#include "bits/stdc++.h"
using namespace std;
#define int long long
#define i_1 jakcjacjl
struct Fish {
int col, row;
int weight;
};
bool operator < (const Fish& a, const Fish& b) {
if (a.col != b.col) return a.col < b.col;
return a.row < b.row;
}
void upMax(int& f, int val) {
if (val > f) f = val;
}
// sub1 - 3 {{{
// fishes are on even columns -> build piers on odd columns
// & catch all fishes
int sub1(const std::vector<Fish>& fishes) {
int res = 0;
for (const auto& fish : fishes) {
res += fish.weight;
}
return res;
}
// fishes are on first 2 columns
int sub2(int n, const std::vector<Fish>& fishes) {
std::vector<int> zeroes(n); // prefix sum of fish weights at column == 0
std::vector<int> ones(n); // prefix sum of fish weights at column == 1
for (const auto& fish : fishes) {
if (fish.col == 0) zeroes[fish.row] += fish.weight;
if (fish.col == 1) ones[fish.row] += fish.weight;
}
std::partial_sum(zeroes.begin(), zeroes.end(), zeroes.begin());
std::partial_sum(ones.begin(), ones.end(), ones.begin());
int res = ones.back(); // init: only catch fishes at column == 1
for (int i = 0; i < n; ++i) {
// build pier until at column 1, row 0-i
if (n == 2) upMax(res, zeroes[i]);
else upMax(res, zeroes[i] + ones.back() - ones[i]);
}
return res;
}
// all fishes are on row == 0
int sub3(int n, const std::vector<Fish>& fishes) {
std::vector<int> weights(n); // weights[i] = weight of fish at column i
for (const auto& fish : fishes) {
weights[fish.col] += fish.weight;
}
// f[i] = best strategy if we BUILD PIER AT i, only considering col 0..i
// i-4 i-3 i-2 i-1 i
std::vector<int> f(n);
f[0] = 0;
for (int i = 1; i < n; ++i) {
f[i] = std::max(f[i-1], weights[i-1]);
if (i >= 2) {
upMax(f[i], f[i-2] + weights[i-1]);
}
if (i >= 3) {
upMax(f[i], f[i-3] + weights[i-2] + weights[i-1]);
}
}
int res = 0;
for (int i = 0; i < n; ++i) {
int cur = f[i];
if (i + 1 < n) cur += weights[i+1];
upMax(res, cur);
}
return res;
}
// }}}
// sub 5 N <= 300 {{{
int sub5(int n, const std::vector<Fish>& fishes) {
// Init weights[i][j] = sum of fish on column i, from row 0 -> row j
std::vector<std::vector<int>> weights(n, std::vector<int> (n, 0));
for (const auto& fish : fishes) {
weights[fish.col][fish.row] += fish.weight;
}
for (int col = 0; col < n; ++col) {
std::partial_sum(weights[col].begin(), weights[col].end(), weights[col].begin());
}
// f[c][r] = best strategy if we last BUILD PIER AT column c, row r
// only considering fishes <= (c, r)
// g[c][r] = similar to f[c][r] but consider fishes at column c, in row [r, n-1]
std::vector<std::vector<int>> f(n, std::vector<int> (n, 0)),
g(n, std::vector<int> (n, 0));
// f <= g
for (int c = 1; c < n; ++c) {
for (int r = 0; r < n; ++r) {
// this is first pier
f[c][r] = g[c][r] = weights[c-1][r];
// last pier at column i-1
for (int lastRow = 0; lastRow < n; ++lastRow) {
if (lastRow <= r) {
int cur = std::max(
f[c-1][lastRow] + weights[c-1][r] - weights[c-1][lastRow],
g[c-1][lastRow]);
upMax(f[c][r], cur);
upMax(g[c][r], cur);
} else {
upMax(f[c][r], g[c-1][lastRow]);
upMax(g[c][r], g[c-1][lastRow] + weights[c][lastRow] - weights[c][r]);
}
}
// last pier at column i-2
if (c >= 2) {
for (int lastRow = 0; lastRow < n; ++lastRow) {
int cur = g[c-2][lastRow] + weights[c-1][std::max(lastRow, r)];
upMax(f[c][r], cur);
upMax(g[c][r], cur);
}
}
// last pier at column i-3
if (c >= 3) {
for (int lastRow = 0; lastRow < n; ++lastRow) {
int cur = g[c-3][lastRow] + weights[c-2][lastRow] + weights[c-1][r];
upMax(f[c][r], cur);
upMax(g[c][r], cur);
}
}
}
}
int res = 0;
for (int c = 0; c < n; ++c) {
for (int r = 0; r < n; ++r) {
assert(g[c][r] >= f[c][r]);
int cur = g[c][r];
if (c + 1 < n) {
cur += weights[c+1][r];
}
upMax(res, cur);
}
}
return res;
}
// }}}
// RMQ {{{
//
// Sparse table
// Usage:
// RMQ<int, _min> st(v);
//
// Note:
// - doesn't work for empty range
//
// Tested:
// - https://judge.yosupo.jp/problem/staticrmq
template<class T, T (*op) (T, T)> struct RMQ {
RMQ() = default;
RMQ(const vector<T>& v) : t{v}, n{(int) v.size()} {
for (int k = 1; (1<<k) <= n; ++k) {
t.emplace_back(n - (1<<k) + 1);
for (int i = 0; i + (1<<k) <= n; ++i) {
t[k][i] = op(t[k-1][i], t[k-1][i + (1<<(k-1))]);
}
}
}
// get range [l, r-1]
// doesn't work for empty range
T get(int l, int r) const {
assert(0 <= l && l < r && r <= n);
int k = __lg(r - l);
return op(t[k][l], t[k][r - (1<<k)]);
}
T get_all() const {
return get(0, n);
}
private:
vector<vector<T>> t;
int n;
};
template<class T> T _min(T a, T b) { return b < a ? b : a; }
template<class T> T _max(T a, T b) { return a < b ? b : a; }
// }}}
// N <= 3000 {{{
int sub6(int n, const std::vector<Fish>& fishes) {
// Init weights[i][j] = sum of fish on column i, from row 0 -> row j
std::vector<std::vector<int>> weights(n, std::vector<int> (n, 0));
for (const auto& fish : fishes) {
weights[fish.col][fish.row] += fish.weight;
}
for (int col = 0; col < n; ++col) {
std::partial_sum(weights[col].begin(), weights[col].end(), weights[col].begin());
}
// f[c][r] = best strategy if we last BUILD PIER AT column c, row r
// only considering fishes <= (c, r)
// g[c][r] = similar to f[c][r] but consider fishes at column c, in row [r, n-1]
std::vector<std::vector<int>> f(n, std::vector<int> (n, 0)),
g(n, std::vector<int> (n, 0));
std::vector<RMQ<int, _max>> rmq_g(n), rmq_g_with_next_col(n), rmq_f_with_next_col(n);
// f <= g
for (int c = 0; c < n; ++c) {
// compute {{{
if (c > 0) {
for (int r = 0; r < n; ++r) {
// this is first pier
f[c][r] = weights[c-1][r];
// last pier at column i-3
if (c >= 3) {
upMax(f[c][r], rmq_g_with_next_col[c-3].get_all() + weights[c-1][r]);
}
// last pier at column i-2
if (c >= 2) {
upMax(f[c][r], std::max(
rmq_g[c-2].get_all() + weights[c-1][r],
rmq_g_with_next_col[c-2].get_all()));
}
g[c][r] = f[c][r];
// last pier at column i-1
if (c >= 1) {
// last row <= r
int cur = std::max(
rmq_g[c-1].get(0, r+1),
rmq_f_with_next_col[c-1].get(0, r+1) + weights[c-1][r]);
upMax(f[c][r], cur);
upMax(g[c][r], cur);
// last row > r
if (r + 1 < n) {
upMax(f[c][r], rmq_g[c-1].get(r+1, n));
upMax(g[c][r], rmq_g_with_next_col[c-1].get(r+1, n) - weights[c][r]);
}
}
}
}
// }}}
// aggregate {{{
rmq_g[c] = RMQ<int, _max> (g[c]);
if (c + 1 < n) {
std::vector<int> g_with_next_col(n);
for (int r = 0; r < n; ++r) {
g_with_next_col[r] = g[c][r] + weights[c+1][r];
}
rmq_g_with_next_col[c] = RMQ<int, _max> (g_with_next_col);
std::vector<int> f_with_next_col(n);
for (int r = 0; r < n; ++r) {
f_with_next_col[r] = f[c][r] - weights[c][r];
}
rmq_f_with_next_col[c] = RMQ<int, _max> (f_with_next_col);
}
// }}}
}
int res = 0;
for (int c = 0; c < n; ++c) {
for (int r = 0; r < n; ++r) {
assert(g[c][r] >= f[c][r]);
int cur = g[c][r];
if (c + 1 < n) {
cur += weights[c+1][r];
}
upMax(res, cur);
}
}
return res;
}
// }}}
#undef int
long long max_weights(
int n, int nFish,
std::vector<int> xs,
std::vector<int> ys,
std::vector<int> ws) {
std::vector<Fish> fishes;
for (int i = 0; i < nFish; ++i) {
fishes.push_back({xs[i], ys[i], ws[i]});
}
if (std::all_of(xs.begin(), xs.end(), [] (int x) { return x % 2 == 0; })) {
return sub1(fishes);
}
if (*std::max_element(xs.begin(), xs.end()) <= 1) {
return sub2(n, fishes);
}
if (*std::max_element(ys.begin(), ys.end()) == 0) {
return sub3(n, fishes);
}
return sub6(n, fishes);
}
