1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
|
//------------------------------------------------------------------------------
// GraphBLAS/Demo/Source/tricount.c: count the number of triangles in a graph
//------------------------------------------------------------------------------
// SuiteSparse:GraphBLAS, Timothy A. Davis, (c) 2017-2020, All Rights Reserved.
// http://suitesparse.com See GraphBLAS/Doc/License.txt for license.
//------------------------------------------------------------------------------
// Given a symmetric graph A with no-self edges, tricount counts the exact
// number of triangles in the graph.
// One of 5 methods are used. Each computes the same result, ntri:
// 0: minitri: ntri = nnz (A*E == 2) / 3
// 1: Burkhardt: ntri = sum (sum ((A^2) .* A)) / 6
// 2: Cohen: ntri = sum (sum ((L * U) .* A)) / 2
// 3: Sandia: ntri = sum (sum ((L * L) .* L))
// 4: Sandia2: ntri = sum (sum ((U * U) .* U))
// 5: SandiaDot: ntri = sum (sum ((L * U') .* L)). Note that L=U'.
// 6: SandiaDot2: ntri = sum (sum ((U * L') .* U))
// All matrices are assumed to be in CSR format (GxB_BY_ROW).
// Method 0 can take a huge amount of memory, for all of A*E. As a result,
// it often fails for large problems.
// Methods 1 and 2 are much more memory efficient as compare to Method 0,
// taking memory space the same size as A. But they are slower than methods 3
// and 4.
// Methods 3 and 4 take a little less memory than methods 1 and 2, are by far
// the fastest methods in general. The two methods compute the same
// intermediate matrix (U*U), and differ only in the way the matrix
// multiplication is done. Method 3 uses an outer-product method (Gustavson's
// method). Method 5 uses dot products and does not explicitly transpose U.
// They are called the "Sandia" method since matrices in the KokkosKernels
// are stored in compressed-sparse row form, so (L*L).*L in the KokkosKernel
// method is equivalent to (L*L).*L in SuiteSparse:GraphBLAS when the matrices
// in SuiteSparse:GraphBLAS are in their default format (also by row).
// A is a binary square symmetric matrix. E is the edge incidence matrix of A.
// L=tril(A), and U=triu(A). See GraphBLAS/Demo/tricount.m for a complete
// definition of each method and the matrices A, E, L, and U, and citations of
// relevant references.
// All input matrices should have binary values (0 and 1). Any type will work,
// but int32 is recommended for fastest results since that is the type used
// here for the semiring. GraphBLAS will do typecasting internally, but that
// takes extra time.
// This method has been updated as of Version 2.2 of SuiteSparse:GraphBLAS. It
// now assumes the matrix is held by row (GxB_BY_ROW), not by column
// (GxB_BY_COL). Both methods work fine, but with matrices stored by column,
// C<M>=A'*B uses the dot product method by default, whereas C<M>=A*B' uses the
// dot product method if the matrices are stored by row.
#include "GraphBLAS.h"
#define FREE_ALL \
GrB_UnaryOp_free (&Two) ; \
GrB_Descriptor_free (&d) ; \
GrB_Matrix_free (&S) ; \
GrB_Matrix_free (&C) ;
#undef GB_PUBLIC
#define GB_LIBRARY
#include "graphblas_demos.h"
//------------------------------------------------------------------------------
// two: unary function for GrB_apply
//------------------------------------------------------------------------------
void two (int32_t *z, const int32_t *x)
{
(*z) = (double) (((*x) == 2) ? 1 : 0) ;
}
//------------------------------------------------------------------------------
// tricount: count the number of triangles in a graph
//------------------------------------------------------------------------------
GB_PUBLIC
GrB_Info tricount // count # of triangles
(
int64_t *p_ntri, // # of trianagles
const int method, // 0 to 6, see above
const GrB_Matrix A, // adjacency matrix for methods 0, 1, and 2
const GrB_Matrix E, // edge incidence matrix for method 0
const GrB_Matrix L, // L=tril(A) for methods 2, 3, 5, and 6
const GrB_Matrix U, // U=triu(A) for methods 2, 4, 5, and 6
double t [2] // t [0]: multiply time, t [1]: reduce time
)
{
//--------------------------------------------------------------------------
// check inputs
//--------------------------------------------------------------------------
double tic [2] ;
simple_tic (tic) ;
GrB_Info info ;
int64_t ntri ;
GrB_Index n, ne ;
GrB_UnaryOp Two = NULL ;
GrB_Matrix S = NULL, C = NULL ;
GrB_Descriptor d = NULL ;
OK (GrB_Descriptor_new (&d)) ;
GrB_Semiring semiring = GrB_PLUS_TIMES_SEMIRING_INT32 ;
GrB_Type ctype = GrB_INT32 ;
switch (method)
{
case 0: // minitri: ntri = nnz (A*E == 2) / 3
OK (GrB_Matrix_nrows (&n, A)) ;
OK (GrB_Matrix_ncols (&ne, E)) ;
OK (GrB_Matrix_new (&C, ctype, n, ne)) ;
// mxm: outer product method, no mask
OK (GxB_Desc_set (d, GxB_AxB_METHOD, GxB_AxB_GUSTAVSON)) ;
OK (GrB_mxm (C, NULL, NULL, GxB_PLUS_TIMES_UINT32, A, E, d)) ;
t [0] = simple_toc (tic) ;
simple_tic (tic) ;
OK (GrB_UnaryOp_new (&Two, (GxB_unary_function) two, ctype, ctype));
OK (GrB_Matrix_new (&S, ctype, n, ne)) ;
OK (GrB_Matrix_apply (S, NULL, NULL, Two, C, NULL)) ;
OK (GrB_Matrix_reduce_INT64 (&ntri, NULL, GrB_PLUS_MONOID_INT64,
S, NULL)) ;
ntri /= 3 ;
break ;
case 1: // Burkhardt: ntri = sum (sum ((A^2) .* A)) / 6
OK (GrB_Matrix_nrows (&n, A)) ;
OK (GrB_Matrix_new (&C, ctype, n, n)) ;
// mxm: outer product method, with mask
OK (GxB_Desc_set (d, GxB_AxB_METHOD, GxB_AxB_GUSTAVSON)) ;
OK (GrB_mxm (C, A, NULL, semiring, A, A, d)) ;
t [0] = simple_toc (tic) ;
simple_tic (tic) ;
OK (GrB_Matrix_reduce_INT64 (&ntri, NULL, GrB_PLUS_MONOID_INT64,
C, NULL)) ;
ntri /= 6 ;
break ;
case 2: // Cohen: ntri = sum (sum ((L * U) .* A)) / 2
OK (GrB_Matrix_nrows (&n, A)) ;
OK (GrB_Matrix_new (&C, ctype, n, n)) ;
// mxm: outer product method, with mask
OK (GxB_Desc_set (d, GxB_AxB_METHOD, GxB_AxB_GUSTAVSON)) ;
OK (GrB_mxm (C, A, NULL, semiring, L, U, d)) ;
t [0] = simple_toc (tic) ;
simple_tic (tic) ;
OK (GrB_Matrix_reduce_INT64 (&ntri, NULL, GrB_PLUS_MONOID_INT64,
C, NULL)) ;
ntri /= 2 ;
break ;
case 3: // Sandia: ntri = sum (sum ((L * L) .* L))
OK (GrB_Matrix_nrows (&n, L)) ;
OK (GrB_Matrix_new (&C, ctype, n, n)) ;
OK (GxB_Desc_set (d, GxB_AxB_METHOD, GxB_AxB_GUSTAVSON)) ;
OK (GrB_mxm (C, L, NULL, semiring, L, L, d)) ;
t [0] = simple_toc (tic) ;
simple_tic (tic) ;
OK (GrB_Matrix_reduce_INT64 (&ntri, NULL, GrB_PLUS_MONOID_INT64,
C, NULL)) ;
break ;
case 4: // Sandia2: ntri = sum (sum ((U * U) .* U))
OK (GrB_Matrix_nrows (&n, U)) ;
OK (GrB_Matrix_new (&C, ctype, n, n)) ;
// mxm: outer product method, with mask
OK (GxB_Desc_set (d, GxB_AxB_METHOD, GxB_AxB_GUSTAVSON)) ;
OK (GrB_mxm (C, U, NULL, semiring, U, U, d)) ;
t [0] = simple_toc (tic) ;
simple_tic (tic) ;
OK (GrB_Matrix_reduce_INT64 (&ntri, NULL, GrB_PLUS_MONOID_INT64,
C, NULL)) ;
break ;
case 5: // SandiaDot: ntri = sum (sum ((L * U') .* L))
OK (GrB_Matrix_nrows (&n, U)) ;
OK (GrB_Matrix_new (&C, ctype, n, n)) ;
OK (GrB_Descriptor_new (&d)) ;
OK (GxB_Desc_set (d, GrB_INP1, GrB_TRAN)) ;
// mxm: dot product method, with mask
OK (GxB_Desc_set (d, GxB_AxB_METHOD, GxB_AxB_DOT)) ;
OK (GrB_mxm (C, L, NULL, semiring, L, U, d)) ;
t [0] = simple_toc (tic) ;
simple_tic (tic) ;
OK (GrB_Matrix_reduce_INT64 (&ntri, NULL, GrB_PLUS_MONOID_INT64,
C, NULL)) ;
break ;
case 6: // SandiaDot2: ntri = sum (sum ((U * L') .* U))
OK (GrB_Matrix_nrows (&n, U)) ;
OK (GrB_Matrix_new (&C, ctype, n, n)) ;
OK (GrB_Descriptor_new (&d)) ;
OK (GxB_Desc_set (d, GrB_INP1, GrB_TRAN)) ;
// mxm: dot product method, with mask
OK (GxB_Desc_set (d, GxB_AxB_METHOD, GxB_AxB_DOT)) ;
OK (GrB_mxm (C, U, NULL, semiring, U, L, d)) ;
t [0] = simple_toc (tic) ;
simple_tic (tic) ;
OK (GrB_Matrix_reduce_INT64 (&ntri, NULL, GrB_PLUS_MONOID_INT64,
C, NULL)) ;
break ;
default: // invalid method
return (GrB_INVALID_VALUE) ;
break ;
}
FREE_ALL ;
t [1] = simple_toc (tic) ;
(*p_ntri) = ntri ;
return (GrB_SUCCESS) ;
}
|
