#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;
}
// }}}
// SegTree, copied from AtCoder library {{{
// AtCoder doc: https://atcoder.github.io/ac-library/master/document_en/segtree.html
//
// Notes:
// - Index of elements from 0 -> n-1
// - Range queries are [l, r-1]
//
// Tested:
// - (binary search) https://atcoder.jp/contests/practice2/tasks/practice2_j
// - https://oj.vnoi.info/problem/gss
// - https://oj.vnoi.info/problem/nklineup
// - (max_right & min_left for delete position queries) https://oj.vnoi.info/problem/segtree_itstr
// - https://judge.yosupo.jp/problem/point_add_range_sum
// - https://judge.yosupo.jp/problem/point_set_range_composite
int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (unsigned int)(n)) x++;
return x;
}
template<
class T, // data type for nodes
T (*op) (T, T), // operator to combine 2 nodes
T (*e)() // identity element
>
struct SegTree {
SegTree() : SegTree(0) {}
explicit SegTree(int n) : SegTree(vector<T> (n, e())) {}
explicit SegTree(const vector<T>& v) : _n((int) v.size()) {
log = ceil_pow2(_n);
size = 1<<log;
d = vector<T> (2*size, e());
for (int i = 0; i < _n; i++) d[size+i] = v[i];
for (int i = size - 1; i >= 1; i--) {
update(i);
}
}
// 0 <= p < n
void set(int p, T x) {
assert(0 <= p && p < _n);
p += size;
d[p] = x;
for (int i = 1; i <= log; i++) update(p >> i);
}
// 0 <= p < n
T get(int p) const {
assert(0 <= p && p < _n);
return d[p + size];
}
// Get product in range [l, r-1]
// 0 <= l <= r <= n
// For empty segment (l == r) -> return e()
T prod(int l, int r) const {
assert(0 <= l && l <= r && r <= _n);
T sml = e(), smr = e();
l += size;
r += size;
while (l < r) {
if (l & 1) sml = op(sml, d[l++]);
if (r & 1) smr = op(d[--r], smr);
l >>= 1;
r >>= 1;
}
return op(sml, smr);
}
T all_prod() const {
return d[1];
}
// Binary search on SegTree to find largest r:
// f(op(a[l] .. a[r-1])) = true (assuming empty array is always true)
// f(op(a[l] .. a[r])) = false (assuming op(..., a[n]), which is out of bound, is always false)
template <bool (*f)(T)> int max_right(int l) const {
return max_right(l, [](T x) { return f(x); });
}
template <class F> int max_right(int l, F f) const {
assert(0 <= l && l <= _n);
assert(f(e()));
if (l == _n) return _n;
l += size;
T sm = e();
do {
while (l % 2 == 0) l >>= 1;
if (!f(op(sm, d[l]))) {
while (l < size) {
l = (2 * l);
if (f(op(sm, d[l]))) {
sm = op(sm, d[l]);
l++;
}
}
return l - size;
}
sm = op(sm, d[l]);
l++;
} while ((l & -l) != l);
return _n;
}
// Binary search on SegTree to find smallest l:
// f(op(a[l] .. a[r-1])) = true (assuming empty array is always true)
// f(op(a[l-1] .. a[r-1])) = false (assuming op(a[-1], ..), which is out of bound, is always false)
template <bool (*f)(T)> int min_left(int r) const {
return min_left(r, [](T x) { return f(x); });
}
template <class F> int min_left(int r, F f) const {
assert(0 <= r && r <= _n);
assert(f(e()));
if (r == 0) return 0;
r += size;
T sm = e();
do {
r--;
while (r > 1 && (r % 2)) r >>= 1;
if (!f(op(d[r], sm))) {
while (r < size) {
r = (2 * r + 1);
if (f(op(d[r], sm))) {
sm = op(d[r], sm);
r--;
}
}
return r + 1 - size;
}
sm = op(d[r], sm);
} while ((r & -r) != r);
return 0;
}
int _n, size, log;
vector<T> d;
void update(int k) {
d[k] = op(d[2*k], d[2*k+1]);
}
};
// }}}
// SegTree examples {{{
// Examples: Commonly used SegTree ops: max / min / sum
struct MaxSegTreeOp {
static int op(int x, int y) {
return max(x, y);
}
static int e() {
return INT_MIN;
}
};
struct MinSegTreeOp {
static int op(int x, int y) {
return min(x, y);
}
static int e() {
return INT_MAX;
}
};
struct SumSegTreeOp {
static long long op(long long x, long long y) {
return x + y;
}
static long long e() {
return 0;
}
};
// Example
// SegTree<int, MaxSegTreeOp::op, MaxSegTreeOp::e> seg_tree(a);
// SegTree<int, MinSegTreeOp::op, MinSegTreeOp::e> seg_tree(a);
// }}}
// AC {{{
int sub7(int n, const std::vector<Fish>& fishes) {
std::vector<std::vector<int>> rows(n); // rows[c] = important coordinates at col c
std::vector<std::vector<int>> weights(n); // prefix sum of weights
std::vector<std::vector<std::pair<int,int>>> fishesAt(n); // stores {row, weight}
for (const auto& fish : fishes) {
int c = fish.col;
rows[c].push_back(fish.row);
if (c > 0) rows[c-1].push_back(fish.row);
if (c + 1 < n) rows[c+1].push_back(fish.row);
fishesAt[c].push_back({fish.row, fish.weight});
}
for (int c = 0; c < n; ++c) {
rows[c].push_back(-1);
std::sort(rows[c].begin(), rows[c].end());
rows[c].erase(std::unique(rows[c].begin(), rows[c].end()), rows[c].end());
std::sort(fishesAt[c].begin(), fishesAt[c].end());
weights[c].resize(rows[c].size());
int fish_id = 0;
for (int i = 0; i < (int) rows[c].size(); ++i) {
if (i > 0) weights[c][i] = weights[c][i-1];
while (fish_id < (int) fishesAt[c].size()
&& fishesAt[c][fish_id].first <= rows[c][i]) {
weights[c][i] += fishesAt[c][fish_id].second;
++fish_id;
}
}
}
// 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), g(n);
std::vector<SegTree<int, MaxSegTreeOp::op, MaxSegTreeOp::e>> st_g(n), st_g_with_next_col(n), st_f_with_next_col(n);
for (int c = 0; c < n; ++c) {
int sz = static_cast<int> (rows[c].size());
f[c] = g[c] = std::vector<int> (sz, 0);
// compute {{{
if (c > 0) {
for (int i = 0; i < sz; ++i) {
int i_1 = std::upper_bound(rows[c-1].begin(), rows[c-1].end(), rows[c][i])
- rows[c-1].begin() - 1;
// this is first pier
f[c][i] = weights[c-1][i_1];
// last pier at column i-3
if (c >= 3) {
upMax(f[c][i], st_g_with_next_col[c-3].all_prod() + weights[c-1][i_1]);
}
// last pier at column i-2
if (c >= 2) {
upMax(f[c][i], std::max(
st_g[c-2].all_prod() + weights[c-1][i_1],
st_g_with_next_col[c-2].all_prod()));
}
g[c][i] = f[c][i];
// last pier at column i-1
if (c >= 1) {
// last row <= r
int cur = std::max(
st_g[c-1].prod(0, i_1+1),
st_f_with_next_col[c-1].prod(0, i_1+1) + weights[c-1][i_1]);
upMax(f[c][i], cur);
upMax(g[c][i], cur);
// last row > r
upMax(f[c][i], st_g[c-1].prod(i_1+1, (int) g[c-1].size()));
upMax(g[c][i], st_g_with_next_col[c-1].prod(i_1+1, (int) g[c-1].size()) - weights[c][i]);
}
}
}
// }}}
// aggregate {{{
st_g[c] = SegTree<int, MaxSegTreeOp::op, MaxSegTreeOp::e> (g[c]);
if (c + 1 < n) {
std::vector<int> g_with_next_col(sz), f_with_next_col(sz);
for (int i = 0; i < sz; ++i) {
int i_1 = std::upper_bound(rows[c+1].begin(), rows[c+1].end(), rows[c][i])
- rows[c+1].begin() - 1;
g_with_next_col[i] = g[c][i] + weights[c+1][i_1];
f_with_next_col[i] = f[c][i] - weights[c][i];
}
st_g_with_next_col[c] = SegTree<int, MaxSegTreeOp::op, MaxSegTreeOp::e> (g_with_next_col);
st_f_with_next_col[c] = SegTree<int, MaxSegTreeOp::op, MaxSegTreeOp::e> (f_with_next_col);
}
// }}}
}
int res = 0;
for (int c = 0; c < n; ++c) {
int sz = rows[c].size();
for (int i = 0; i < sz; ++i) {
int cur = g[c][i];
if (c + 1 < n) {
int i_1 = std::upper_bound(rows[c+1].begin(), rows[c+1].end(), rows[c][i])
- rows[c+1].begin() - 1;
cur += weights[c+1][i_1];
}
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 sub7(n, fishes);
}
