CF1388E-Uncle Bogdan and Projections
题目:
题目描述:
After returning to shore, uncle Bogdan usually visits the computer club “The Rock”, to solve tasks in a pleasant company. One day, uncle Bogdan met his good old friend who told him one unusual task…
There are $ n $ non-intersecting horizontal segments with ends in integers points on the plane with the standard cartesian coordinate system. All segments are strictly above the $ OX $ axis. You can choose an arbitrary vector ( $ a $ , $ b $ ), where $ b < 0 $ and coordinates are real numbers, and project all segments to $ OX $ axis along this vector. The projections shouldn’t intersect but may touch each other.
Find the minimum possible difference between $ x $ coordinate of the right end of the rightmost projection and $ x $ coordinate of the left end of the leftmost projection.
输入格式:
The first line contains the single integer $ n $ ( $ 1 \le n \le 2000 $ ) — the number of segments.
The $ i $ -th of the next $ n $ lines contains three integers $ xl_i $ , $ xr_i $ and $ y_i $ ( $ -10^6 \le xl_i < xr_i \le 10^6 $ ; $ 1 \le y_i \le 10^6 $ ) — coordinates of the corresponding segment.
It’s guaranteed that the segments don’t intersect or touch.
输出格式:
Print the minimum possible difference you can get.
Your answer will be considered correct if its absolute or relative error doesn’t exceed $ 10^{-6} $ .
Formally, if your answer is $ a $ and jury’s answer is $ b $ then your answer will be considered correct if $ \frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6} $ .
样例:
样例输入 1:
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1 6 2
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4 6 6
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样例输出 1:
样例输入 2:
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| 3
2 5 1
4 6 4
7 8 2
|
样例输出 2:
样例输入 3:
样例输出 3:
思路:
实现:
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| #include "ybwhead/ios.h"
#define pb push_back
#define x first
#define y second
using namespace std;
const int N = 1e5 + 10;
const double eps = 1e-9;
double xl[N], xr[N], y[N], pi = acos(-1), mn_x, mx_x;
int ind_l, ind_r;
double point_pr(double x, double y, double ctg)
{
return x - y * ctg;
}
int main()
{
int n;
vector<pair<double, int>> q;
vector<pair<double, pair<int, int>>> events, prom_left, prom_right;
yin >> n;
pair<double, double> mx, mn;
mx = {-1.0, -1.0};
mn = {2e9, 2e9};
mn_x = 2e9;
mx_x = -2e9;
for (int i = 0; i < n; ++i)
{
yin >> xl[i] >> xr[i] >> y[i];
if (xl[i] < mn_x)
{
mn_x = xl[i];
}
if (xr[i] > mx_x)
{
mx_x = xr[i];
}
if (mx.y < y[i])
{
mx.y = y[i];
mx.x = xl[i];
ind_l = i;
}
else if (mx.y == y[i] && mx.x > xl[i])
{
mx.y = y[i];
mx.x = xl[i];
ind_l = i;
}
if (mn.y > y[i])
{
mn.y = y[i];
mn.x = xl[i];
ind_r = i;
}
else if (mn.y == y[i] && mn.x < xl[i])
{
mn.y = y[i];
mn.x = xl[i];
ind_r = i;
}
}
double a1, a2;
for (int i = 0; i < n; ++i)
{
for (int j = 0; j < n; ++j)
{
if (y[i] > y[j])
{
a1 = (xr[i] - xl[j]) / (y[i] - y[j]);
a2 = (xl[i] - xr[j]) / (y[i] - y[j]);
q.pb({a1, 1});
q.pb({a2, 2});
a1 = (xl[i] - xl[j]) / (y[i] - y[j]);
a2 = (xr[i] - xr[j]) / (y[i] - y[j]);
events.pb({a1, {i, j}});
events.pb({a2, {-i - 1, j}});
}
}
}
if (q.empty())
{
yout << mx_x - mn_x << endl;
return 0;
}
sort(q.rbegin(), q.rend());
int cnt = 0;
double last = 0;
for (auto i : q)
{
if (i.y == 2)
{
--cnt;
if (!cnt)
{
events.pb({i.x, {-1e9, -1e9}});
}
}
else
{
if (!cnt)
{
events.pb({i.x, {-1e9, -1e9}});
}
++cnt;
}
}
sort(events.rbegin(), events.rend());
double ans = 1e18, ang;
last = -1e18;
for (auto i : events)
{
if (i.y.x == i.y.y)
{
unordered_set<int> s;
vector<int> to_check;
for (auto j : prom_left)
{
s.insert(j.y.x);
if (j.y.x == ind_l)
{
to_check.pb(j.y.y);
}
}
prom_left.clear();
for (auto j : to_check)
{
if (!s.count(j))
{
ind_l = j;
break;
}
}
s.clear();
to_check.clear();
for (auto j : prom_right)
{
s.insert(j.y.y);
if (j.y.y == ind_r)
{
to_check.pb(-j.y.x - 1);
}
}
prom_right.clear();
for (auto j : to_check)
{
if (!s.count(j))
{
ind_r = j;
break;
}
}
s.clear();
to_check.clear();
double res = point_pr(xr[ind_r], y[ind_r], i.x) - point_pr(xl[ind_l], y[ind_l], i.x);
if (ans > res)
{
ans = res;
ang = i.x;
}
}
else if (i.y.x < 0)
{
if (abs(i.x - last) > eps)
{
unordered_set<int> s;
vector<int> to_check;
for (auto j : prom_right)
{
s.insert(j.y.y);
if (j.y.y == ind_r)
{
to_check.pb(-j.y.x - 1);
}
}
prom_right.clear();
for (auto j : to_check)
{
if (!s.count(j))
{
ind_r = j;
break;
}
}
s.clear();
to_check.clear();
}
prom_right.pb(i);
}
else
{
if (abs(i.x - last) > eps)
{
unordered_set<int> s;
vector<int> to_check;
for (auto j : prom_left)
{
s.insert(j.y.x);
if (j.y.x == ind_l)
{
to_check.pb(j.y.y);
}
}
prom_left.clear();
for (auto j : to_check)
{
if (!s.count(j))
{
ind_l = j;
break;
}
}
s.clear();
to_check.clear();
}
prom_left.pb(i);
}
last = i.x;
}
yout << ans << endl;
return 0;
}
|