Graham Scan Algorithm
The Graham Scan Algorithm is a popular and efficient method used in computational geometry to determine the convex hull of a finite set of points on a plane. The convex hull is the smallest convex polygon that contains all the given points, and can be thought of as a rubber band stretched around the points, tightly enclosing them. The algorithm was proposed by Ronald Graham in 1972 and has been widely used in several applications, such as pattern recognition, image processing, and geographical information systems.
The Graham Scan Algorithm works by first selecting the point with the lowest y-coordinate as the anchor point to start the scan. If there are multiple points with the same y-coordinate, the one with the lowest x-coordinate is chosen. The rest of the points are then sorted based on their polar angle with respect to the anchor point, in counter-clockwise order. The algorithm proceeds by iterating through the sorted points, and for each point, it checks whether moving from the two previously considered points to the current point constitutes a left or right turn. If it's a right turn, the middle point is removed from the convex hull, as it is considered an internal point. This process continues until all points have been examined, and the remaining points form the vertices of the convex hull. The time complexity of the algorithm is O(n log n), primarily due to the sorting step, making it one of the most efficient algorithms for solving the convex hull problem.
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package com.algorithmexamples.geometry.convexhull
import com.algorithmexamples.datastructures.Stack
import com.algorithmexamples.geometry.Point
class GrahamScan: ConvexHullAlgorithm {
override fun convexHull(points: Array<Point>): Collection<Point> {
if (points.size < 3) throw IllegalArgumentException("there must be at least 3 points")
val hull = Stack<Point>()
// Find the leftmost point
points.sortBy { it.y }
// Sort points by polar angle with p
points.sortWith( Comparator { q1, q2 ->
val dx1 = q1.x - points[0].x
val dy1 = q1.y - points[0].y
val dx2 = q2.x - points[0].x
val dy2 = q2.y - points[0].y
if (dy1 >= 0 && dy2 < 0)
-1 // q1 above; q2 below
else if (dy2 >= 0 && dy1 < 0)
+1 // q1 below; q2 above
else if (dy1 == 0 && dy2 == 0) { // 3-collinear and horizontal
if (dx1 >= 0 && dx2 < 0)
-1
else if (dx2 >= 0 && dx1 < 0)
+1
else
0
} else
-Point.orientation(points[0], q1, q2) // both above or below
// Note: ccw() recomputes dx1, dy1, dx2, and dy2
})
hull.push(points[0])
hull.push(points[1])
for (i in IntRange(2, points.size - 1)) {
var top = hull.poll()
while (Point.orientation(hull.peek(), top, points[i]) <= 0) {
top = hull.poll()
}
hull.push(top)
hull.push(points[i])
}
return hull
}
}
