The purpose of the Travelling Salesperson^{1} Problem (TSP) assignment is to practice implementing linked lists. The specific goals are to:
A travelling salesperson 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 salesperson 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 salesperson problem. You will also implement a simpler method which creates a tour by traveling to points in order.
1,000 points | Optimal tour through the same 1,000 points |
The travelling salesperson 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.
In this assignment, you will write a Tour
class that models a tour using a linked list of Point
objects.
You will implement the following methods to insert points into a tour.
Open HW7 on Codio and look at the files provided, or you can download them here. The provided files contain files you can use for testing as well as some helper classes and interfaces you will need:
TourInterface.java
outlines the set of methods that your Tour class will have to implement.Point.java
is the class for point objects. Each node in the linked list (Tour) will hold a point object. Point objects are further explained in the next section.Node.java
represents nodes in your Tour
’s linked list and is also described in the next section.VisualizeTour.java
is a program that helps 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 which will appear when you run it.Review the class material and textbook chapters on linked lists.
In this section, you will write Tour
, implementing TourInterface
. The Tour is represented by a linked list which contains nodes. Each node contains a point which is essentially just a coordinate. Make sure you understand the distinction between the Tour, nodes, and points after reading section 1.
Point
ClassThe Point
class file that represents a point, contained by a node, in a tour. Open it in Codio and study it carefully. The API is as follows:
public class Point
----------------------------------------------------------------------------------------
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
Tour
ClassCreate 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 for methods with non-void return types must each return a dummy value so that Tour.java
compiles.
Add appropriate header comments and method comments.
Node
ClassThe Node
class we have provided will form the basis of your Tour
class’s linked list structure. Each Node
stores a single Point
in your Tour
’s path and a reference to the next Node
in the path. Its API is as follows:
public class Node
--------------------------------------------------------------------------------------
Node(Point p) // create a Node containing Point p
Node(Node n, Point p) // create a Node containing Point p and
// with n as its next Node in the list
Declare (do not initialize yet) the following private fields in your Tour class:
Node
named head
. This will represent the first Node
in your Tour
.Node
named lastNode
. This will represent the last Node
in your Tour
. Whenever you append (add) a Node
to the end of your linked list, it will be placed just before this Node
. Essentially the lastNode
never changes once it is initialized, but the node in the penultimate position of the tour can change.When your Tour
class’s linked list is not empty, both head
and lastNode
must be different instances of the Node
class (two different objects) yet each Node must store the same Point
object. That is, there will be two different Nodes in memory each of which contains a reference to one common Point in memory. Note that this is different from having each Node
refer to distinct Point
objects with equivalent values. The purpose of this is to represent the cyclical nature of the salesperson’s route: it begins and ends at the salesperson’s home. This is a required implementation detail.
You may not use Java’s built-in LinkedList
class to implement your linked list. You should also not write your own class called LinkedList
since your Tour
class will handle all linked list functionality.
toString()
Implement a single constructor for your Tour
class that takes no arguments and creates an empty Tour
. This means that both head
and lastNode
will be null.
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
. Your output must match this description exactly in order to pass the autograder tests.
If the Tour
is empty (has no Nodes), toString()
should return the empty String.
Required Testing: Add a main
in which you create an empty tour and print it out. Your program should now simply print a blank line.
insertInOrder()
To facilitate testing, you will need to implement insertInOrder()
so you can add Nodes to your tour.
insertInOrder(Point p)
adds a node storing Point p
as the “last” node of the Tour
.
Remember that your Tour
class should always maintain lastNode
at the end of the linked list referring to the same Point
as the first node in the tour.
If the Tour
is initially empty, make sure that after this method finishes, your linked list contains two Node
objects, both referring to the same Point
.
If you need to iterate over your linked list without specifically doing anything to it, you can do the following:
Node curr = head;
while (curr.next != null) {
curr = curr.next
}
Required Testing: Add code to main
to create the following four points and add them to your tour using insertInOrder()
:
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)
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 Point
s in the Tour
, (counting the point in Head and lastNode only once).
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 entire Tour
from Point
to Point
using functions that call PennDraw
code. Both edges adjacent to the input Point p
should be 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
. As we have provided VisualizeTour
to handle setting up PennDraw’s canvas for drawing your Tour
, all this function needs to do is call the drawTo
method to draw every line segment between adjacent Points
. If an empty Tour
calls the draw
method, the method should simply return without drawing anything.
The image below shows what our reference draws when we call tour.draw(a)
(refer to section E for the value of a
). tour
is the four-point Tour
we 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.
As you debug your code, you may find this Java execution visualizer helpful. (It was created by daveagp.)
VisualizeTour
The VisualizeTour
program included provides a user interface for you to test the methods you have written in Tour
. Run it with a filename argument (one of the files we provide) in the terminal 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 produce.
You might find it helpful to write a helper function that, given a point, inserts a new Node after a given Node
.
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 the reference outputs at the bottom of the page. Do not continue until insertInOrder
works for all these cases!
insertNearest()
insertNearest(Point p)
adds a Node storing the Point p
to the Tour
after the closest Point
(Node) already in the Tour
.
If there are multiple closest Point
s 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.
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.
insertSmallest()
insertSmallest(Point p)
adds a Node storing 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 Point
s s
and t
is the sum of the distances from s
to p
and from p
to t
, minus 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 Point
s. 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.
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. Note that for the files containing large quantities of points, such as mona-50k.txt, your program may take a long time to build the tour. You may have to wait for several moments, staring at a blank white PennDraw canvas, before your tour is visualized.
File | In-Order Insertion('o' ) |
Nearest-Neighbor Heuristic('n' ) |
Smallest-Increase Heuristic('s' ) |
---|---|---|---|
tsp0.txt |
Size: 0 Distance: 0.0000 |
Size: 0 Distance: 0.0000 |
Size: 0 Distance: 0.0000 |
tsp1.txt |
Size: 1 Distance: 0.0000 |
Size: 1 Distance: 0.0000 |
Size: 1 Distance: 0.0000 |
tsp2.txt |
Size: 2 Distance: 632.46 |
Size: 2 Distance: 632.46 |
Size: 2 Distance: 632.46 |
tsp3.txt |
Size: 3 Distance: 832.46 |
Size: 3 Distance: 832.46 |
Size: 3 Distance: 832.46 |
tsp4.txt |
Size: 4 Distance: 963.44 |
Size: 4 Distance: 956.06 |
Size: 4 Distance: 839.83 |
tsp5.txt |
Size: 5 Distance: 2595.1 |
Size: 5 Distance: 2595.1 |
Size: 5 Distance: 1872.8 |
tsp8.txt |
Size: 8 Distance: 3898.9 |
Size: 8 Distance: 3378.8 |
Size: 8 Distance: 2545.6 |
tsp10.txt |
Size: 10 Distance: 2586.7 |
Size: 10 Distance: 1566.1 |
Size: 10 Distance: 1655.7 |
tsp100.txt |
Size: 100 Distance: 25547 |
Size: 100 Distance: 7389.9 |
Size: 100 Distance: 4887.2 |
tsp1000.txt |
Size: 1000 Distance: 3.2769e+05 |
Size: 1000 Distance: 27869 |
Size: 1000 Distance: 17266 |
bier127.txt |
Size: 127 Distance: 21743 |
Size: 127 Distance: 6494.0 |
Size: 127 Distance: 4536.8 |
circuit1290.txt |
Size: 1290 Distance: 4.303e+05 |
Size: 1290 Distance: 25030 |
Size: 1290 Distance: 14596 |
germany15112.txt |
Size: 15112 Distance: 4.2116e+06 |
Size: 15112 Distance: 93119 |
Size: 15112 Distance: 55754 |
mona-20k.txt |
Size: 20000 Distance: 4.9650e+06 |
Size: 20000 Distance: 94894 |
Size: 20000 Distance: 56334 |
mona-50k.txt |
Size: 50000 Distance: 1.2366e+07 |
Size: 50000 Distance: 1.6168e+05 |
Size: 50000 Distance: 95598 |
mona-100k.txt |
Size: 100001 Distance: 2.4795e+07 |
Size: 100001 Distance. 2.6272e+05 |
Size: 100001 Distance: 1.5472e+05 |
usa13509.txt |
Size: 13509 Distance: 3.9108e+06 |
Size: 13509 Distance: 77450 |
Size: 13509 Distance: 45075 |
For extra credit, implement a better heuristic in a class TourEC
that implements the TourECInterface
interface. You are not required to use the Tour
or Point
classes for your extra credit solution. If you use a modified version of these classes to implement TourEC
, include them in your extra.zip; otherwise, your TA may be unable to compile your code.
Be warned that this is a relatively difficult extra credit, although it gives an opportunity to learn a great deal about an extremely important problem. Try to write a TourEC
that implements one of the heuristics below.
Here are some heuristics you may choose to implement.
Farthest insertion The farthest insertion heuristic is just like the smallest increase insertion heuristic described in the assignment, except that the Point
s need not be inserted in the same order as the input. Start with a Tour
consisting of the two Point
s that are farthest apart. Repeat the following:
Point
s not in the Tour
, choose the one that is farthest from any Point
already in the Tour
.Point
into the Tour
in the position where it causes the smallest increase in the distance.You will have to store all of the unused Point
s in an appropriate data structure, until they get inserted into the Tour
. If your code takes a long time, your algorithm probably performs approximately n3steps. If you’re careful and clever, this can be improved to n2 steps.
Node interchange local search Run the original greedy heuristic (or any other heuristic). Then, repeat the following:
Point
s.Point
s in if this improves the Tour
. For example if the original greedy heuristic returns 1-5-6-2-3-4-1, you might consider swapping 5 and 3 if the Tour
1-3-6-2-5-4-1 has a smaller distance.Writing a function to swap two nodes in a linked list provides great practice with coding linked lists. Be careful, it can be a little trickier that you might first expect (e.g., make sure your code handles the case when the two Point
s occur consecutively in the original Tour
).
Edge interchange local search Run the original greedy heuristic (or any other heuristic). Then, repeat the following:
Tour
1-3-6-2-5-4-1 has a smaller distance.This requires some care, as you will have to reverse the orientation of the links in the original Tour
between Node
s 3 and 2. After performing this heuristic, there will be no crossing edges in the Tour
, although it need not be optimal.
tsp1000.txt
is a solution of distance 15476.519, which was found using the Concorde TSP solver.Complete readme_tsp.txt
in the same way that you have done for previous assignments.
Submit Tour.java
and readme_tsp.txt
on gradescope.
You may also submit a TourEC.java
file for extra credit. If your TourEC.java
requires any additional files, including a modified Point.java
or Tour.java
, make sure to submit those as well.
If you want to research this problem, it’s historically been called the Travelling Salesman Problem. ↩