Write a program to estimate the value of the percolation threshold via Monte Carlo simulation.
Install a Java programming environment. Install a Java programming environment on your computer by following these step-by-step instructions for your operating system [ Mac OS X · Windows · Linux ]. After following these instructions you will have stdlib.jar and algs4.jar in your Java classpath: the former contains libraries for reading data from standard input, writing data to standard output, drawing results to standard draw, generating random numbers, computing statistics, and timing programs; the latter contains all of the algorithms in the textbook.
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.)
The 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, 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 with the following API.
public class Percolation {
public Percolation(int N) // create N-by-N grid, with all sites blocked
public void open(int i, int j) // open site (row i, col j) if it is not already
public boolean isOpen(int i, int j) // is site (row i, col j) open?
public boolean isFull(int i, int j) // is site (row i, col j) full?
public boolean percolates() // does the system percolate?
}
Use the union-find data structure (as described and implemented
in lecture and provided below) to efficiently implement the Percolation data type.
By convention, the indices i and j
are integers between 1 and N, where (1, 1) is the upper-left cell:
Throw a java.lang.IndexOutOfBoundsException
if either i or j is outside this range.
Visualization. Once you have implemented the API above, write a client program PercolationVisualizer.java to perform the following visualization:
% more input20.txt 20 7 11 18 11 12 5 9 5 5 9 ... % java PercolationVisualizer < input20.txt
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This visualization serves as a large-scale trace. Knowing that it works properly for a 20-by-20 grid gives us confidence that it works for a 100-by-100 grid.
Monte Carlo simulation. To estimate the percolation threshold, consider the following computational experiment:
By repeating this computation experiment T times and averaging the results, we can obtain a more accurate estimate of the percolation threshold. Let xt be the fraction of open sites in computational experiment t. The sample mean μ provides an estimate of the percolation threshold. The sample standard deviation σ 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:
Write a client program PercolationStats.java that takes two command-line arguments N and T, performs T independent computational experiments (discussed above) on an N-by-N grid, and prints out the mean, standard deviation, and the 95% confidence interval for the percolation threshold. Use standard random from our standard libraries to generate random numbers; use standard statistics to compute the sample mean and standard deviation.
% java PercolationStats 200 100 mean = 0.5929934999999997 stddev = 0.00876990421552567 95% confidence interval = 0.5912745987737567, 0.5947124012262428 % java PercolationStats 200 100 mean = 0.592877 stddev = 0.009990523717073799 95% confidence interval = 0.5909188573514536, 0.5948351426485464 % java PercolationStats 2 10000 mean = 0.666925 stddev = 0.11776536521033558 95% confidence interval = 0.6646167988418774, 0.6692332011581226 % java PercolationStats 2 100000 mean = 0.6669475 stddev = 0.11775205263262094 95% confidence interval = 0.666217665216461, 0.6676773347835391
Analysis of running time and memory usage. Implement Percolation.java using the quick-find algorithm QuickFindUF.java from algs4.jar.
Now, implement Percolation.java using the weighted quick-union algorithm WeightedQuickUnionUF.java from algs4.jar. Answer the questions in the previous paragraph.
Deliverables. Submit Percolation.java (using the weighted quick-union algorithm), PercolationVisualizer.java, and PercolationStats.java. We will supply stdlib.jar and algs4.jar. Also, submit a readme.txt file, answering all questions. You will need to read the COS 226 Collaboration Policy in order to answer the related questions in your readme file.