Graham Scan Algorithm

The convex hull is the minimum closed area which can cover all given data points.

Graham’s Scan algorithm will find the corner points of the convex hull. In this algorithm, at first, the lowest point is chosen. That point is the starting point of the convex hull. Remaining n-1 vertices are sorted based on the anti-clockwise direction from the start point. If two or more points are forming the same angle, then remove all points of the same angle except the farthest point from start.

From the remaining points, push them into the stack. And remove items from stack one by one, when orientation is not anti-clockwise for stack top point, second top point and newly selected point points[i], after checking, insert points[i] into the stack.

Input and Output

Input:
Set of points: {(-7,8), (-4,6), (2,6), (6,4), (8,6), (7,-2), (4,-6), (8,-7),(0,0), (3,-2),(6,-10),(0,-6),(-9,-5),(-8,-2),(-8,0),(-10,3),(-2,2),(-10,4)}
Output:
Boundary points of convex hull are:
(-9, -5) (-10, 3) (-10, 4) (-7, 8) (8, 6) (8, -7) (6, -10)

Algorithm

findConvexHull(points, n)

Input − The set of points, number of points.

Output − The boundary points of the convex hull.

Begin
   minY := points[0].y
   min := 0

   for i := 1 to n-1 do
      y := points[i].y
      if y < minY or minY = y and points[i].x < points[min].x, then
         minY := points[i].y
         min := i
   done

   swap points[0] and points[min]
   p0 := points[0]
   sort points from points[1] to end
   arrSize := 1

   for i := 1 to n, do
      when i < n-1 and (p0, points[i], points[i+1]) are collinear, do
         i := i + 1
      done
      points[arrSize] := points[i]
      arrSize := arrSize + 1
   done

   if arrSize < 3, then
      return cHullPoints
   push points[0] into stack
   push points[1] into stack
   push points[2] into stack

   for i := 3 to arrSize, do
      while top of stack, item below the top and points[i] is not in
         anticlockwise rotation, do
         delete top element from stack
      done
      push points[i] into stack
   done

   while stack is not empty, do
      item stack top element into cHullPoints
      pop from stack
   done
End

Example

#include<iostream>
#include<stack>
#include<algorithm>
#include<vector>
using namespace std;

struct point {    //define points for 2d plane
   int x, y;
};

point p0;    //used to another two points

point secondTop(stack<point>&stk) {
   point tempPoint = stk.top(); stk.pop();
   point res = stk.top();    //get the second top element
   stk.push(tempPoint);    //push previous top again
   return res;
}

int squaredDist(point p1, point p2) {
   return ((p1.x-p2.x)*(p1.x-p2.x) + (p1.y-p2.y)*(p1.y-p2.y));
}

int direction(point a, point b, point c) {
   int val = (b.y-a.y)*(c.x-b.x)-(b.x-a.x)*(c.y-b.y);
   if (val == 0)
      return 0;     //colinear
   else if(val < 0)
      return 2;    //anti-clockwise direction
      return 1;    //clockwise direction
}

int comp(const void *point1, const void*point2) {
   point *p1 = (point*)point1;
   point *p2 = (point*)point2;
   int dir = direction(p0, *p1, *p2);

   if(dir == 0)
      return (squaredDist(p0, *p2) >= squaredDist(p0, *p1))?-1 : 1;
   return (dir==2)? -1 : 1;  
}

vector<point>findConvexHull(point points[], int n) {
   vector<point> convexHullPoints;
   int minY = points[0].y, min = 0;

   for(int i = 1; i<n; i++) {
      int y = points[i].y;
      //find bottom most or left most point
      if((y < minY) || (minY == y) && points[i].x < points[min].x) {
         minY = points[i].y;
         min = i;  
      }
   }

   swap(points[0], points[min]);    //swap min point to 0th location
   p0 = points[0];
   qsort(&points[1], n-1, sizeof(point), comp);    //sort points from 1 place to end
   
   int arrSize = 1;    //used to locate items in modified array
   for(int i = 1; i<n; i++) {

      //when the angle of ith and (i+1)th elements are same, remove points
      while(i < n-1 && direction(p0, points[i], points[i+1]) == 0)
         i++;
      points[arrSize] = points[i];
      arrSize++;
   }

   if(arrSize < 3)
      return convexHullPoints;    //there must be at least 3 points, return empty list.
         
      //create a stack and add first three points in the stack
      stack<point> stk;
      stk.push(points[0]); stk.push(points[1]); stk.push(points[2]);
   
      for(int i = 3; i<arrSize; i++) {    //for remaining vertices
         while(direction(secondTop(stk), stk.top(), points[i]) != 2)
            stk.pop();    //when top, second top and ith point are not making left turn, remove point
         stk.push(points[i]);
      }

      while(!stk.empty()) {
         convexHullPoints.push_back(stk.top());    //add points from stack
         stk.pop();
      }
}

int main() {
   point points[] = {{-7,8},{-4,6},{2,6},{6,4},{8,6},{7,-2},{4,-6},{8,-7},{0,0},
                     {3,-2},{6,-10},{0,-6},{-9,-5},{-8,-2},{-8,0},{-10,3},{-2,2},{-10,4}};
   int n = 18;
   vector<point> result;
   result = findConvexHull(points, n);
   cout << "Boundary points of convex hull are: "<<endl;
   vector<point>::iterator it;

   for(it = result.begin(); it!=result.end(); it++)
      cout << "(" << it->x << ", " <<it->y <<") ";
}

Output

Boundary points of convex hull are:
(-9, -5) (-10, 3) (-10, 4) (-7, 8) (8, 6) (8, -7) (6, -10)