Simple geometry (mathematical formula + convex hull) UVA 11168 Airport

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Test instructions: Find a straight line, so that the rest of the points are in the same side of the line, and so that the average distance to get straight lines shortest.

Analysis: The training guide P274, the convex hull first, if each side is counted side, is O (n ^ 2), however, according to the formula known straight line ax + by + C = 0. The distance from the point (x0, y0) to the line is: Fabs (ax0+by0+c)/sqrt (a*a+ B*B).

So as long as the x's and and Y's and, can be calculated at O (1) all distances and.

The two-point linear equation P1 (x1, y1), p2 (x2, y2) is converted to A general linear equation: A = y1-y2, B = x2-x1, C =-A * x1-b * y1;

/************************************************* author:running_time* Created time:2015/11/10 Tuesday 11:24:09* Fi Le Name:UVA_11168.cpp ************************************************/#include <bits/stdc++.h>using namespace std; #define Lson L, Mid, RT << 1#define Rson mid + 1, R, RT << 1 | 1typedef long ll;const int N = 1e5 + 10;const int INF = 0x3f3f3f3f;const int MOD = 1e9 + 7;const Double EPS = 1e-10;c    Onst Double PI = ACOs ( -1.0), int dcmp (double x) {if (Fabs (x) < EPS) return 0; else return x < 0? -1:1;}    struct point {double x, y;        Point () {} A (double x, double y): X (x), Y (y) {} point operator-(const point &r) const {//Vector subtraction    Return point (X-r.x, Y-R.Y);    } Point operator * (double p) const {//vector multiplied by a scalar return point (x * p, Y * p);    } point operator/(double p) const {//vector divided by scalar return point (x/p, y/p); } point operator + (const-point & r) Const {return point (x + r.x, y + r.y); } BOOL operator < (const point &r) const {return x < r.x | |    (x = = r.x && y < r.y);    } bool operator = = (Const point &r) const {return dcmp (x-r.x) = = 0 && dcmp (y-r.y) = = 0; }};typedef Point Vector;    Point Read_point (void) {double x, y;    scanf ("%lf%lf", &x, &y); Return point (x, y);} Double dot (vector A, vector B) {//vector dot product return a.x * b.x + a.y * B.Y;} Double Cross (vector A, vector B) {//vector fork product return a.x * B.Y-A.Y * b.x;} Double Length (Vector v) {return sqrt (dot (v, v));}    Vector<point> convex_hull (vector<point> PS) {sort (Ps.begin (), Ps.end ());    Ps.erase (Unique (Ps.begin (), Ps.end ()), Ps.end ());    int n = ps.size (), k = 0;    Vector<point> QS (n * 2);         for (int i=0; i<n; ++i) {when (K > 1 && Cross (qs[k-1]-qs[k-2], Ps[i]-qs[k-2]) <= 0) k--; qs[k++] = ps[I]; } for (int t=k, i=n-2; i>=0;-I.) {while (K > t && Cross (Qs[k-1]-qs[k-2], Ps[i]-qs[k-2]) &lt        ; = 0) k--;    qs[k++] = ps[i];    } qs.resize (k-1); return QS;} Double Sqr (double x) {return x * x;}  int main (void) {int T, cas = 0;    scanf ("%d", &t);  while (t--) {int n;        scanf ("%d", &n);        Vector<point> PS;        Double x, y, sx = 0, sy = 0;            for (int i=0; i<n; ++i) {scanf ("%lf%lf", &x, &y);    SX + = x;            sy + = y;        Ps.push_back (Point (x, y));        } vector<point> qs = Convex_hull (PS);    if (int) qs.size () <= 2) {printf ("Case #%d:%.3f\n", ++cas, 0.0);        Continue        } double mn = 1e9;        Qs.push_back (Qs[0]);            for (int i=0; i<qs.size ()-1; ++i) {Double A = qs[i].y-qs[i+1].y;            Double B = qs[i+1].x-qs[i].x;    Double C =-A * qs[i].x-b * QS[I].Y;        Double tmp = fabs (A * SX + B * sy + n * C)/sqrt (SQR (a) + SQR (B));        if (MN > tmp) MN = tmp;    } printf ("Case #%d:%.3f\n", ++cas, mn/n);    }//cout << "time Elapsed:" << 1.0 * Clock ()/clocks_per_sec << "s.\n"; return 0;}

　　

Simple geometry (mathematical formula + convex hull) UVA 11168 Airport