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--- gibson:teaching:fall-2014:math445:lab5 [2014/11/12 18:12]
gibson
+++ gibson:teaching:fall-2014:math445:lab5 [2015/03/09 13:39] (current)
gibson
@@ Line 25: / Line 25: @@
 **(a)**  Go to the website [[http://www.mathworks.com/moler/chapters.html|Numerical Computing with Matlab]]
-by Clive Moler and download the “surfer.m” program. Set ''N=100'' and run the command
+by Clive Moler and download the “surfer.m” program. Set ''M=100'' and run the command
-''[U,L] = surfer('http://www.unh.edu',N);''. This might take a few minutes.
+''[U,L] = surfer('http://www.unh.edu',M);''. This might take a few minutes.
 **(b)**  After some time a vector ''U'' of websites will be generated and a matrix ''L'' of links will be generated. All the entries of ''L'' should be either 0 or 1. But ''L'' will have 100^2 == 10,000 entries, so you can't check this by eye. Write a short piece of Matlab code that double-checks that all entries of ''L'' are either 0 or 1.
@@ Line 37: / Line 37: @@
 ''G = (1-alpha) T + alpha **1**/M;''
-where **1** is a matrix of ones. Rumor has it that Google uses alpha = 0.15, so use that value and calculate ''T''.
+where **1** is a matrix of ones. Rumor has it that Google uses alpha = 0.15, so use that value and calculate ''G''.
-**(e)** Double-check that the sum of each column of ''T'' is 1. Again, be clever and get Matlab to do the work, rather than listing out the sums of all the columns and verifying manually that each of the 100 numbers is 1!
+**(e)** Double-check that the sum of each column of ''G'' is 1. Again, be clever and get Matlab to do the work, rather than listing out the sums of all the columns and verifying manually that each of the 100 numbers is 1!
 **(f)** Let’s assume we start at the first webpage, ''p=zeros(N,1); p(1)=1;''. If we applying  ''G'' to ''p'' many times (say 1000 times), the resulting vector will give the probability that we end up on a particular website and after a long session of random web surfing. We can do this with a for-loop
@@ Line 46: / Line 46: @@
 N=100;
 for n=1:N;
-  p=T*p;
+  p=G*p;
 end
 </code>
-or equivalently
-<code>
-N=100;
-p=(G^N)*p;
-</code>
 After that is complete the elements of ''p'' give the probabilities that a random web surfer will end up at the web pages listed in ''U'', and you can rank the pages in ''U'' according to their probabilities in ''p''. Look up the ''sort'' command in the Help menu and find a way to print the list of websites in order of their rank. Turn in the list of the top 10, the fraction of the time a random web surfer would spend in each one of those 10, and your code to complete the Lab.
@@ Line 72: / Line 66: @@
 ----
-** Problem 4:** In 2(f) above, it was claimed that
-<code>
+**Problem 4:** This lab has mentioned three different algorithms for calculating the page rank. For this problem, you will compare the execution time T of these three algorithms as a function of M, the number of pages in your web of links. The end result will be a plot of execution time versus M, with three lines, one for each algorithm.
-m=1000;
-p=(T^m)*p;
+It will be easiest to do the comparisons if you convert your page rank script to a Matlab function and then write three different algorithms as three different versions of that function.
+^function^algorithm^
+| ''pagerank_fullpower''     | matrix power method with full matrix |
+| ''pagerank_fulliterate''   | iterated matrix multiplication with full matrix |
+| ''pagerank_sparseiterate'' | iterated matrix multiplication with sparse matrix |
+** 1. matrix power method with full matrix**
+Write a Matlab function
+  function [i,p] = pagerank_fullpower(L);
+that takes a matrix of links L as an input and returns the indices i of webpages sorted by their page ranks, and the page ranks p sorted the same way (so that p(1) >= p(2) etc).
+To be sure that the function operates with full matrices, make this the first line of the function body
+  L = full(L);
+Compute T and G as discussed above, and then compute the page rank G^N p by the straightforward Matlab code
+  N = 100;
+  p = G^N p;
+** 2. iterated matrix multiplication with full matrix **
+Write a Matlab function
+  function [i,p] = pagerank_fulliterate(L);
+just as before, except compute G^N p with the for-loop
+  N=100;
+  for n=1:N
+    p = G*p
+  end
+** 3. iterated matrix multiplication with sparse matrix **
+Write a Matlab function
+  function [i,p] = pagerank_sparseiterate(L);
+that converts the input L to a sparse matrix in the first line with
+  L = sparse(L)
+and computes G^N p with the for-loop
+  N = 100;
+  for n=1:N
+    p = (1-alpha)*(T*p) + alpha*ones(M,1)/M;
+  end
+or equivalently
+  N = 100;
+  one_alpha_T = (1-alpha)*T;
+  alpha_ones  = alpha*ones(M,1)/M;
+  for n=1:N
+    p = one_alpha_T*p + alpha_ones;
+  end
+Note that the sparse algorithm avoids the matrix of 1's that is present in the other two algorithms, because that matrix is necessarily full.
+Now write a script that tests and times these algorithms on link matrices L of sizes M = 32, 64, 128, 256, ...., up to whatever power of two your computer can handle. It's smart to develop and test your code on a smaller set of M, perhaps just M = 32, 64, 128. As a starting point, here's the code we developed in class to measure execution time of full versus sparse matrices.
+<code matlab>
+% script to compare full versus sparse matrix multiplication
+for n=1:12
+  % set up random M x M Ax=b problem
+  m = 2^n;
+  A = randomL(m);
+  x = randn(m,1);
+  % measure execution time of sparse matrix multiplication
+  tic;
+  b = A*x;
+  Tsparse(n) = toc;
+  % measure execution time of full matrix multiplication
+  A = full(A);
+  tic;
+  b = A*x;
+  Tfull(n) = toc;
+  M(n) = m;
+end
 </code>
-is faster than
+You can use this Matlab function to produce synthetic data for testing any given value of M
-<code>
+<code matlab>
-m=1000;
+function L = randomL(M);
-for j=1:m;
+% return random link matrix for testing pagerank algorithms
-  p=T*p;
+  maxlinks = 10;  % maximum of number of links per page
+  L = sparse(M,M);
+  for j=1:M
+    i = randi(M, maxlinks, 1); % get vector of random numbers between 1 and M
+    for n = 1:maxlinks
+      L(i(n),j) = 1;           % create links from page j to the random pages i
+    end
+  end
 end
 </code>
-as a way of computing the steady-state distribution of websurfers.
+Produce a plot of execution time T versus M for each of your three pagerank functions using Matlab's ''tic'' and ''toc''. From the plot, estimate the explicit function that relates T to M for each of the three algorithms.
+Which algorithm is fastest? Why? How long do you estimate each algorithm would take to rank M=10^6 web pages? Provide your answer in easily understandable units of time (days, weeks, years, etc.)

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