Percolation

Problem Statement

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 is 0, the system does not percolate; when p is 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 programs to:

1. visualize the percolation process
2. estimate p*
3. compare brute force (depth-first search) to union-find for finding connected open sites

Visualizing the Percolation Process

50 open sites

100 open sites

150 open sites

204 open sites

Percolation data type

public interface IPercolate { public abstract void draw(); // Draw the current state of the grid public abstract void step(); // Find a blocked site and mark it as open public abstract boolean percolates(); // Returns true iff open path from to bottom }

Estimating p*

• Initialize all sites to be blocked.

• Repeat the following until the system percolates:

  • Choose a blocked site (row i, column j) uniformly at random among all blocked sites.

  • Open the site (row i, column j).

• The fraction of sites that are opened until the system percolates provides an estimate of the percolation threshold.

To obtain an accurate estimate of the percolation threshold, repeat the experiment T times and average the results. Let x_t be the fraction of open sites in experiment t. The mean μ provides an estimate of the percolation threshold. The standard deviation σ measures the sharpness of the threshold.

mean percolation threshold  = 0.5920965000000004
stddev                      = 0.009811413646870666
95% confidence interval     = [0.5901734629252137, 0.594019537074787]

total time                  = 2.074
mean time per experiment    = 0.02073999999999999
stddev                      = 0.0037646248153036512

Comparing IPercolate Implementations

• How does doubling N affect the running time?
• How does doubling T affect the running time?
• Measure running time (using calls to System.currentTimeMillis) of the 3 versions of your program (DFS, Quick Find, and weighted quick union with path compression).
• Give a formula (using Big-Oh notation) of the running time on your computer (in seconds) as a function of N and T.
• Give a formula (using Big-Oh notation) that describes the amount of memory (in bytes) that your program consumes as a function of N.

Extra Credit

1. Write a program UnionFindVisualizer.java to visualize the average path length from each node to the root using the quick union and weighted quick union with path compression data structures. Organize your program so that it takes N from the user, performs one experiment on an N-by-N grid, and plots the average path length for each of the two data structures. Explain your results.

2. Sedgewick & Wayne: Problem 2.4.17

3. Sedgewick & Wayne: Problem 2.4.18

4. Sedgewick & Wayne: Problem 2.4.19

Grading