Project 1 Solution

Description

Problem1. (Percolation.java)

1. Write a program to estimate the value of the percolation threshold via Monte Carlo simulation.

2. Use the .java files attached

3. To access a class in algs4.jar, you need an import statement, such as the ones below:

import edu.princeton.cs.algs4.StdRandom; import edu.princeton.cs.algs4.StdStats;

import edu.princeton.cs.algs4.WeightedQuickUnionUF;

4. To run the files (Windows, Mac), use java-algs4

Percolation. 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.)

CS146 Project 1

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, the system does not percolate; when p equals 1, 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).

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*.

Percolation data type. To model a percolation system, create a data type Percolation with the following API:

public class Percolation {

• create n-by-n grid, with all sites blocked public Percolation(int n)

• open site (row, col) if it is not open already

public	void	open(int row, int col)
public boolean		isOpen(int row, int col)	// is site (row, col) open?
public boolean		isFull(int row, int col)	// is site (row, col) full?
public	int	numberOfOpenSites()	// number of open sites
public boolean		percolates()	// does the system percolate?
public static void main(String[] args)			// test client (optional)

}

CS146 Project 1

Throw a java.lang.IllegalArgumentException 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 n2; all methods should take constant time plus a constant number of calls to the union–find methods union(),

find(), connected(), and count().

$java-algs4 Percolation < input10.txt

56 open sites

percolates

$java-algs4 Percolation < input10-no.txt 55 open sites

does not percolate

Problem2. (PercolationStats.java)

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:

• 15. Choose a site uniformly at random among all blocked sites. o 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 lattice 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.

CS146 Project 1

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

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

To perform a series of computational experiments, create a data type PercolationStats with the following API.

public class PercolationStats {

• perform trials independent experiments on an n-by-n grid public PercolationStats(int n, int trials)

• sample mean of percolation threshold public double mean()

• sample standard deviation of percolation threshold public double stddev()

public double confidenceLo() // low endpoint of 95% confidence interval public double confidenceHi() // high endpoint of 95% confidence interval

public static void main(String[] args) // test client (described below)

}

The constructor should throw a java.lang.IllegalArgumentException if either n ≤ 0 or trials ≤ 0.

Also, the main() method that takes two command-line arguments n and T, performs T independent computational experiments (discussed above) on an n-by-n grid, and prints the sample mean, sample standard deviation, and the 95% confidence interval for the percolation threshold.

Use StdRandom to generate random numbers; use StdStats to compute the sample mean and sample standard deviation.

Corner cases. The constructor should throw a java.lang.IllegalArgumentException if either N ≤ 0 or T ≤ 0.

CS146 Project 1

• java-algs4 PercolationStats 200 100

mean	=	0.5929934999999997
stddev	=	0.00876990421552567
95% confidence interval =		[0.5912745987737567, 0.5947124012262428]

• java-algs4 PercolationStats 200 100

mean	= 0.592877
stddev	=	0.009990523717073799
95% confidence interval =		[0.5909188573514536, 0.5948351426485464]

• java-algs4 PercolationStats 2 10000

mean	=	0.666925
stddev	= 0.11776536521033558
95% confidence interval =		[0.6646167988418774, 0.6692332011581226]

Acknowledgements This project is an adaptation of the Percolation assignment developed at Princeton University by Robert Sedgewick and Kevin Wayne.

Submitting Information:

• Use the code I provided for each problem. DON’T DELETE ANY FUNCTION

• You should write your code in (Percolation.java, PercolationStats.java)

• Submit your work on Canvas.

• The deadline is Thursday, Sep 25th at 6:00PM

• Follow the guidelines in homework rubric P1