Homework 8: TSP

0. Background

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A. Goals

The purpose of the Travelling Salesman Problem assignment is to practice implementing linked lists. The specific goals are to:

• implement and use a linked list, a type of recursive data structure
• learn about the Travelling Salesman Problem, an important theoretical problem in computer science

B. Background

A travelling salesman needs to visit each of n cities exactly once, and arrive back home, keeping the total distance travelled as short as possible. In this assignment, you will write a program to find a path connecting n points that passes through each point exactly once.

The travelling salesman problem is a notoriously difficult combinatorial optimization problem. There does not exist an efficient algorithm to find the optimal tour, the tour of smallest distance. The only way to find the optimal tour is to use brute force: to compute the distance of all possible tours. The problem with this is that there are n! (n factorial) possible tours; enumerating them all and computing their distance would be very slow.

However, there are efficient ways to find a tour that is near-optimal; these methods are called heuristics. You will implement two heuristics to find good (but not optimal) solutions to the traveling salesman problem.

1,000 points	optimal tour through the same 1,000 points

The travelling salesman problem has a wealth of applications such as vehicle routing, circuit board drilling, circuit board design, robot control, X-ray crystallography, machine scheduling, and computational biology.

C. Your Program

In this assignment, you will write a Tour class that models a tour by a linked list of Points.

You will implement the following methods to insert points into a tour.

• In-order Insertion: Insert each point at the end of the current tour. This is the easiest to implement.
• Nearest-Neighbor Heuristic: Insert each point into the tour after the closest point that is already in the tour.
• Smallest-Increase Heuristic: Insert each point into the tour where it would cause the smallest increase in the tour distance.

D. Designing the Requirements and Interface

Download data.zip and decompress it into your folder for this homework assignment. This zip file contains data files you can use for testing as well as some helper classes and interfaces you will need:

• TourInterface.java is described above.
• Point.java represents points in your tour, and is described in detail in the next tab.
• VisualizeTour.java is a program that help you graphically test your Tour class as you write it. It takes a single command-line argument — the name of the data file to use; directions for using it are displayed in the window when you run it.

Review the class material and textbook chapters on linked lists.

1. Tour and Point Classes

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A. The Point Class

data.zip contains the Point class that represents a point in a tour. Open it in DrJava and study it carefully. The API is as follows:

public class Point
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       Point(double x, double y)    // create the Point (x, y)
String toString()                   // return String representation
  void draw()                       // draw Point using PennDraw
  void drawTo(Point that)           // draw line segment between this Point and that
double distanceTo(Point that)       // return Euclidean distance between this Point and that

B. The Tour Class

Create a skeleton for your Tour class, which must implement TourInterface:

public interface TourInterface
-----------------------------------------------------------------------------------------------------------
String toString()                 // create a String representation of the Tour
  void draw(Point p)              // draw the Tour using PennDraw
                                  // any edge starting or ending at p should be in a distinct color
   int size()                     // number of Points in this Tour
double distance()                 // return the total distance of the Tour
  void insertInOrder(Point p)     // insert p at the end of the Tour
  void insertNearest(Point p)     // insert p using the nearest neighbor heuristic
  void insertSmallest(Point p)    // insert p using the smallest increase heuristic

Write method stubs for each method declaration in the TourInterface interface. The stubs must each return a dummy value if necessary so that Tour.java compiles.

Add appropriate header comments and method comments.

C. Linked List Node Class

Your Tour class must use a linked list using a private inner class Node. Each Node holds a reference to a Point.

You can create an inner class by declaring a class inside another. For instance, one can write

class OuterClass {
    // ...
    class InnerClass {
        // ...
    }
}

Having an inner classes is useful when the class is only used in a certain context, leading to more encapsulation and greater code maintainability.

Just like fields, inner classes can be declared as public or private. private inner classes can only be used within the surrounding class. Inner classes are able to access the fields of its surrounding class. However, because inner class instances must be associated with an instance of the outer class, inner classes cannot declare static members, and they cannot be instantiated independently of the instance of the outer class.

Declare a private field head in your Tour class to hold the first Node in the Tour.

Your linked list should contain two Nodes that contain the first Point of the Tour – one of these Nodes at the head, and the other Node at the end. Both of these two Nodes must refer to the same Point object (as opposed to them referring to distinct Point object instances with the same field values). This is a required implementation detail.

You may not use Java's built-in LinkedList class to implement your linked list.

D. Constructor and toString()

Implement a single constructor for your Tour class that takes no arguments and creates an empty Tour.

toString() returns a String representation of the Tour (the first Point should show up at the end as well, just like it does in your linked list structure). Call toString() on each Point to get a String representation of the Point. Add a line break character, '\n', after each Point. Your output must match this description exactly in order to pass our grading scripts.

If the Tour is empty (has no Points), toString() should return the empty String.

Required Testing: Add a main that creates an empty tour and prints it out. Your program should now simply print a blank line. Once this works, submit and make sure you also pass the empty tour submission test.

E. insertInOrder()

To facilitate testing, you will need to implement insertInOrder() so you can add Points to your tour.

insertInOrder(Point p) adds the Point p as the last Point of the Tour.

Remember that your Tour class should maintain one Node at the end of the linked list referring to the first Point.

If the Tour is empty, make sure that after this method finishes, your linked list contains two Node objects, both referring to the same Point.

If you wish, you may write a helper function that adds a given Point after a given Node. (Helper functions should be declared private.)

Required Testing: Add code to main to create the following four points and add them to your tour using insertInOrder():

• a = (0, 0)
• b = (1, 0)
• c = (1, 1)
• d = (0, 1)

The following image shows the structure of the link list these insertions should create:

Print out the tour using System.out.println You should see the following output (including a blank line at the end):

(0.0, 0.0)
(1.0, 0.0)
(1.0, 1.0)
(0.0, 1.0)
(0.0, 0.0)

Once this test passes, submit your code and make sure it passes our toString() tests for insertInOrder() as well. (At this point, your code will still fail the tests for size() and distance().)

F. Utility Methods

Implement the size(), distance(), and draw() methods of Tour. There are many good ways to implement these methods, using for loops, while loops, or recursion. The choice is up to you.

size() returns the number of Points in the Tour, (without including the first Point twice).

distance() returns the total length of the Tour from Point to Point. Use the distanceTo(Point p) method of a Point to find its distance to p. An empty Tour has a distance of 0.0.

draw(Point p) draws the Tour from Point to Point using PennDraw. Both edges adjacent to Point p are drawn in a different color (if p is null, none of the edges should be in a different color). Use the drawTo(Point q) method of a Point to draw a line from it to q.

The image below shows what our reference draws when we call tour.draw(a). tour is the four-point tour you created for testing.

Required Testing: Add code to main to test each of these methods on an empty tour, a tour containing only one point, a tour containing two points, and a tour containing four points. We encourage you to include additional tests as well. When you are certain all of your own tests work, submit and make sure our tests pass as well.

As you debug your code, you may find this Java execution visualizer helpful. (It was created by daveagp.)

2. Tour Insertion Heuristics

Implement the insertNearest() and insertSmallest() methods.

A. Testing with VisualizeTour

The client VisualizeTour program included in data.zip provides a user interface for you to test the methods you have written in Tour. Run it with a filename argument to animate the construction of your Tour. In the table at the bottom of this page, we have listed the values of size() and distance() that your methods should obtain for each insert method, as well as the PennDraw output that draw() should give.

Required Testing: Check that your in-order insertion method works for at least the input files tsp0.txt, tsp1.txt, tsp2.txt, tsp3.txt, tsp4.txt, tsp5.txt, tsp8.txt, tsp10.txt, and tsp100.txt. Both the drawing itself, and the size and distance, need to match. Do not continue until insertInOrder works for all these cases!

B. insertNearest()

insertNearest(Point p) adds the Point p to the Tour after the closest Point already in the Tour.

If there are multiple closest Points with equal distances to p, insert p after the first such Point in the linked list.

Your method must behave as insertInOrder() does when the linked list is empty.

If you wrote a helper function in the previous section that inserts a given Point after a given Node, you may find it useful again here.

Required Testing: Make sure your VisualizeTour results match the figures below for the Nearest-Neighbor Heuristic for all test cases through tsp100.txt. Both the drawing itself, and the size and distance, need to match. Then submit and make sure it passes our submission tests as well.

C. insertSmallest()

insertSmallest(Point p) adds the Point p to the Tour in the position where it would cause the smallest increase in the Tour's distance.

Do not compute the entire Tour distance for each position of p. Instead, compute the incremental distance: the change in distance from adding p between Points s and t is the sum of the distances from s to p and from p to t, less the original distance from s to t.

If there are multiple positions for p that cause the same minimal increase in distance, insert p in the first such position.

Your method must behave as insertInOrder() does when the linked list is empty.

If you wrote a helper function when writing insertInOrder() that inserts a given Point after a given Node, you may find it useful again here.

Comment out all print statements in Tour before running VisualizeTour on a file of more than 100 Points. Otherwise, you will be waiting for a long time for VisualizeTour to finish.

Required Testing: Make sure your VisualizeTour results match the figures below for all test cases through tsp100.txt. Both the drawing itself, and the size and distance, need to match. Then submit and make sure it passes our submission tests as well.

D. Reference Output

Test your nearest-neighbor heuristic and smallest-increase heuristic methods using VisualizeTour. The following are the values and PennDraw output that your Tour methods should give for each of the provided input files.