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recalculating2.cpp
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recalculating2.cpp
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// Copyright (c) 2020 kamyu. All rights reserved.
/*
* Google Code Jam 2020 Round 3 - Problem D. Recalculating
* https://codingcompetitions.withgoogle.com/codejam/round/000000000019ff7e/00000000003775e9
*
* Time: O(N^2 * logN)
* Space: O(N^2)
*
* slower but more general in calculation of rect area
*
*/
#include <iostream>
#include <functional>
#include <array>
#include <string>
#include <vector>
#include <deque>
#include <utility>
#include <tuple>
#include <unordered_set>
#include <unordered_map>
#include <algorithm>
using std::ios_base;
using std::cin;
using std::cout;
using std::endl;
using std::function;
using std::array;
using std::string;
using std::to_string;
using std::vector;
using std::deque;
using std::pair;
using std::tuple;
using std::tie;
using std::get;
using std::unordered_set;
using std::unordered_map;
using std::sort;
using Groups = unordered_map<int64_t, vector<tuple<int64_t, int64_t, int64_t, int64_t>>>;
using Points = vector<array<int64_t, 2>>;
static const int64_t MOD = 1e9 + 7;
static const int64_t P = 113;
uint64_t gcd(uint64_t a, uint64_t b) {
while (b != 0) {
const auto tmp = b;
b = a % b;
a = tmp;
}
return a;
}
template <typename T>
class SegmentTree {
public:
explicit SegmentTree(
int N,
const function<void(vector<T> *, int)>& query_fn,
const function<void(T *, int64_t)>& update_fn)
: N_(N),
tree_(2 * N),
query_fn_(query_fn),
update_fn_(update_fn) {
for (int i = tree_.size() - 1; i >= 1; --i) {
query_fn_(&tree_, i);
}
}
void update(int L, int R, int val) {
L += N_; R += N_;
int L0 = L, R0 = R;
for (; L <= R; L >>= 1, R >>= 1) {
if ((L & 1) == 1) {
apply(L++, val);
}
if ((R & 1) == 0) {
apply(R--, val);
}
}
pull(L0); pull(R0);
}
T query() {
return tree_[1];
}
private:
void apply(int x, int val) {
update_fn_(&tree_[x], val);
query_fn_(&tree_, x);
}
void pull(int x) {
while (x > 1) {
x >>= 1;
query_fn_(&tree_, x);
}
}
int N_;
vector<T> tree_;
const function<void(vector<T> *, int)> query_fn_;
const function<void(T *, int64_t)> update_fn_;
};
Groups group_rects(const Points& points, int64_t D) {
unordered_set<int64_t> x_set, y_set;
for (auto& p : points) {
x_set.emplace(p[0] - D);
x_set.emplace(p[0] + D);
y_set.emplace(p[1] - D);
y_set.emplace(p[1] + D);
}
vector<int64_t> xs(cbegin(x_set), cend(x_set)), ys(cbegin(y_set), cend(y_set));
sort(begin(xs), end(xs)), sort(begin(ys), end(ys));
vector<int64_t> exp(1, 1);
while (exp.size() < points.size()) {
exp.emplace_back(exp.back() * P * P % MOD);
}
Groups groups;
for (int j = 0; j < ys.size() - 1; ++j) {
int64_t rolling_hash = 0;
deque<int> dq;
int left = 0, right = 0;
for (int i = 0; i < xs.size() - 1; ++i) {
for (; right < points.size() && points[right][0] <= xs[i] + D; ++right) {
if (ys[j + 1] - D <= points[right][1] && points[right][1] <= ys[j] + D) {
if (!dq.empty()) {
auto a = dq.back(), b = right;
auto x = ((points[b][0] - points[a][0]) % MOD + MOD) % MOD;
auto y = ((points[b][1] - points[a][1]) % MOD + MOD) % MOD;
auto delta = (x * P + y) % MOD;
rolling_hash = (rolling_hash * P * P + delta) % MOD;
}
dq.emplace_back(right);
}
}
for (; left < points.size() && points[left][0] < xs[i + 1] - D; ++left) {
if (ys[j + 1] - D <= points[left][1] && points[left][1] <= ys[j] + D) {
auto a = dq.front(); dq.pop_front();
if (dq.size() >= 1) {
auto b = dq.front();
auto x = ((points[b][0] - points[a][0]) % MOD + MOD) % MOD;
auto y = ((points[b][1] - points[a][1]) % MOD + MOD) % MOD;
auto delta = ((x * P + y) * exp[dq.size() - 1]) % MOD;
rolling_hash = ((rolling_hash - delta) % MOD + MOD) % MOD;
}
}
}
if (dq.empty()) {
continue;
}
// the rectangle is fully covered by ordered repair centers in dq,
// normalized by being relative to the first repair center
int64_t x0 = xs[i] - points[dq.front()][0], y0 = ys[j] - points[dq.front()][1];
int64_t x1 = xs[i + 1] - points[dq.front()][0], y1 = ys[j + 1] - points[dq.front()][1];
groups[rolling_hash].emplace_back(x0, y0, x1, y1);
}
}
return groups;
}
pair<uint64_t, uint64_t> calc_areas(const Groups& groups) {
using Event = tuple<int64_t, int64_t, int64_t, int64_t>;
int64_t unique = 0, total = 0;
for (const auto& kvp : groups) {
vector<Event> intervals;
for (const auto& rect : kvp.second) {
int64_t x0, y0, x1, y1;
tie(x0, y0, x1, y1) = rect;
intervals.emplace_back(x0, y0, y1, +1);
intervals.emplace_back(x1, y0, y1, -1);
}
sort(begin(intervals), end(intervals)); // at most O(N^2) intervals, total time: O(N^2 * logN)
unordered_set<int64_t> y_set;
for (const auto& interval : intervals) {
int64_t x, y0, y1, v;
tie(x, y0, y1, v) = interval;
y_set.emplace(y0);
y_set.emplace(y1);
}
vector<int64_t> ys(cbegin(y_set), cend(y_set));
sort(begin(ys), end(ys));
unordered_map<int, int> height_to_idx;
for (int i = 0; i < ys.size(); ++i) {
height_to_idx[ys[i]] = i;
}
// Node: [sum_len_of_covered, len_of_1_or_up_covered, len_of_2_or_up_covered, len_of_0_or_up_covered, count_of_covered]
using Node = array<int64_t, 5>; // define customized operations of segment tree
const auto& query = [&ys](vector<Node> *tree, int x) {
int N = tree->size() / 2;
if (x >= N) { // leaf node
(*tree)[x][3] = ys[(x - N) + 1] - ys[(x - N)];
(*tree)[x][0] = (*tree)[x][3] * (*tree)[x][4];
for (int i = 1; i <= 2; ++i) {
(*tree)[x][i] = (i - (*tree)[x][4] > 0) ? 0 : (*tree)[x][3];
}
} else {
(*tree)[x][3] = (*tree)[2 * x][3] + (*tree)[2 * x + 1][3];
(*tree)[x][0] = (*tree)[x][3] * (*tree)[x][4] + (*tree)[2 * x][0] + (*tree)[2 * x + 1][0];
for (int i = 1; i <= 2; ++i) {
(*tree)[x][i] = (i - (*tree)[x][4] > 0) ? (*tree)[2 * x][i - (*tree)[x][4]] + (*tree)[2 * x + 1][i - (*tree)[x][4]] : (*tree)[x][3];
}
}
};
const auto& update = [](Node *x, int64_t val) {
(*x)[4] += val;
};
SegmentTree<Node> segment_tree(ys.size() - 1, query, update); // init segment tree with customized operations
for (int i = 0; i < intervals.size() - 1; ++i) {
int64_t x, y0, y1, v;
tie(x, y0, y1, v) = intervals[i];
segment_tree.update(height_to_idx[y0], height_to_idx[y1] - 1, v); // at most O(N^2) intervals, total time: O(N^2 * logN)
unique += (get<0>(intervals[i + 1]) - x) * (segment_tree.query()[1] - segment_tree.query()[2]);
total += (get<0>(intervals[i + 1]) - x) * segment_tree.query()[0];
}
}
return {unique, total};
}
string recalculating() {
int N; int64_t D;
cin >> N >> D;
Points points;
for (int i = 0; i < N; ++i) {
int64_t x, y;
cin >> x >> y;
points.push_back({x + y, x - y});
}
sort(begin(points), end(points));
const auto& groups = group_rects(points, D); // Time: O(N^2)
uint64_t unique, total;
tie(unique, total) = calc_areas(groups); // Time: O(N^2 * logN)
const auto& g = gcd(unique, total);
return to_string(unique / g) + " " + to_string(total / g);
}
int main() {
ios_base::sync_with_stdio(false), cin.tie(nullptr);
int T;
cin >> T;
for (int test = 1; test <= T; ++test) {
cout << "Case #" << test << ": " << recalculating() << '\n';
}
return 0;
}