Problem:
Score : 500 points

Problem Statement

Given are a sequence of N positive integers A_1, A_2, \ldots, A_N, and a positive integer K.

Find the number of non-empty contiguous subsequences in A such that the remainder when dividing the sum of its elements by K is equal to the number of its elements. We consider two subsequences different if they are taken from different positions, even if they are equal sequences.

Constraints

All values in input are integers.

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

1 \leq K \leq 10^9

1 \leq A_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \cdots A_N

Output

Print the number of subsequences that satisfy the condition.

Sample Input 1

5 4
1 4 2 3 5

Sample Output 1

4

Four sequences satisfy the condition: (1), (4,2), (1,4,2), and (5).

Sample Input 2

8 4
4 2 4 2 4 2 4 2

Sample Output 2

7

(4,2) is counted four times, and (2,4) is counted three times.

Sample Input 3

10 7
14 15 92 65 35 89 79 32 38 46

Sample Output 3

8

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

#![allow(unused_imports, unused_macros, non_snake_case)]
use std::cmp::{max, min, Ordering};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::io::{self, BufReader, BufWriter, Read, Write};

//{{{ macros missing in 1.15.1
#[macro_use]
mod macros {
    macro_rules! eprint {
        ($($t:tt)*) => {{
            use ::std::io::Write;
            let _ = write!(::std::io::stderr(), $($t)*);
        }};
    }
    macro_rules! eprintln {
        () => { eprintln!(""); };
        ($($t:tt)*) => {{
            use ::std::io::Write;
            let _ = writeln!(::std::io::stderr(), $($t)*);
        }};
    }
    macro_rules! dbg {
        ($val:expr) => {
            match $val {
                tmp => {
                    eprintln!(
                        "[{}:{}] {} = {:#?}",
                        file!(),
                        line!(),
                        stringify!($val),
                        &tmp
                    );
                    tmp
                }
            }
        };
    }
}
//}}}

fn solve<R: Read, W: Write>(mut sc: Scanner<R>, mut w: BufWriter<W>) {
    let n: usize = sc.read();
    let k: usize = sc.read();
    let mut count = HashMap::new();

    let mut cur = 0;
    *count.entry(cur).or_insert(0) += 1;

    let v: Vec<usize> = sc.read_vec(n);
    let mut s: Vec<usize> = vec![0; n + 1];
    let mut answer = 0usize;

    for i in 0..n {
        let x = v[i];
        cur += x + k - 1;
        cur %= k;
        s[i + 1] = cur;
        if i + 1 >= k {
            *count.entry(s[i + 1 - k]).or_insert(0) -= 1;
        }
        answer += *count.entry(cur).or_insert(0);
        *count.entry(cur).or_insert(0) += 1;
    }

    writeln!(w, "{}", answer);
}

fn run() {
    let stdin = io::stdin();
    let stdout = io::stdout();

    let input = stdin.lock();
    let output = BufWriter::new(stdout.lock());

    solve(Scanner::new(input), output);
}

fn main() {
    std::thread::Builder::new()
        .name("run".to_string())
        .stack_size(256 * 1024 * 1024)
        .spawn(run)
        .unwrap()
        .join()
        .unwrap()
}

//{{{ utils
pub struct Tokenizer<R: std::io::Read> {
    reader: R,
}

impl<R: std::io::Read> Tokenizer<R> {
    pub fn new(r: R) -> Tokenizer<R> {
        Tokenizer { reader: r }
    }
}

fn is_whitespace(b: u8) -> bool {
    b == b' ' || b == b'\n' || b == b'\r' || b == b'\t'
}

impl<R: std::io::Read> std::iter::Iterator for Tokenizer<R> {
    type Item = String;
    fn next(&mut self) -> Option<Self::Item> {
        let buf = std::io::Read::bytes(self.reader.by_ref())
            .map(|b| b.unwrap_or(b' '))
            .skip_while(|&b| is_whitespace(b))
            .take_while(|&b| !is_whitespace(b))
            .collect::<Vec<_>>();
        if buf.is_empty() {
            None
        } else {
            Some(unsafe { String::from_utf8_unchecked(buf) })
        }
    }
}

pub struct Scanner<R: std::io::Read> {
    tokenizer: Tokenizer<R>,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn new(r: R) -> Scanner<R> {
        Scanner {
            tokenizer: Tokenizer::new(r),
        }
    }
}

pub trait FromTokens {
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self;
}

macro_rules! from_tokens_primitives {
    ($($t:ty),*) => { $(
        impl FromTokens for $t {
            fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
                tokens
                    .next()
                    .expect("no more token")
                    .parse()
                    .ok()
                    .expect("Parse error")
            }
        }
    )* }
}

// FIXME: Can it be implemented by FromStr?
from_tokens_primitives! {
    String,
    bool,
    f32,
    f64,
    isize,
    i8,
    i16,
    i32,
    i64,
    usize,
    u8,
    u16,
    u32,
    u64
}

impl<T1: FromTokens, T2: FromTokens> FromTokens for (T1, T2) {
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (T1::from_tokens(tokens), T2::from_tokens(tokens))
    }
}

impl<T1: FromTokens, T2: FromTokens, T3: FromTokens> FromTokens for (T1, T2, T3) {
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (
            T1::from_tokens(tokens),
            T2::from_tokens(tokens),
            T3::from_tokens(tokens),
        )
    }
}

impl<T1: FromTokens, T2: FromTokens, T3: FromTokens, T4: FromTokens> FromTokens
    for (T1, T2, T3, T4)
{
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (
            T1::from_tokens(tokens),
            T2::from_tokens(tokens),
            T3::from_tokens(tokens),
            T4::from_tokens(tokens),
        )
    }
}

impl<T1: FromTokens, T2: FromTokens, T3: FromTokens, T4: FromTokens, T5: FromTokens> FromTokens
    for (T1, T2, T3, T4, T5)
{
    fn from_tokens(tokens: &mut Iterator<Item = String>) -> Self {
        (
            T1::from_tokens(tokens),
            T2::from_tokens(tokens),
            T3::from_tokens(tokens),
            T4::from_tokens(tokens),
            T5::from_tokens(tokens),
        )
    }
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: FromTokens>(&mut self) -> T {
        T::from_tokens(&mut self.tokenizer)
    }

    pub fn read_vec<T: FromTokens>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }

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

pub trait SetMinMax {
    fn set_min(&mut self, v: Self) -> bool;
    fn set_max(&mut self, v: Self) -> bool;
}

impl<T> SetMinMax for T
where
    T: PartialOrd,
{
    fn set_min(&mut self, v: T) -> bool {
        *self > v && {
            *self = v;
            true
        }
    }
    fn set_max(&mut self, v: T) -> bool {
        *self < v && {
            *self = v;
            true
        }
    }
}

#[derive(PartialEq, Eq, Debug, Copy, Clone, Default, Hash)]
pub struct Reverse<T>(pub T);

impl<T: PartialOrd> PartialOrd for Reverse<T> {
    #[inline]
    fn partial_cmp(&self, other: &Reverse<T>) -> Option<Ordering> {
        other.0.partial_cmp(&self.0)
    }

    #[inline]
    fn lt(&self, other: &Self) -> bool {
        other.0 < self.0
    }
    #[inline]
    fn le(&self, other: &Self) -> bool {
        other.0 <= self.0
    }
    #[inline]
    fn ge(&self, other: &Self) -> bool {
        other.0 >= self.0
    }
    #[inline]
    fn gt(&self, other: &Self) -> bool {
        other.0 > self.0
    }
}

impl<T: Ord> Ord for Reverse<T> {
    #[inline]
    fn cmp(&self, other: &Reverse<T>) -> Ordering {
        other.0.cmp(&self.0)
    }
}
//}}}