Mini-Project 3: Percolation

Lectures needed for this project:

• Lecture 13 (Disjoint Sets), or Chapter 14 of the textbook.

Partner policy: No partners. Discussing ideas with other students is allowed, but code sharing is not allowed, and the solutions you submit should be your own work! More details on the policies page.

In this assignment, we’ll use what we’ve learned about disjoint sets to model percolation.

By the end of this assignment, you will…

• Gain an understanding of applying the disjoint sets data structure to a real-world situation.

Setup

Follow the Assignment Workflow Guide to get started with this assignment. This starter code is in the proj2 folder.

On this project, you are limited to submitting to Gradescope 4 times every 24 hours. In other words, you have 4 submission tokens, each with a refresh rate of 24 hours.

Unlike previous assignments, not all tests will be provided locally, so it is up to you to write tests to verify the correctness of your own code.

You must follow the style guide for Percolation.java, and you can check style yourself in IntelliJ as many times as you want. (The style checker is turned off for any tests you write.)

Task 1: Percolation

In this program, we’ll write a program to estimate the value of the percolation threshold via Monte Carlo simulation.

Intro Video

An introductory video for this project can be found at this link. It is broken into three parts: Intro, Implementation Spoilers, and Optimization Spoilers. Feel free to ignore these spoilers for a more difficult challenge. If you’d prefer to watch a eight year old video that Professor Hug made when he was at Princeton, see this link. Note that these videos mention implementing PercolationStats.java; we’ve now given you the full solution for this class, so you don’t need to worry about this step.

Slides

Slides for this project can be found here. Because this is a mini-project, we give spoilers for how to approach it. You may ignore them if you’d like a bigger challenge.

Who cares?

Given a composite systems comprised of randomly distributed insulating and metallic materials: what fraction of the materials need to be metallic so that the composite system is an electrical conductor? Given a porous landscape with water on the surface (or oil below), under what conditions will the water be able to drain through to the bottom (or the oil to gush through to the surface)? Scientists have defined an abstract process known as percolation to model such situations.

The Model

We model a percolation system using an N-by-N grid of sites. Each site is either open or blocked. A full site is an open site that can be connected to an open site in the top row via a chain of neighboring (left, right, up, down) open sites. We say the system percolates if there is a full site in the bottom row. In other words, a system percolates if we fill all open sites connected to the top row and that process fills some open site on the bottom row. (For the insulating/metallic materials example, the open sites correspond to metallic materials, so that a system that percolates has a metallic path from top to bottom, with full sites conducting. For the porous substance example, the open sites correspond to empty space through which water might flow, so that a system that percolates lets water fill open sites, flowing from top to bottom.)

In the diagrams below, you can see that in the system on the left, the water is able to start in a site on the top row and trickle down through empty sites until it reaches an empty site on the bottom row.

Whereas on the right, the water in the site on the top row has no way of trickling down to an open site on the bottom row.

Percolation.java

To model a percolation system, complete the Percolation class with the following API:

public class Percolation {
  public Percolation(int N)                // create N-by-N grid, with all sites initially blocked
  public void open(int row, int col)       // open the site (row, col) if it is not open already
  public boolean isOpen(int row, int col)  // is the site (row, col) open?
  public boolean isFull(int row, int col)  // is the site (row, col) full?
  public int numberOfOpenSites()           // number of open sites
  public boolean percolates()              // does the system percolate?
}

Fill out Percolation.java by implementing all the methods given in the above API.

The Percolation Problem

In a famous scientific problem, researchers are interested in the following question: if sites are independently set to be open with probability $p$ (and therefore blocked with probability $1 − p$), what is the probability that the system percolates? When $p$ equals 0 (no site is open), the system does not percolate; when $p$ equals 1 (all sites are open), the system percolates. The plots below show the site vacancy probability $p$ versus the percolation probability for 20-by-20 random grid (left) and 100-by-100 random grid (right).

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When $N$ is sufficiently large, there is a threshold value $p^*$ such that when $p < p^*$ a random N-by-N grid almost never percolates, and when $p > p^*$, a random N-by-N grid almost always percolates. No mathematical solution for determining the percolation threshold $p^*$ has yet been derived. Your task is to write a computer program to estimate $p^*$.

The WeightedQuickUnionUF class

The WeightedQuickUnionUF class is provided by Princeton’s algs4 library, and has the following API:

public class WeightedQuickUnionUF {
  public WeightedQuickUnionUF(int n)      // creates a UF with n elements
  public int count()                      // number of disjoint sets
  public int find(int p)                  // the root of p's set
  public boolean connected(int p, int q)  // whether p and q are in the same set
  public void union(int p, int q)         // join p and q's sets together
}

Your code must use the WeightedQuickUnionUF class! It has already been implemented for you, so do not reimplement the Union Find ADT.

Tips

IMPORTANT: We’ve released a set of videos that contain some hints for how to approach this homework! Please do watch them; they might save you a lot of time on this assignment.

• Video 1: A Small Hint
• Video 2: A Larger Hint

Corner Cases

By convention, the row and column indices are integers between 0 and N − 1, where (0, 0) is the upper-left site: Throw a java.lang.IndexOutOfBoundsException if any argument to open(), isOpen(), or isFull() is outside its prescribed range. The constructor should throw a java.lang.IllegalArgumentException if N ≤ 0.

Performance Requirements

The constructor should take time proportional to $N^2$; all methods should take constant time plus a constant number of calls to the union-find methods union(), find(), connected(), and count(). Meeting these requirements is somewhat tricky! You might consider creating a solution that simply works, before figuring out a way to make it faster. For tips on meeting the speed requirements, see the video at the beginning of this spec. Your numberOfOpenSites() method must take constant time. Part of the goal of this assignment is to learn how to cast one problem (Percolation) in terms of an already solved problem (Disjoint Sets, a.k.a Union Find).

If you are at this part of the spec and are still unsure of how to start, go through these slides and watch this video.

Testing

You will find InteractivePercolationVisualizer to be quite helpful as a sanity check that your Percolation class is implemented properly. Clicking on cells should open (and fill, if necessary) those sites, if your implementation is correct. There will also be a note for the number of open sites and whether the system percolates.

In the tests folder, we have provided two basic tests for you. You should add more tests to PercolationTest.java to test your code.

We also provide two clients that serve as large-scale visual traces. We highly recommend using them for testing and debugging your Percolation implementation.

Sample Data Files

The inputFiles directory contains some sample files for use with the visualization client, including input20.txt, which we will use below.

See Outputs for the expected output images corresponding to each input file.

File-Based Visualization Client

PercolationPicture.java visualizes by performing the following steps:

• Read the grid size N from the file.
• Create an N-by-N grid of sites (initially all blocked).
• Read in a sequence of sites (row i, column j) to open from the file. After each site is opened, draw full sites in light blue, open sites (that aren’t full) in white, and blocked sites in black using standard draw, with site (0, 0) in the upper left-hand corner.

For the input file, input20.txt, the program should produce output as shown in the following images. The images correspond to the 50, 100, 150, 204, and 250 sites being open, respectively.

You can run the visualizer on this input by passing in the correct arguments to the IntelliJ program’s Run -> Edit Configurations -> + -> Application tab. Here, set the “Main class” to PercolationPicture and set the “Program arguments” to an input file (for example, inputFiles/input20.txt). Finally, hit the “Run” button to begin running the visualizer.

Open PercolationPicture.java and follow the steps above to run the visualizer. Use this tool to help you debug your Percolation.java methods!

Interactive Visualization Client

InteractivePercolationVisualizer.java animates the results of opening sites in a percolation system, using the mouse as input. It takes a command-line integer N that specifies the grid size. As a bonus, it prints out the sequence of sites opened in the same format used by PercolationPicture (described above), so you can use it to prepare interesting files for testing. If you design an interesting data file, feel free to share it on Ed.

Open InteractivePercolationVisualizer.java and follow the same steps as PercolationPicture to run the interactive visualizer. Note that for the interactive version, you do not need to provide a program argument.

Use this tool to help you debug your Percolation.java methods!

Note: If you are using IntelliJ, you can choose which class to run (i.e. choose whose main method you want to run) in Run -> Edit Configurations. For more information, please check the FAQ

Task 2: Backwash

As you start testing Percolation, you might realize that after the system has percolated, PercolationVisualizer colors in light blue all sites connected to open sites on the bottom (in addition to those connected to open sites on the top).

Edit your implementation to prevent backwash from occurring.

While the Percolation API should not officially allow backwash, it requires quite a bit of ingenuity to fix and it leads to only a small deduction if you don’t. If you need a hint, see Tips.

Task 3: Percolation Threshold (Ungraded)

This part of the project will not be graded, but it is highly recommended that you at least read though and think about the following points.

Monte Carlo simulation. To estimate the percolation threshold, consider the following computational experiment:

• Initialize all sites to be blocked.
• Repeat the following until the system percolates:
  • Choose a site uniformly at random among all blocked sites.
  • Open the site.
• The fraction of sites that are opened when the system percolates provides an estimate of the percolation threshold.

For example, if sites are opened in a 20-by-20 grid according to the snapshots below, then our estimate of the percolation threshold is 204/400 = 0.51 because the system percolates when the 204th site is opened. The images correspond to the 50, 100, 150, and 204 sites being open, respectively.

If you’re interested in the math behind the following calculations, click here.

By repeating this computation experiment $T$ times and averaging the results, we obtain a more accurate estimate of the percolation threshold. Let $x_t$ be the fraction of open sites in computational experiment $t$. The sample mean $\mu$ provides an estimate of the percolation threshold; the sample standard deviation $\sigma$ measures the sharpness of the threshold.

$\mu = \frac{x_1 + x_2 + … + x_T}{T}$, $\sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + … + (x_T - \mu) ^2}{T-1}$

Assuming $T$ is sufficiently large (say, at least 30), the following provides a 95% confidence interval for the percolation threshold:

$[\mu - \frac{1.96\sigma}{\sqrt{T}}, \mu + \frac{1.96\sigma}{\sqrt{T}}]$

To perform a series of computational experiments, we’ve given you a PercolationStats data type.

The constructor takes two arguments N and T, and performs T independent computational experiments ( discussed above) on an N-by-N grid. Using this experimental data, it calculates the mean, standard deviation, and the 95% confidence interval for the percolation threshold.

Open PercolationStats.java and take a look at the provided constructor and methods. Then, run the main method, and interpret the results. What do these numbers tell you about the solution to the Percolation Problem?

If you’ve implemented Percolation.java correctly, you should now have a 95% confidence interval for the percolation threshold p.

Task 4: Runtime Analysis (Ungraded)

This part of the project will also not be graded, but it is highly recommended that you at least read though and think about the following points.

Implement the Percolation data type using the quick-find algorithm in QuickFindUF. Use Stopwatch to measure the total running time of PercolationStats for various values of N and T. How does doubling N affect the total running time? How does doubling T affect the total running time?

Now, implement the Percolation data type using the weighted quick-union algorithm in WeightedQuickUnionUF. What are your answers to the same questions from before?

Submission

Once you’ve written local tests and passed them, try submitting to the autograder. You may or may not pass everything.

• If you fail a correctness test, this means that there is a case that your local tests did not cover.
• If you fail any of the timing tests, it means that your implementation does not meet the timing constraints described above.
• The autograder will not run unless you fix all your style errors. Reminder that you can check style in IntelliJ as often as you’d like:
• You will have a token limit of 4 tokens every 24 hours. We will not reinstate tokens for failing to add/commit/push your code, run style, etc.

Scoring:

1. Percolation Functionality (40%)
2. Randomized (40%)
3. Timing (20%)

The score you receive on Gradescope is your final score for this assignment (assuming you followed the collaboration policy).

This assignment originally developed by Kevin Wayne and Bob Sedgewick at Princeton University, with autograder built by Josh Hug for the Princeton Algorithms course.

Outputs

greeting57

heart25

input10-no

input10

input1-no

input1

input20

input2-no

input2

input3

input4

input50

input5

input6

input7

input8-dups

input8-no

input8

jerry47

sedgewick60

snake101

snake13

wayne98