Percolation

Princeton Algorithm Programming Assignment 1 Write a program to estimate the value of the percolationthreshold via Monte Carlo simulation.

Percolation. Given a composite systems comprised ofrandomly distributed insulating and metallic materials: whatfraction of the materials need to be metallic so that the compositesystem is an electrical conductor? Given a porous landscape withwater on the surface (or oil below), under what conditions will thewater be able to drain through to the bottom (or the oil to gushthrough to the surface)? Scientists have defined an abstractprocess known aspercolation to model such situations.

The model. We model a percolation system using anN-by-N grid ofsites. Each site iseitheropen or blocked. A full site isan open site that can be connected to an open site in the top rowvia a chain of neighboring (left, right, up, down) open sites. Wesay the systempercolates if there is a full site in thebottom row. In other words, a system percolates if we fill all opensites connected to the top row and that process fills some opensite on the bottom row. (For the insulating/metallic materialsexample, the open sites correspond to metallic materials, so that asystem that percolates has a metallic path from top to bottom, withfull sites conducting. For the porous substance example, the opensites correspond to empty space through which water might flow, sothat a system that percolates lets water fill open sites, flowingfrom 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 Percolation with the following API:

public class Percolation {
   public Percolation(int n)                // create n-by-n grid, with all sites blocked
   public    void open(int row, int col)    // open site (row, col) if it is not open already
   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)
}

Corner cases. By convention, the row and column indices are integers between 1 and n, where (1, 1) is the upper-left site: 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().

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 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.

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 {
   public PercolationStats(int n, int trials)    // perform trials independent experiments on an n-by-n grid
   public double mean()                          // sample mean of percolation threshold
   public double stddev()                        // sample standard deviation of percolation threshold
   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, include a 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.

% 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]

% java-algs4 PercolationStats 2 100000
mean                    = 0.6669475
stddev                  = 0.11775205263262094
95% confidence interval = [0.666217665216461, 0.6676773347835391]

Analysis of running time and memory usage (optional and not graded). 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? Give a formula (using tilde notation) of the total running time on your computer (in seconds) as a single function of both n and T.

Using the 64-bit memory-cost model from lecture, give the total memory usage in bytes (using tilde notation) that a Percolation object uses to model an n-by-n percolation system. Count all memory that is used, including memory for the union–find data structure.

Now, implement the Percolation data type using the weighted quick union algorithm in WeightedQuickUnionUF. Answer the questions in the previous paragraph.

Deliverables. Submit only Percolation.java (using the weighted quick-union algorithm from WeightedQuickUnionUF) and PercolationStats.java. We will supply algs4.jar. Your submission may not call library functions except those in StdIn, StdOut, StdRandom, StdStats, WeightedQuickUnionUF, and java.lang.

For fun. Create your own percolation input file and share it in the discussion forums. For some inspiration, do an image search for "nonogram puzzles solved."

This assignment was developed by Bob Sedgewick and Kevin Wayne. Copyright © 2008.