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 C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)


#include <algorithm>
#include <iostream>

using namespace std;

constexpr static int MAXN = 1e5;
constexpr static int MAXK = 2000;

int N, K;
int x[MAXN], y[MAXN];
char c[MAXN];

int scnt[MAXK], ecnt[MAXK];

int main() {
    cin >> N >> K;
    for (int i = 0; i < N; i++) {
        cin >> x[i] >> y[i] >> c[i];
        x[i] %= (2 * K);
        y[i] %= (2 * K);
    }

    int best = 0;
    for (int dy = 0; dy < 2 * K; dy++) {
        for (int dx = 0; dx < 2 * K; dx++)
            scnt[dx] = ecnt[dx] = 0;

        //cout << "dy: " << dy << endl;
        int y0 = dy;
        int y1 = (dy + K) % (2 * K);
        for (int i = 0; i < N; i++) {
            int x0 = x[i];
            int x1 = (x[i] + K) % (2 * K);
            bool flip = false;
            if ((y0 < y1 && y[i] >= y0 && y[i] < y1)
                    || (y0 > y1 && (y[i] >= y0 || y[i] < y1))) {
                if (c[i] == 'W') {
                    scnt[x0]++;
                    ecnt[x1]++;
                }
                else {
                    scnt[x1]++;
                    ecnt[x0]++;
                    flip = true;
                }
            }
            else {
                if (c[i] == 'W') {
                    scnt[x1]++;
                    ecnt[x0]++;
                    flip = true;
                }
                else {
                    scnt[x0]++;
                    ecnt[x1]++;
                }
            }
            //cout << x[i] << " " << y[i] << " " << c[i] << " | ";
            //cout << x0 << " " << x1 << " " << flip << endl;
        }
        //for (int dx = 0; dx < 2 * K; dx++)
        //    cout << dx << " " << scnt[dx] << " " << ecnt[dx] << endl;

        int cur = 0;
        for (int dx = K; dx < 2 * K; dx++)
            cur += scnt[dx];
        for (int dx = 0; dx < 2 * K; dx++) {
            cur -= ecnt[dx];
            cur += scnt[dx];
            best = max(best, cur);
        }
    }
    cout << best << endl;
}