C++で分割統治アルゴリズムを使用した凸包

このチュートリアルでは、特定の点のセットの凸包を見つけるプログラムについて説明します。

凸包は、図形の内側の境界上に指定されたすべての点を含む最小の多角形の凸包です。

このプログラムでは、ブルートフォースを使用して、指定されたポイントをより小さなセグメントに分割し、最後に後続のポイントをマージして凸包を構築します。

例

#include<bits/stdc++.h>
using namespace std;
//storing the center point of polygon
pair<int, int> mid;
//calculating the quadrant of
//a particular point
int quad(pair<int, int> p){
   if (p.first >= 0 && p.second >= 0)
   return 1;
   if (p.first <= 0 && p.second >= 0)
   return 2;
   if (p.first <= 0 && p.second <= 0)
   return 3;
   return 4;
}
//if line is touching the polygon
int calc_line(pair<int, int> a, pair<int, int> b,
pair<int, int> c){
   int res = (b.second-a.second)*(c.first-b.first) -
   (c.second-b.second)*(b.first-a.first);
   if (res == 0)
   return 0;
   if (res > 0)
   return 1;
   return -1;
}
//comparing functions for sorting
bool compare(pair<int, int> p1, pair<int, int> q1){
   pair<int, int> p = make_pair(p1.first - mid.first,
   p1.second - mid.second);
   pair<int, int> q = make_pair(q1.first - mid.first,
   q1.second - mid.second);
   int one = quad(p);
   int two = quad(q);
   if (one != two)
   return (one < two);
   return (p.second*q.first < q.second*p.first);
}
//finding the upper tangent for both polygons
vector<pair<int, int>> merger(vector<pair<int, int> > a,
vector<pair<int, int> > b){
   int n1 = a.size(), n2 = b.size();
   int ia = 0, ib = 0;
   for (int i=1; i<n1; i++)
   if (a[i].first > a[ia].first)
   ia = i;
   //calculating leftmost point of b
   for (int i=1; i<n2; i++)
   if (b[i].first < b[ib].first)
   ib=i;
   int inda = ia, indb = ib;
   bool done = 0;
   while (!done){
      done = 1;
      while (calc_line(b[indb], a[inda], a[(inda+1)%n1]) >=0)
      inda = (inda + 1) % n1;
      while (calc_line(a[inda], b[indb], b[(n2+indb-1)%n2]) <=0){
         indb = (n2+indb-1)%n2;
         done = 0;
      }
   }
   int uppera = inda, upperb = indb;
   inda = ia, indb=ib;
   done = 0;
   int g = 0;
   //calculating the lower tangent
   while (!done){
      done = 1;
      while (calc_line(a[inda], b[indb], b[(indb+1)%n2])>=0)
      indb=(indb+1)%n2;
      while (calc_line(b[indb], a[inda], a[(n1+inda-1)%n1])<=0){
      inda=(n1+inda-1)%n1;
      done=0;
   }
}
int lowera = inda, lowerb = indb;
vector<pair<int, int>> ret;
//merging the two polygons to get convex hull
int ind = uppera;
ret.push_back(a[uppera]);
while (ind != lowera){
   ind = (ind+1)%n1;
   ret.push_back(a[ind]);
}
ind = lowerb;
ret.push_back(b[lowerb]);
while (ind != upperb){
   ind = (ind+1)%n2;
   ret.push_back(b[ind]);
}
return ret;
}
//brute force algo to find convex hull
vector<pair<int, int>> bruteHull(vector<pair<int, int>> a){
   set<pair<int, int> >s;
   for (int i=0; i<a.size(); i++){
      for (int j=i+1; j<a.size(); j++){
         int x1 = a[i].first, x2 = a[j].first;
         int y1 = a[i].second, y2 = a[j].second;
         int a1 = y1-y2;
         int b1 = x2-x1;
         int c1 = x1*y2-y1*x2;
         int pos = 0, neg = 0;
         for (int k=0; k<a.size(); k++){
            if (a1*a[k].first+b1*a[k].second+c1 <= 0)
            neg++;
            if (a1*a[k].first+b1*a[k].second+c1 >= 0)
            pos++;
         }
         if (pos == a.size() || neg == a.size()){
            s.insert(a[i]);
            s.insert(a[j]);
         }
      }
   }
   vector<pair<int, int>>ret;
   for (auto e:s)
   ret.push_back(e);
   //moving through anti clockwise direction
   mid = {0, 0};
   int n = ret.size();
   for (int i=0; i<n; i++){
      mid.first += ret[i].first;
      mid.second += ret[i].second;
      ret[i].first *= n;
      ret[i].second *= n;
   }
   sort(ret.begin(), ret.end(), compare);
   for (int i=0; i<n; i++)
   ret[i] = make_pair(ret[i].first/n, ret[i].second/n);
   return ret;
}
//returning the value of convex hull
vector<pair<int, int>> divide(vector<pair<int, int>> a){
   if (a.size() <= 5)
   return bruteHull(a);
   // left contains the left half points
   // right contains the right half points
   vector<pair<int, int>>left, right;
   for (int i=0; i<a.size()/2; i++)
   left.push_back(a[i]);
   for (int i=a.size()/2; i<a.size(); i++)
   right.push_back(a[i]);
   vector<pair<int, int>>left_hull = divide(left);
   vector<pair<int, int>>right_hull = divide(right);
   //merging the convex hulls
   return merger(left_hull, right_hull);
}
int main(){
   vector<pair<int, int> > a;
   a.push_back(make_pair(0, 0));
   a.push_back(make_pair(1, -4));
   a.push_back(make_pair(-1, -5));
   a.push_back(make_pair(-5, -3));
   a.push_back(make_pair(-3, -1));
   a.push_back(make_pair(-1, -3));
   a.push_back(make_pair(-2, -2));
   a.push_back(make_pair(-1, -1));
   a.push_back(make_pair(-2, -1));
   a.push_back(make_pair(-1, 1));
   int n = a.size();
   sort(a.begin(), a.end());
   vector<pair<int, int> >ans = divide(a);
   cout << "Convex Hull:\n";
   for (auto e:ans)
   cout << e.first << " "<< e.second << endl;
   return 0;
}

出力

Convex Hull:
-5 -3
-1 -5
1 -4
0 0
-1 1