Convex Hull Using Divide and Conquer Algorithm in C++

In this tutorial, we will be discussing a program to find the convex hull of a given set of points.

Convex hull is the smallest polygon convex figure containing all the given points either on the boundary on inside the figure.

In this program, we will use brute force to divide the given points into smaller segments and then finally merging the ones that follow on to construct the convex hull.

Example

 Live Demo

#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;
}

Output

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