < seg1.max.0); self.set.insert(Segment { min: Min(x), ..seg1 }); } Ok(brace) => { if brace.max.0 < seg1.max.0 { self.set.insert(Segment { min: Min(brace.max.0), ..seg1 }); } self.set.insert(brace); } } } let min = Min(self .set .range(..p) .next_back() .map_or(MIN, |seg1| seg1.max.0)); let max = Max(self.set.range(p..).next().map_or(MAX, |seg1| seg1.min.0)); if min.0 < max.0 { self.set.insert(Segment { line, min, max }); } true } } #[derive(Clone, Debug, Default, Hash, PartialEq, Eq, PartialOrd, Ord, Copy)] struct Min(i64); #[derive(Clone, Debug, Default, Hash, PartialEq, Eq, PartialOrd, Ord, Copy)] struct Max(i64); #[derive(Clone, Debug, Default, Hash, PartialEq, Eq, PartialOrd, Ord, Copy)] struct Line { p: i64, q: i64, } impl Line { fn eval(&self, x: i64) -> i64 { self.p * x - self.q } fn brace(self, other: Self) -> Result { let Self { p: p0, q: q0 } = self; let Self { p: p1, q: q1 } = other; debug_assert!(p0 < p1); let x0 = (q1 - q0).div_euclid(p1 - p0); if x0 * (p1 - p0) == (q1 - q0) { return Err(x0); } let x1 = x0 + 1; let p = (p1 * x1 - p0 * x0) - (q1 - q0); let q = (p1 - p0) * x0 * x1 - q1 * x0 + q0 * x1; debug_assert_eq!(p * x0 - q, p0 * x0 - q0); debug_assert_eq!(p * x1 - q, p1 * x1 - q1); Ok(Segment { line: Self { p, q }, min: Min(x0), max: Max(x1), }) } } #[derive(Clone, Default, Debug, Hash, PartialEq, Eq, PartialOrd, Ord, Copy)] struct Segment { line: Line, min: Min, max: Max, } impl Borrow for Segment { fn borrow(&self) -> &i64 { &self.line.p } } impl Borrow for Segment { fn borrow(&self) -> &Min { &self.min } } impl Borrow for Segment { fn borrow(&self) -> &Max { &self.max } } } // }}} // ngtio {{{ #[allow(dead_code)] mod ngtio { mod i { pub use self::{ multi_token::{Leaf, Parser, ParserTuple, RawTuple, Tuple, VecLen}, token::{Token, Usize1}, }; use std::{ io::{self, BufRead}, iter, }; pub fn with_stdin() -> Tokenizer> { io::BufReader::new(io::stdin()).tokenizer() } pub fn with_str(src: &str) -> Tokenizer<&[u8]> { src.as_bytes().tokenizer() } pub struct Tokenizer { queue: Vec, // FIXME: String のみにすると速そうです。 scanner: S, } macro_rules! prim_method { ($name:ident: $T:ty) => { pub fn $name(&mut self) -> $T { <$T>::leaf().parse(self) } }; ($name:ident) => { prim_method!($name: $name); }; } macro_rules! prim_methods { ($name:ident: $T:ty; $($rest:tt)*) => { prim_method!($name:$T); prim_methods!($($rest)*); }; ($name:ident; $($rest:tt)*) => { prim_method!($name); prim_methods!($($rest)*); }; () => () } impl Tokenizer { pub fn token(&mut self) -> String { self.load(); self.queue.pop().expect("入力が終了したのですが。") } pub fn new(scanner: S) -> Self { Self { queue: Vec::new(), scanner, } } fn load(&mut self) { while self.queue.is_empty() { let mut s = String::new(); let length = self.scanner.read_line(&mut s).unwrap(); // 入力が UTF-8 でないときにエラーだそうです。 if length == 0 { break; } self.queue = s.split_whitespace().rev().map(str::to_owned).collect(); } } pub fn skip_line(&mut self) { assert!( self.queue.is_empty(), "行の途中で呼ばないでいただきたいです。現在のトークンキュー: {:?}", &self.queue ); self.load(); } pub fn end(&mut self) { self.load(); assert!(self.queue.is_empty(), "入力はまだあります!"); } pub fn parse(&mut self) -> T::Output { T::parse(&self.token()) } pub fn parse_collect(&mut self, n: usize) -> B where B: iter::FromIterator, { iter::repeat_with(|| self.parse::()).take(n).collect() } pub fn tuple(&mut self) -> ::Output { T::leaf_tuple().parse(self) } pub fn vec(&mut self, len: usize) -> Vec { T::leaf().vec(len).parse(self) } pub fn vec_tuple( &mut self, len: usize, ) -> Vec<::Output> { T::leaf_tuple().vec(len).parse(self) } pub fn vec2(&mut self, height: usize, width: usize) -> Vec> { T::leaf().vec(width).vec(height).parse(self) } pub fn vec2_tuple( &mut self, height: usize, width: usize, ) -> Vec::Output>> where T: RawTuple, { T::leaf_tuple().vec(width).vec(height).parse(self) } prim_methods! { u8; u16; u32; u64; u128; usize; i8; i16; i32; i64; i128; isize; f32; f64; char; string: String; } } mod token { use super::multi_token::Leaf; use std::{any, fmt, marker, str}; pub trait Token: Sized { type Output; fn parse(s: &str) -> Self::Output; fn leaf() -> Leaf { Leaf(marker::PhantomData) } } impl Token for T where T: str::FromStr, ::Err: fmt::Debug, { type Output = Self; fn parse(s: &str) -> Self::Output { s.parse().unwrap_or_else(|_| { panic!("Parse error!: ({}: {})", s, any::type_name::(),) }) } } pub struct Usize1 {} impl Token for Usize1 { type Output = usize; fn parse(s: &str) -> Self::Output { usize::parse(s) .checked_sub(1) .expect("Parse error! (Zero substruction error of Usize1)") } } } mod multi_token { use super::{Token, Tokenizer}; use std::{io::BufRead, iter, marker}; pub trait Parser: Sized { type Output; fn parse(&self, server: &mut Tokenizer) -> Self::Output; fn vec(self, len: usize) -> VecLen { VecLen { len, elem: self } } } pub struct Leaf(pub(super) marker::PhantomData); impl Parser for Leaf { type Output = T::Output; fn parse(&self, server: &mut Tokenizer) -> T::Output { server.parse::() } } pub struct VecLen { pub len: usize, pub elem: T, } impl Parser for VecLen { type Output = Vec; fn parse(&self, server: &mut Tokenizer) -> Self::Output { iter::repeat_with(|| self.elem.parse(server)) .take(self.len) .collect() } } pub trait RawTuple { type LeafTuple: Parser; fn leaf_tuple() -> Self::LeafTuple; } pub trait ParserTuple { type Tuple: Parser; fn tuple(self) -> Self::Tuple; } pub struct Tuple(pub T); macro_rules! impl_tuple { ($($t:ident: $T:ident),*) => { impl<$($T),*> Parser for Tuple<($($T,)*)> where $($T: Parser,)* { type Output = ($($T::Output,)*); #[allow(unused_variables)] fn parse(&self, server: &mut Tokenizer) -> Self::Output { match self { Tuple(($($t,)*)) => { ($($t.parse(server),)*) } } } } impl<$($T: Token),*> RawTuple for ($($T,)*) { type LeafTuple = Tuple<($(Leaf<$T>,)*)>; fn leaf_tuple() -> Self::LeafTuple { Tuple(($($T::leaf(),)*)) } } impl<$($T: Parser),*> ParserTuple for ($($T,)*) { type Tuple = Tuple<($($T,)*)>; fn tuple(self) -> Self::Tuple { Tuple(self) } } }; } impl_tuple!(); impl_tuple!(t1: T1); impl_tuple!(t1: T1, t2: T2); impl_tuple!(t1: T1, t2: T2, t3: T3); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6, t7: T7); impl_tuple!( t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6, t7: T7, t8: T8 ); } trait Scanner: BufRead + Sized { fn tokenizer(self) -> Tokenizer { Tokenizer::new(self) } } impl Scanner for R {} } pub use self::i::{with_stdin, with_str}; mod prelude { pub use super::i::{Parser, ParserTuple, RawTuple, Token, Usize1}; } } // }}} // template {{{ #[cfg(not(feature = "dbg"))] #[allow(unused_macros)] #[macro_export] macro_rules! lg { ($($expr:expr),*) => {}; } // }}}