Project 4: Convex Hull (170 pts)
Due at 11:59pm Tuesday, Apr 10
1. Project Overview
In this project, you are asked to implement Graham’s scan and Javis’ march to construct the convex hull of an input set of points in the plane, and compare their execution times. For a description of the algorithm, please refer to the lecture notes “convex-hull.pptx” on Monday Mar 19th.
We make the following two assumptions:
- All the input points have integral coordinates ranging between −50 and 50.
- The input points may have duplicates.
In a), integral coordinates are assumed to avoid issues with floating-point arithmetic. The rectangular range [−50, 50] × [−50, 50] is big enough to contain 10,201 points with integral coordinates. In b), duplicates may appear since the input points will be either generated as pseudo-random points or read from an input file.
The classes PureStack and ArrayBasedStack were from one of the lectures and should not
be modified. Also, no need to modify the classes Segment and Plot.
1.1 Point Class
The Point class implements the Comparable interface. Its compareTo() method compares the
𝑦-coordinates of two points first, and then their 𝑥-coordinates only in case of a tie.
1.2 PointAngleComparator Class
Point comparison can be also done using an object of the PolarAngleComparator class, which you are required to implement. The polar angle is with respect to a reference point stored in the instance variable referencePoint. The compare() method in this class must be implemented using cross and dot products not any trigonometric or square root functions. You need to handle a special situation where multiple points have the same polar angle. Please read the Javadoc for the compare() method carefully.
The class PolarAngleComparator has a second instance variable flag used for breaking a tie between two points which have the same polar angle with respect to referencePoint. When flag == true, the point closer to referencePoint is considered “smaller”; when flag ==
false, the further point is considered “smaller”. The two values of flag will be set for Graham’s scan and Jarvis’ march, respectively.
2. ConvexHull Class
Convex hull construction is implemented by the abstract class ConvexHull, which has two constructors:
public ConvexHull(Point pts) throws IllegalArgumentException public ConvexHull(String inputFileName) throws FileNotFoundException,
The first constructor takes an array pts of points and copy them over to the array points. The array pts may consist of random points, or more precisely, points whose coordinates are pseudo-random numbers within the range [−50, 50] × [−50, 50]. (Random point generation is described in Section 6.)
The second constructor reads points from an input file of integers. Every pair of integers represents the 𝑥 and 𝑦-coordinates of a point. A FileNotFoundException will be thrown if no file with the inputFileName exists, and an InputMismathException will be thrown if the file consists of an odd number of integers. (There is no need to check if the input file contains unneeded characters like letters since they can be taken care of by the hasNextInt() and nextInt() methods of a Scanner object.)
For example, suppose a file points.txt has the following content:
0 0 -3 -9 0 -10
8 4 3 3
7 3 10 5
-7 -10 0 8
There are 34 integers in the file. A constructor call ConvexHull(“points.txt”) will initialize the array points to store 17 points below (aligned with five points per row just for display clarity here):
|(0, 0)||(-3,||-9)||(0, -10)||(8, 4)||(3, 3)|
|(-6, 3)||(-2,||1)||(10, 5)||(-7, -10)||(5, -2)|
|(7, 3)||(10,||5)||(-7, -10)||(0, 8)||(-1, -6)|
(-10, 0) (5, 5)
Note that the points (-7, -10) and (10, 5) each appears twice in the input, and thus their second appearances are duplicates. The 15 distinct points are plotted in Fig.1 using Mathematica.
There is a non-negligible chance that duplicates occur among the input points, whether they are from the input file or randomly generated from the range [−50, 50] × [−50, 50]. For a fair comparison between Graham’s scan and Javis’ march, both assuming their input points to be distinct, all the duplicates should be eliminated before the convex hull construction. This is done by the constructors via calling the method removeDuplicates().
The method removeDuplicates() performs quicksort on all the points by 𝒚–coordinate. After the sorting, equal points will appear next to each other in points. The method creates an object of the class QuickSortPoints to carry out quicksort using a provided Comparator<Point> object, which invokes the compareTo() method in the class Point. After the sorting, identical points will appear together, and duplicates will be easily removed. Distinct points are then saved to the array pointsNoDuplicate with the element at index 0 assigned to lowestPoint.
Fig. 1. Input set of 15 different points.
In the previous example, quicksort produces the following sequence:
|(-7, -10)||(-7, -10)||(0,||-10)||(-3, -9)||(-1,||-6)|
|(5, -2)||(-10, 0)||(0,||0)||(-2, 1)||(-6,||3)|
|(3, 3)||(7, 3)||(8,||4)||(5, 5)||(10,||5)|
|(10, 5)||(0, 8)|
The two (-7, -10)s appear together, so do the two (10, 5)s. After removal of duplicates,
the remaining points are copied over to the array pointsNoDuplicate:
|(-7, -10)||(0,||-10)||(-3, -9)||(-1, -6)||(5,||-2)|
|(-10, 0)||(0,||0)||(-2, 1)||(-6, 3)||(3,||3)|
|(7, 3)||(8,||4)||(5, 5)||(10, 5)||(0,||8)|
The variable lowestPoint is set to (-7, -10).
The array pointsNoDuplicate will be the input of a convex hull algorithm. The class ConvexHull has an abstract method constructHull() which will be implemented by its subclasses to carry out convex hull construction using either Graham’s scan or Jarvis’ march.
public abstract void constructHull();
The vertices of the constructed convex hull will be stored in the array hullVertices in counterclockwise order starting with lowestPoint.
3. Convex Hull Construction
Two algorithms, Graham’s scan and Jarvis’ march, are respectively implemented by the subclasses GrahamScan and JarvisMarch of the abstract class ConvexHull. Both subclasses must handle the special case of one or two points only in the array pointsNoDuplicates, where the corresponding convex hull is the sole point or the segment connecting the two points.
3.1. Graham’s scan
The class GrahamScan sorts all the points in points by polar angle with respect to lowestPoint. Point comparison is done using a Comparator<Point> object generated by the constructor call PolarAngleComparator(lowestPoint, true). The second argument true ensures that points with the same polar angle are ordered in increasing distance. (In case your previous implementation did not work well, this is a chance to make it up. ) The compare() method must be implemented using cross and dot products not any trigonometric or square root functions. (Please read the Javadoc for the compare() method carefully.)
Point sorting above is carried out by quicksort as follows. Create an object of the QuickSortPoint class and have it call the quicksort() method using the PolarAngleComparator object mentioned in the above paragraph. The comparator uses lowestPoint as the reference point. Note that quicksort has the expected running time
𝑂(𝑛 log 𝑛) and the worst-case running time 𝑂(𝑛2 ), which will respectively be the expected and worst-case running times for this implementation of Graham’s scan for convex hull construction.
Sorting is performed within the method setUpScan(). In the array pointsNoDuplicate, (- 1, -6) and (5, -2) have the same polar angle with respect to (-7, -10). That (-1, -6) appears
before (5, -2) is because it is closer to (-7, -10).
Graham’s scan is performed within the method constructHull() on the array pointsNoDuplicate using a private stack vertexStack. As the scan terminates, the vertices of the constructed convex hull are on vertexStack. Pop them out one by one and store them in a new array hullVertices, starting at the highest index. When the stack becomes empty, the elements in the array, in increasing index, are the hull vertices in counterclockwise order.
3.2. Jarvis’ March
This algorithm of the gift wrapping style is implemented in the class JarvisMarch. There are three private instance variables.
private Point highestPoint; private PureStack<Point> leftChain;
private PureStack<Point> rightChain;
The algorithm builds the right chain from lowerestPoint to highestPoint and then the left chain from highestPoint downward to lowestPoint. Two stacks, leftChain and rightChain, are used respectively to store the chains during the construction. The convex hull vertices on the stacks are later merged into the array hullVertices.
The convex hull construction is described with more details in the PowerPoint notes “convex hull.pptx”. At each step, the method
private Point nextVertex(Point v)
determines, given the current vertex v, the next vertex to be the point which has the smallest polar angle with respect to v. In the situation where multiple points attain the smallest polar angle, the one that is the furthest from v is the next vertex. Hence, the comparator used in this construction step should be generated by the call PolarAngleCompartor(v, false)
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