Problem:
Score : 500 points

Problem Statement

AtCoDeer is thinking of painting an infinite two-dimensional grid in a checked pattern of side K.
Here, a checked pattern of side K is a pattern where each square is painted black or white so that each connected component of each color is a K × K square.
Below is an example of a checked pattern of side 3:

AtCoDeer has N desires.
The i-th desire is represented by x_i, y_i and c_i.
If c_i is B, it means that he wants to paint the square (x_i,y_i) black; if c_i is W, he wants to paint the square (x_i,y_i) white.
At most how many desires can he satisfy at the same time?

Constraints

1 ≤ N ≤ 10^5

1 ≤ K ≤ 1000

0 ≤ x_i ≤ 10^9

0 ≤ y_i ≤ 10^9

If i ≠ j, then (x_i,y_i) ≠ (x_j,y_j).

c_i is B or W.

N, K, x_i and y_i are integers.

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1 c_1
x_2 y_2 c_2
:
x_N y_N c_N

Output

Print the maximum number of desires that can be satisfied at the same time.

Sample Input 1

4 3
0 1 W
1 2 W
5 3 B
5 4 B

Sample Output 1

4

He can satisfy all his desires by painting as shown in the example above.

Sample Input 2

2 1000
0 0 B
0 1 W

Sample Output 2

2

Sample Input 3

6 2
1 2 B
2 1 W
2 2 B
1 0 B
0 6 W
4 5 W

Sample Output 3

4

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

//extern crate rand;

use std::io;
use std::cmp;
//use rand::Rng;

fn main() {
    let (n, k) = {
        let v = read_values::<usize>();
        (v[0], v[1])
    };

    let xyc = (0..n).map(|_| {
        let v = read_values::<String>();
        let x = v[0].parse::<usize>().unwrap();
        let y = v[1].parse::<usize>().unwrap();
        let c = (v[2] == "B");
        if !c {
            assert_eq!(v[2], "W");
        }
        (x % (2 * k), y % (2 * k), c)
    }).collect::<Vec<_>>();
    println!("{}", solve(&xyc, k));
//    println!("{}", naive(&xyc, k));
//    test();
}


//fn test() {
//    let mut rng = rand::thread_rng();
//    loop {
//        let n = 2;
//        let k = 3;
//        let xyc: Vec<(usize, usize, bool)> = (0..n).map(|_| {
//            (rng.gen::<usize>() % (2 * k), rng.gen::<usize>() % (2 * k), rng.gen())
//        }).collect::<Vec<_>>();
//
//
//        let ans = solve(&xyc, k);
//        let ans2 = naive(&xyc, k);
//        if ans != ans2 {
//            println!("{} {}", n, k);
//            for t in &xyc {
//                let (x, y, c) = *t;
//                let color = if c { "B" } else { "W" };
//                println!("{} {} {}", x, y, color);
//            }
//            panic!();
//        }
//    }
//}

fn solve(xyc: &Vec<(usize, usize, bool)>, k: usize) -> usize {
    let n = xyc.len();
    let mut b = vec![vec![0; k]; k];
    let mut w = vec![vec![0; k]; k];

    for i in 0..n {
        let (x, y, c) = xyc[i];

        if x >= k && y >= k {
            if c {
                b[x % k][y % k] += 1;
            } else {
                w[x % k][y % k] += 1;
            }
        } else if x >= k {
            if c {
                w[x % k][y] += 1;
            } else {
                b[x % k][y] += 1;
            }
        } else if y >= k {
            if c {
                w[x][y % k] += 1;
            } else {
                b[x][y % k] += 1;
            }
        } else {
            if c {
                b[x][y] += 1;
            } else {
                w[x][y] += 1;
            }
        }
    }

    let b_sum = CumulativeSum::new(&b);
    let w_sum = CumulativeSum::new(&w);

    let w_all = w_sum.get_sum(0, 0, k - 1, k - 1);
    let b_all = b_sum.get_sum(0, 0, k - 1, k - 1);
    assert_eq!(w_all + b_all, n);

    let mut ans = cmp::max(w_all, b_all);
    for i in 0..k {
        for j in 0..k {
            let b_up = b_sum.get_sum(i, j, k - 1, k - 1);
            let b_low = if i == 0 || j == 0 {
                0
            } else {
                b_sum.get_sum(0, 0, i - 1, j - 1)
            };

            let w_up = w_sum.get_sum(i, j, k - 1, k - 1);
            let w_low = if i == 0 || j == 0 {
                0
            } else {
                w_sum.get_sum(0, 0, i - 1, j - 1)
            };

//            println!("i={}, j={}, w_up={}, w_low={}, b_up={}, b_low={}", i, j, w_up, w_low, b_up, b_low);

            let ok = b_up + b_low + w_all - w_up - w_low;
            assert!(w_all >= w_up + w_low);
            ans = cmp::max(ans, ok);

            let ok = w_up + w_low + (n - w_all) - b_up - b_low;
            ans = cmp::max(ans, ok);
        }
    }
    /*
    010
    000
    000

    000
    000
    001
    */
    assert!(ans <= n);
    ans
}

fn naive(xyc: &Vec<(usize, usize, bool)>, k: usize) -> usize {
    let t = k * 2;
    let n = xyc.len();
    let mut x = vec![0; n];
    let mut y = vec![0; n];
    let mut bw = vec![false; n];

    for i in 0..n {
        let (a, b, c) = xyc[i];
        x[i] = a % t;
        y[i] = b % t;
        bw[i] = c;
    }

    let mut ans = 0;
    for i in 0..t {
        let mut cur = 0;
        let mut cnt = vec![0 as i64; t + 1];
        for j in 0..n {
            let f = (if x[j] < i { x[j] + t - i } else { x[j] - i }) < k;
            if f ^ (y[j] < k) {
                if bw[j] {
                    cur += 1;
                }
            } else {
                if !bw[j] {
                    cur += 1;
                }
            }
            cnt[y[j]] += if f ^ (!bw[j]) { 1 } else { -1 };
        }
        for j in 0..t {
            ans = cmp::max(ans, cur as usize);
            cur += cnt[j] - cnt[(j + k) % t];
        }
    }
    ans
}

struct CumulativeSum {
    ny: usize,
    nx: usize,
    sum: Vec<Vec<usize>>,
}

impl CumulativeSum {
    fn new(a: &Vec<Vec<usize>>) -> CumulativeSum {
        let ny = a.len();
        let nx = a[0].len();
        let mut sum = vec![vec![0; nx + 1]; ny + 1];
        for i in 0..ny {
            for j in 0..nx {
                sum[i + 1][j + 1] = a[i][j] + sum[i][j + 1] + sum[i + 1][j] - sum[i][j];
            }
        }
        CumulativeSum { ny: ny, nx: nx, sum: sum }
    }
    fn get_sum(&self, y1: usize, x1: usize, y2: usize, x2: usize) -> usize {
        if y1 > y2 || x1 > x2 {
            return 0;
        }
        let y2 = cmp::min(y2, self.ny - 1);
        let x2 = cmp::min(x2, self.nx - 1);
        return self.sum[y2 + 1][x2 + 1] + self.sum[y1][x1] - self.sum[y1][x2 + 1] - self.sum[y2 + 1][x1];
    }
}

fn read_line() -> String {
    let stdin = io::stdin();
    let mut buf = String::new();
    stdin.read_line(&mut buf).unwrap();
    buf
}

fn read_values<T>() -> Vec<T> where T: std::str::FromStr, T::Err: std::fmt::Debug {
    read_line()
        .split(' ')
        .map(|a| a.trim().parse().unwrap())
        .collect()
}