001
002
003
004
005
006
007
008
009
010
011
012
013
014
015
016
017
018
019
020
021
022
023
024
025
026
027
028
029
030
031
032
033
034
035
036
037
038
039
040
041
042
043
044
045
046
047
048
049
050
051
052
053
054
055
056
057
058
059
060
061
062
063
064
065
066
067
068
069
070
071
072
073
074
075
076
077
078
079
080
081
082
083
084
085
086
087
088
089
090
091
092
093
094
095
096
097
098
099
100
101
102
103
104
105
106
107
108
package algs9; // section 9.8
import stdlib.*;
import java.util.Arrays;
import algs12.Point2D;
import algs13.Stack;
/* ***********************************************************************
 *  Compilation:  javac GrahamaScanNondegenerate.java
 *  Execution:    java GrahamNondegenerate < input.txt
 *  Dependencies: Point2D.java Stack.java
 *
 *  Read points from standard input and compute their convex hull
 *  using Graham's algorithm.
 *
 *  Returns the extreme points of the convex hull in counterclockwise
 *  order (starting with the point with smallest y-coordinate).
 *
 *  Non-degeneracy assumption
 *  -------------------------
 *   -  at least 3 points
 *   -  no coincident points
 *   -  no 3 collinear points
 *
 *  GrahamScan.java removes these degeneracy assumptions.
 *
 *************************************************************************/

public class XGrahamScanNondegenerate {
  private final Stack<Point2D> hull = new Stack<>();

  public XGrahamScanNondegenerate(Point2D[] points) {
    // defensive copy
    int N = points.length;
    if (N <= 2) throw new Error("Requires at least 3 points");
    Point2D[] p = new Point2D[N];
    for (int i = 0; i < N; i++)
      p[i] = points[i];

    // preprocess so that p[0] has lowest y-coordinate; break ties by x-coordinate
    // p[0] is an extreme point of the convex hull
    // (could do easily in linear time)
    Arrays.sort(p, Point2D.Y_ORDER);

    // sort by polar angle with respect to base point p[0].
    // (no ties because of general position assumption)
    Arrays.sort(p, 1, N, p[0].POLAR_ORDER);

    // p[0] and p[1] are extreme points (p[1] because of general position)
    hull.push(p[0]);
    hull.push(p[1]);

    // Graham scan
    for (int i = 2; i < N; i++) {
      Point2D top = hull.pop();
      // could replace >= with > since no three collinear
      // could replace unnecessary popping/pushing with peekpeek()
      while (Point2D.ccw(hull.peek(), top, p[i]) <= 0) {
        top = hull.pop();
      }
      hull.push(top);
      hull.push(p[i]);
    }

    assert isConvex();
  }


  // return extreme points on convex hull in counterclockwise order as an Iterable
  // (no need to reverse if we want to return in clockwise order)
  public Iterable<Point2D> hull() {
    Stack<Point2D> s = new Stack<>();
    for (Point2D p : hull) s.push(p);
    return s;
  }

  // check that boundary of hull is strictly convex
  private boolean isConvex() {
    int N = hull.size();
    Point2D[] points = new Point2D[N];
    int n = 0;
    for (Point2D p : hull()) {
      points[n++] = p;
    }

    // needs to check N = 1 and N = 2 cases if not in general position

    for (int i = 0; i < N; i++) {
      if (Point2D.ccw(points[i], points[(i+1) % N], points[(i+2) % N]) <= 0) {
        return false;
      }
    }
    return true;
  }

  // test client
  public static void main(String[] args) {
    int N = StdIn.readInt();
    Point2D[] points = new Point2D[N];
    for (int i = 0; i < N; i++) {
      int x = StdIn.readInt();
      int y = StdIn.readInt();
      points[i] = new Point2D(x, y);
    }
    XGrahamScanNondegenerate graham = new XGrahamScanNondegenerate(points);
    for (Point2D p : graham.hull())
      StdOut.println(p);
  }

}