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Completely new implementation of sprandsym
author Jordi Gutiérrez Hermoso <jordigh@octave.org>
date Sat, 24 Sep 2011 03:46:36 -0500
parents bae887ebea48
children 0257eb36e15a
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13204:be7bfd59300a 13205:abf1e00111dd
1 ## Copyright (C) 2004-2011 David Bateman and Andy Adler 1 ## Copyright (C) 2004-2011 David Bateman and Andy Adler
2 ## Copyright (C) 2011 Jordi Gutiérrez Hermoso
2 ## 3 ##
3 ## This file is part of Octave. 4 ## This file is part of Octave.
4 ## 5 ##
5 ## Octave is free software; you can redistribute it and/or modify it 6 ## Octave is free software; you can redistribute it and/or modify it
6 ## under the terms of the GNU General Public License as published by 7 ## under the terms of the GNU General Public License as published by
21 ## @deftypefnx {Function File} {} sprandsym (@var{s}) 22 ## @deftypefnx {Function File} {} sprandsym (@var{s})
22 ## Generate a symmetric random sparse matrix. The size of the matrix will be 23 ## Generate a symmetric random sparse matrix. The size of the matrix will be
23 ## @var{n} by @var{n}, with a density of values given by @var{d}. 24 ## @var{n} by @var{n}, with a density of values given by @var{d}.
24 ## @var{d} should be between 0 and 1. Values will be normally 25 ## @var{d} should be between 0 and 1. Values will be normally
25 ## distributed with mean of zero and variance 1. 26 ## distributed with mean of zero and variance 1.
26 ##
27 ## Note: sometimes the actual density may be a bit smaller than @var{d}.
28 ## This is unlikely to happen for large really sparse matrices.
29 ## 27 ##
30 ## If called with a single matrix argument, a random sparse matrix is 28 ## If called with a single matrix argument, a random sparse matrix is
31 ## generated wherever the matrix @var{S} is non-zero in its lower 29 ## generated wherever the matrix @var{S} is non-zero in its lower
32 ## triangular part. 30 ## triangular part.
33 ## @seealso{sprand, sprandn} 31 ## @seealso{sprand, sprandn}
53 51
54 if (d < 0 || d > 1) 52 if (d < 0 || d > 1)
55 error ("sprand: density D must be between 0 and 1"); 53 error ("sprand: density D must be between 0 and 1");
56 endif 54 endif
57 55
58 m1 = floor (n/2); 56 ## Actual number of nonzero entries
59 n1 = m1 + rem (n, 2); 57 k = round (n^2*d);
60 mn1 = m1*n1;
61 k1 = round (d*mn1);
62 idx1 = unique (fix (rand (min (k1*1.01, k1+10), 1) * mn1)) + 1;
63 ## idx contains random numbers in [1,mn] generate 1% or 10 more
64 ## random values than necessary in order to reduce the probability
65 ## that there are less than k distinct values; maybe a better
66 ## strategy could be used but I don't think it's worth the price.
67 58
68 ## Actual number of entries in S. 59 ## Diagonal nonzero entries, same parity as k
69 k1 = min (length (idx1), k1); 60 r = pick_rand_diag (n, k);
70 j1 = floor ((idx1(1:k1)-1)/m1);
71 i1 = idx1(1:k1) - j1*m1;
72 61
73 n2 = ceil (n/2); 62 ## Off diagonal nonzero entries
74 nn2 = n2*n2; 63 m = (k - r)/2;
75 k2 = round (d*nn2);
76 idx2 = unique (fix (rand (min (k2*1.01, k1+10), 1) * nn2)) + 1;
77 k2 = min (length (idx2), k2);
78 j2 = floor ((idx2(1:k2)-1)/n2);
79 i2 = idx2(1:k2) - j2*n2;
80 64
81 if (isempty (i1) && isempty (i2)) 65 ondiag = randperm (n, r);
82 S = sparse (n, n); 66 offdiag = randperm (n*(n - 1)/2, m);
83 else 67
84 S1 = sparse (i1, j1+1, randn (k1, 1), m1, n1); 68 ## Do five Newton iterations to solve n(n - 1)/2 = offdiag (this is the
85 S = [tril(S1), sparse(m1,m1); ... 69 ## row index)
86 sparse(i2,j2+1,randn(k2,1),n2,n2), triu(S1,1)']; 70 x = sqrt (offdiag);
87 S = S + tril (S, -1)'; 71 for ii = 1:5
88 endif 72 x = x - (x.^2 - x - 2*offdiag)./(2*x - 1);
73 endfor
74 i = floor(x);
75 i(i.^2 - i - 2*offdiag != 0) += 1;
76
77 ## Column index
78 j = offdiag - (i - 1).*(i - 2)/2;
79
80 diagvals = randn (1, r);
81 offdiagvals = randn (1, m);
82
83 S = sparse ([ondiag, i, j], [ondiag, j, i],
84 [diagvals, offdiagvals, offdiagvals], n, n);
89 85
90 endfunction 86 endfunction
91 87
88 function r = pick_rand_diag (n, k)
89 ## Pick a random number R of entries for the diagonal of a sparse NxN
90 ## square matrix with exactly K nonzero entries, ensuring that this R
91 ## is chosen uniformly over all such matrices.
92 ##
93 ## Let D be the number of diagonal entries and M the number of
94 ## off-diagonal entries. Then K = D + 2*M. Let A = N*(N-1)/2 be the
95 ## number of available entries in the upper triangle of the matrix.
96 ## Then, by a simple counting argument, there is a total of
97 ##
98 ## T = nchoosek (N, D) * nchoosek (A, M)
99 ##
100 ## symmetric NxN matrices with a total of K nonzero entries and D on
101 ## the diagonal. Letting D range from mod (K,2) through min (N,K), and
102 ## dividing by this sum, we obtain the probability P for D to be each
103 ## of those values.
104 ##
105 ## However, we cannot use this form for computation, as the binomial
106 ## coefficients become unmanageably large. Instead, we use the
107 ## successive quotients Q(i) = T(i+1)/T(i), which we easily compute to
108 ## be
109 ##
110 ## (N - D)*(N - D - 1)*M
111 ## Q = -------------------------------
112 ## (D + 2)*(D + 1)*(A - M + 1)
113 ##
114 ## Then the cumprod of these quotients is
115 ##
116 ## C = [ T(2)/T(1), T(3)/T(1), ..., T(N)/T(1) ]
117 ##
118 ## Their sum + 1 is thus S = sum (T)/T(1), and then C(i)/S is the
119 ## desired probability P(i) for i = 2:N. The first P(1) can be
120 ## obtained by the condition that sum (P) = 1, and the cumsum will
121 ## give the distribution function for computing the random number of
122 ## entries on the diagonal R.
92 123
93 ## FIXME: Test for density can't happen until code of sprandsym is improved 124 ## Compute the stuff described above
125 a = n*(n - 1)/2;
126 d = [mod(k,2):2:min(n,k)-2];
127 m = (k - d)/2;
128 q = (n - d).*(n - d - 1).*m ./ (d + 2)./(d + 1)./(a - m + 1);
129 c = cumprod (q);
130 s = sum (c) + 1;
131 p = c/s;
132
133 ## Add missing entries
134 p = [1 - sum(p), p];
135 d(end+1) = d(end) + 2;
136
137 ## Pick a random r using this distribution
138 r = d(sum (cumsum (p) < rand) + 1);
139
140 endfunction
141
94 %!test 142 %!test
95 %! s = sprandsym (10, 0.1); 143 %! s = sprandsym (10, 0.1);
96 %! assert (issparse (s)); 144 %! assert (issparse (s));
97 %! assert (issymmetric (s)); 145 %! assert (issymmetric (s));
98 %! assert (size (s), [10, 10]); 146 %! assert (size (s), [10, 10]);
99 ##%! assert (nnz (s) / numel (s), 0.1, .01); 147 %! assert (nnz (s) / numel (s), 0.1, .01);
100 148
101 %% Test 1-input calling form 149 %% Test 1-input calling form
102 %!test 150 %!test
103 %! s = sprandsym (sparse ([1 2 3], [3 2 3], [2 2 2])); 151 %! s = sprandsym (sparse ([1 2 3], [3 2 3], [2 2 2]));
104 %! [i, j] = find (s); 152 %! [i, j] = find (s);