GraphBLAS: graph algorithms in the language of linear algebra

GraphBLAS is a library for creating graph algorithms based on sparse linear algebraic operations over semirings. Visit http://graphblas.org for more details and resources. See also the SuiteSparse:GraphBLAS User Guide in this package.

http://faculty.cse.tamu.edu/davis

SuiteSparse:GraphBLAS, Timothy A. Davis, (c) 2017-2021, All Rights Reserved. SPDX-License-Identifier: GPL-3.0-or-later

Contents

GraphBLAS: faster and more general sparse matrices for MATLAB

GraphBLAS is not only useful for creating graph algorithms; it also supports a wide range of sparse matrix data types and operations. MATLAB can compute C=A*B with just two semirings: 'plus.times.double' and 'plus.times.complex' for complex matrices. GraphBLAS has 2,518 built-in semirings, such as 'max.plus' (https://en.wikipedia.org/wiki/Tropical_semiring). These semirings can be used to construct a wide variety of graph algorithms, based on operations on sparse adjacency matrices.

MATLAB and GraphBLAS both provide sparse matrices of type double, logical, and double complex. GraphBLAS adds sparse matrices of type: single, int8, int16, int32, int64, uint8, uint16, uint32, uint64, and single complex (with MATLAB matrices, these types can only be held in full matrices).

% reset to the default number of threads
clear all
maxNumCompThreads ('automatic') ;
GrB.clear ;
fprintf ('\n# of threads used by GraphBLAS: %d\n', GrB.threads) ;

format compact
rng ('default') ;
X = 100 * rand (2) ;
G = GrB (X)              % GraphBLAS copy of a matrix X, same type
# of threads used by GraphBLAS: 4

G =

  2x2 GraphBLAS double matrix, full by col
  4 nonzeros, 4 entries

    (1,1)    81.4724
    (2,1)    90.5792
    (1,2)    12.6987
    (2,2)    91.3376

Sparse integer matrices

Here's an int8 version of the same matrix:

S = int8 (G)             % convert G to a full MATLAB int8 matrix
S (1,1) = 0              % add an explicit zero to S
G = GrB (X, 'int8')      % a GraphBLAS full int8 matrix
G (1,1) = 0              % add an explicit zero to G
G = GrB.prune (G)        % a GraphBLAS sparse int8 matrix

try
    S = sparse (S) ;     % MATLAB can't create sparse int8 matrices
catch me
    display (me)
end
S =
  2x2 int8 matrix
   81   13
   91   91
S =
  2x2 int8 matrix
    0   13
   91   91

G =

  2x2 GraphBLAS int8_t matrix, full by col
  4 nonzeros, 4 entries

    (1,1)   81
    (2,1)   91
    (1,2)   13
    (2,2)   91


G =

  2x2 GraphBLAS int8_t matrix, full by col
  3 nonzeros, 4 entries

    (1,1)   0
    (2,1)   91
    (1,2)   13
    (2,2)   91


G =

  2x2 GraphBLAS int8_t matrix, bitmap by col
  3 nonzeros, 3 entries

    (2,1)   91
    (1,2)   13
    (2,2)   91

me = 
  MException with properties:

    identifier: 'MATLAB:sparse:charConversion'
       message: 'Input matrix must be double or logical.'
         cause: {}
         stack: [4x1 struct]
    Correction: []

Sparse single-precision matrices

Matrix operations in GraphBLAS are typically as fast, or faster than MATLAB. Here's an unfair comparison: computing X*X with MATLAB in double precision and with GraphBLAS in single precision. You would naturally expect GraphBLAS to be faster.

CAVEAT: MATLAB R2021a uses SuiteSparse:GraphBLAS v3.3.3 for C=A*B, so on that version of MATLAB, we're comparing 2 versions of GraphBLAS by the same author.

Please wait ...

n = 1e5 ;
X = spdiags (rand (n, 201), -100:100, n, n) ;
G = GrB (X, 'single') ;
G2 = G*G ;  % warmup
tic
G2 = G*G ;
gb_time = toc ;
X2 = X*X ;  % warmup
tic
X2 = X*X ;
matlab_time = toc ;
fprintf ('\nGraphBLAS time: %g sec (in single)\n', gb_time) ;
fprintf ('MATLAB time:    %g sec (in double)\n', matlab_time) ;
fprintf ('Speedup of GraphBLAS over MATLAB: %g\n', ...
    matlab_time / gb_time) ;
fprintf ('\n# of threads used by GraphBLAS: %d\n', GrB.threads) ;
GraphBLAS time: 1.76609 sec (in single)
MATLAB time:    7.92736 sec (in double)
Speedup of GraphBLAS over MATLAB: 4.48864

# of threads used by GraphBLAS: 4

Mixing MATLAB and GraphBLAS matrices

The error in the last computation is about eps('single') since GraphBLAS did its computation in single precision, while MATLAB used double precision. MATLAB and GraphBLAS matrices can be easily combined, as in X2-G2. The sparse single precision matrices take less memory space.

err = norm (X2 - G2, 1) / norm (X2,1)
eps ('single')
whos G G2 X X2
err =
   1.5049e-07
ans =
  single
  1.1921e-07
  Name           Size                    Bytes  Class     Attributes

  G         100000x100000            241879740  GrB                 
  G2        100000x100000            481518540  GrB                 
  X         100000x100000            322238408  double    sparse    
  X2        100000x100000            641756808  double    sparse    

Faster matrix operations

But even with standard double precision sparse matrices, GraphBLAS is typically faster than the built-in MATLAB methods. Here's a fair comparison (caveat: these both use GraphBLAS in MATLAB R2021a):

G = GrB (X) ;
G2 = G*G ;  % warmup
tic
G2 = G*G ;
gb_time = toc ;
err = norm (X2 - G2, 1) / norm (X2,1)
fprintf ('\nGraphBLAS time: %g sec (in double)\n', gb_time) ;
fprintf ('MATLAB time:    %g sec (in double)\n', matlab_time) ;
fprintf ('Speedup of GraphBLAS over MATLAB: %g\n', ...
    matlab_time / gb_time) ;
fprintf ('\n# of threads used by GraphBLAS: %d\n', GrB.threads) ;
err =
     0

GraphBLAS time: 2.01184 sec (in double)
MATLAB time:    7.92736 sec (in double)
Speedup of GraphBLAS over MATLAB: 3.94036

# of threads used by GraphBLAS: 4

A wide range of semirings

MATLAB can only compute C=A*B using the standard '+.*.double' and '+.*.complex' semirings. A semiring is defined in terms of a string, 'add.mult.type', where 'add' is a monoid that takes the place of the additive operator, 'mult' is the multiplicative operator, and 'type' is the data type for the two inputs to the mult operator.

In the standard semiring, C=A*B is defined as:

C(i,j) = sum (A(i,:).' .* B(:,j))

using 'plus' as the monoid and 'times' as the multiplicative operator. But in a more general semiring, 'sum' can be any monoid, which is an associative and commutative operator that has an identity value. For example, in the 'max.plus' tropical algebra, C(i,j) for C=A*B is defined as:

C(i,j) = max (A(i,:).' + B(:,j))

This can be computed in GraphBLAS with:

C = GrB.mxm ('max.+', A, B)
n = 3 ;
A = rand (n) ;
B = rand (n) ;
C = zeros (n) ;
for i = 1:n
    for j = 1:n
        C(i,j) = max (A (i,:).' + B (:,j)) ;
    end
end
C2 = GrB.mxm ('max.+', A, B) ;
fprintf ('\nerr = norm (C-C2,1) = %g\n', norm (C-C2,1)) ;
err = norm (C-C2,1) = 0

The max.plus tropical semiring

Here are details of the "max.plus" tropical semiring. The identity value is -inf since max(x,-inf) = max (-inf,x) = x for any x. The identity for the conventional "plus.times" semiring is zero, since x+0 = 0+x = x for any x.

GrB.semiringinfo ('max.+.double') ;
    GraphBLAS Semiring: max.+.double (built-in)
    GraphBLAS Monoid: semiring->add (built-in)
    GraphBLAS BinaryOp: monoid->op (built-in) z=max(x,y)
    GraphBLAS type: ztype double size: 8
    GraphBLAS type: xtype double size: 8
    GraphBLAS type: ytype double size: 8
    identity: [    -Inf ] terminal: [    Inf ]

    GraphBLAS BinaryOp: semiring->multiply (built-in) z=plus(x,y)
    GraphBLAS type: ztype double size: 8
    GraphBLAS type: xtype double size: 8
    GraphBLAS type: ytype double size: 8

A boolean semiring

MATLAB cannot multiply two logical matrices. MATLAB R2019a converts them to double and uses the conventional +.*.double semiring instead. In GraphBLAS, this is the common Boolean 'or.and.logical' semiring, which is widely used in linear algebraic graph algorithms.

GrB.semiringinfo ('|.&.logical') ;
    GraphBLAS Semiring: |.&.logical (built-in)
    GraphBLAS Monoid: semiring->add (built-in)
    GraphBLAS BinaryOp: monoid->op (built-in) z=or(x,y)
    GraphBLAS type: ztype bool size: 1
    GraphBLAS type: xtype bool size: 1
    GraphBLAS type: ytype bool size: 1
    identity: [   0 ] terminal: [   1 ]

    GraphBLAS BinaryOp: semiring->multiply (built-in) z=and(x,y)
    GraphBLAS type: ztype bool size: 1
    GraphBLAS type: xtype bool size: 1
    GraphBLAS type: ytype bool size: 1
clear
A = sparse (rand (3) > 0.5)
B = sparse (rand (3) > 0.2)
A =
  3x3 sparse logical array
   (2,1)      1
   (2,2)      1
   (3,2)      1
   (1,3)      1
B =
  3x3 sparse logical array
   (1,1)      1
   (2,1)      1
   (3,1)      1
   (1,2)      1
   (2,2)      1
   (3,2)      1
   (1,3)      1
   (2,3)      1
   (3,3)      1
try
    % MATLAB R2019a does this by casting A and B to double
    C1 = A*B
catch
    % MATLAB R2018a throws an error
    fprintf ('MATLAB R2019a required for C=A*B with logical\n') ;
    fprintf ('matrices.  Explicitly converting to double:\n') ;
    C1 = double (A) * double (B)
end
C2 = GrB (A) * GrB (B)
C1 =
   (1,1)        1
   (2,1)        2
   (3,1)        1
   (1,2)        1
   (2,2)        2
   (3,2)        1
   (1,3)        1
   (2,3)        2
   (3,3)        1

C2 =

  3x3 GraphBLAS bool matrix, bitmap by col
  9 nonzeros, 9 entries

    (1,1)   1
    (2,1)   1
    (3,1)   1
    (1,2)   1
    (2,2)   1
    (3,2)   1
    (1,3)   1
    (2,3)   1
    (3,3)   1

Note that C1 is a MATLAB sparse double matrix, and contains non-binary values. C2 is a GraphBLAS logical matrix.

whos
GrB.type (C2)
  Name      Size            Bytes  Class      Attributes

  A         3x3                68  logical    sparse    
  B         3x3               113  logical    sparse    
  C1        3x3               176  double     sparse    
  C2        3x3               784  GrB                  

ans =
    'logical'

GraphBLAS operators, monoids, and semirings

The C interface for SuiteSparse:GraphBLAS allows for arbitrary types and operators to be constructed. However, the MATLAB interface to SuiteSparse:GraphBLAS is restricted to pre-defined types and operators: a mere 13 types, 212 unary operators, 401 binary operators, 77 monoids, 22 select operators (each of which can be used for all 13 types), and 2,518 semirings.

That gives you a lot of tools to create all kinds of interesting graph algorithms. For example:

GrB.bfs    % breadth-first search
GrB.dnn    % sparse deep neural network (http://graphchallenge.org)
GrB.mis    % maximal independent set

See 'help GrB.binopinfo' for a list of the binary operators, and 'help GrB.monoidinfo' for the ones that can be used as the additive monoid in a semiring. 'help GrB.unopinfo' lists the unary operators. 'help GrB.semiringinfo' describes the semirings.

help GrB.binopinfo
 GRB.BINOPINFO list the details of a GraphBLAS binary operator.
 
    GrB.binopinfo
    GrB.binopinfo (op)
    GrB.binopinfo (op, optype)
 
  Binary operators are defined by a string of the form 'op.optype', or
  just 'op', where the optype is inferred from the operands.  Valid
  optypes are 'logical', 'int8', 'int16', 'int32', 'int64', 'uint8',
  'uint16', 'uint32', 'uint64', 'single', 'double', 'single complex',
  'double complex' (the latter can be written as simply 'complex').
 
  For GrB.binopinfo (op), the op must be a string of the form 'op.optype',
  where 'op' is listed below.  The second usage allows the optype to be
  omitted from the first argument, as just 'op'.  This is valid for all
  GraphBLAS operations, since the optype can be determined from the
  operands (see Typecasting, below).  However, GrB.binopinfo does not have
  any operands and thus the optype must be provided, either in the op as
  GrB.binopinfo ('+.double'), or in the second argument as
  GrB.binopinfo ('+', 'double').
 
  The 6 comparator operators come in two flavors.  For the is* operators,
  the result has the same type as the inputs, x and y, with 1 for true and
  0 for false.  For example isgt.double (pi, 3.0) is the double value 1.0.
  For the second set of 6 operators (eq, ne, gt, lt, ge, le), the result
  is always logical (true or false).  In a semiring, the optype of the add
  monoid must exactly match the type of the output of the multiply
  operator, and thus 'plus.iseq.double' is valid (counting how many terms
  are equal).  The 'plus.eq.double' semiring is valid, but not the same
  semiring since the 'plus' of 'plus.eq.double' has a logical type and is
  thus equivalent to 'or.eq.double'.   The 'or.eq' is true if any terms
  are equal and false otherwise (it does not count the number of terms
  that are equal).
 
  The following binary operators are available for most types.  Many have
  equivalent synonyms, so that '1st' and 'first' both define the
  first(x,y) = x operator.
 
    operator name(s) f(x,y)         |   operator names(s) f(x,y)
    ---------------- ------         |   ----------------- ------
    1st first        x              |   iseq              x == y
    2nd second       y              |   isne              x ~= y
    min              min(x,y)       |   isgt              x > y
    max              max(x,y)       |   islt              x < y
    +   plus         x+y            |   isge              x >= y
    -   minus        x-y            |   isle              x <= y
    rminus           y-x            |   ==  eq            x == y
    *   times        x*y            |   ~=  ne            x ~= y
    /   div          x/y            |   >   gt            x > y
    \   rdiv         y/x            |   <   lt            x < y
    |   || or  lor   x | y          |   >=  ge            x >= y
    &   && and land  x & y          |   <=  le            x <= y
    xor lxor         xor(x,y)       |   .^  pow           x .^ y
    pair             1              |   any               pick x or y
 
  All of the above operators are defined for logical operands, but many
  are redundant. 'min.logical' is the same as 'and.logical', for example.
  Most of the logical operators have aliases: ('lor', 'or', '|') are the
  same, as are ('lxnor', 'xnor', 'eq', '==') for logical types.
 
  Positional operators return int32 or int64, and depend only on the position
  of the entry in the matrix.  They do not depend on the values of their
  inputs, but on their position in the matrix instead:
 
    1-based postional ops:          in a semiring:     in ewise operators:
    operator name(s)                f(A(i,k)*B(k,j))   f(A(i,j),B(i,j))
    ----------------                ----------------   ----------------
    firsti1  1sti1 firsti  1sti     i                  i
    firstj1  1stj1 firstj  1stj     k                  j
    secondi1 2ndi1 secondi 2ndi     k                  i
    secondj1 2ndj1 secondj 2ndj     j                  j
 
    0-based postional ops:          in a semiring:     in ewise operators:
    operator name(s)                f(A(i,k)*B(k,j))   f(A(i,j),B(i,j))
    ----------------                ----------------   ----------------
    firsti0  1sti0                  i-1                i-1
    firstj0  1stj0                  k-1                j-1
    secondi0 2ndi0                  k-1                i-1
    secondj0 2ndj0                  j-1                j-1
 
  Comparators (*lt, *gt, *le, *ge) and min/max are not available for
  complex types.
 
  The three logical operators, lor, land, and lxor, can be used with any
  real types.  z = lor.double (x,y) tests the condition (x~=0) || (y~=0),
  and returns the double value 1.0 if true, or 0.0 if false.
 
  The following operators are avaiable for single and double (real); their
  definitions are identical to the ANSI C11 versions of these functions:
  atan2, hypot, fmod, remainder, copysign, ldxep (also called 'pow2').
  All produce the same type as the input, on output.
 
  z = cmplx(x,y) can be computed for x and y as single and double; z is
  single complex or double complex, respectively.
 
  The bitwise ops bitor, bitand, bitxor, bitxnor, bitget, bitset, bitclr,
  and bitshift are available for any signed or unsigned integer type.
 
  Typecasting:  If the optype is omitted from the string (for example,
  GrB.eadd (A, '+', B) or simply C = A+B), then the optype is inferred
  from the type of A and B.  See 'help GrB.optype' for details.
 
  Example:
 
    % valid binary operators
    GrB.binopinfo ('+.double') ;    % also a valid unary operator
    GrB.binopinfo ('1st.int32') ;
    GrB.binopinfo ('cmplx.single') ;
    GrB.binopinfo ('pow2.double') ; % also a valid unary operator
    GrB.unopinfo  ('pow2.double') ;
 
    % invalid binary operator (an error; this is a unary op):
    GrB.binopinfo ('abs.double') ;
 
  See also GrB.descriptorinfo, GrB.monoidinfo, GrB.selectopinfo,
  GrB.semiringinfo, GrB.unopinfo, GrB.optype.

help GrB.monoidinfo
 GRB.MONOIDINFO list the details of a GraphBLAS monoid.
 
    GrB.monoidinfo
    GrB.monoidinfo (monoid)
    GrB.monoidinfo (monoid, type)
 
  For GrB.monoidinfo(op), the op must be a string of the form 'op.type',
  where 'op' is listed below.  The second usage allows the type to be
  omitted from the first argument, as just 'op'.  This is valid for all
  GraphBLAS operations, since the type defaults to the type of the input
  matrices.  However, GrB.monoidinfo does not have a default type and thus
  one must be provided, either in the op as GrB.monoidinfo ('+.double'), or
  in the second argument, GrB.monoidinfo ('+', 'double').
 
  A monoid is any binary operator z=f(x,y) that is commutative and
  associative, with an identity value o so that f(x,o)=f(o,x)=o.  The types
  of z, x, and y must all be identical.  For example, the plus.double
  operator is f(x,y)=x+y, with zero as the identity value (x+0 = 0+x = x).
  The times monoid has an identity value of 1 (since x*1 = 1*x = x).  The
  identity of min.double is +inf.
 
  The valid monoids for real non-logical types are:
        '+', '*', 'max', 'min', 'any'
  For the 'logical' type:
        '|', '&', 'xor', 'eq' (the same as 'xnor'), 'any'
  For complex types:
        '+', '*', 'any'
  For integer types (signed and unsigned):
        'bitor', 'bitand', 'bitxor', 'bitxnor'
 
  Some monoids have synonyms; see 'help GrB.binopinfo' for details.
 
  Example:
 
    % valid monoids
    GrB.monoidinfo ('+.double') ;
    GrB.monoidinfo ('*.int32') ;
    GrB.monoidinfo ('min.double') ;
 
    % invalid monoids
    GrB.monoidinfo ('1st.int32') ;
    GrB.monoidinfo ('abs.double') ;
    GrB.monoidinfo ('min.complex') ;
 
  See also GrB.binopinfo, GrB.descriptorinfo, GrB.selectopinfo,
  GrB.semiringinfo, GrB.unopinfo.

help GrB.unopinfo
 GRB.UNOPINFO list the details of a GraphBLAS unary operator.
 
    GrB.unopinfo
    GrB.unopinfo (op)
    GrB.unopinfo (op, type)
 
  For GrB.unopinfo(op), the op must be a string of the form 'op.type',
  where 'op' is listed below.  The second usage allows the type to be
  omitted from the first argument, as just 'op'.  This is valid for all
  GraphBLAS operations, since the type defaults to the type of the input
  matrix.  However, GrB.unopinfo does not have a default type and thus one
  must be provided, either in the op as GrB.unopinfo ('abs.double'), or in
  the second argument, GrB.unopinfo ('abs', 'double').
 
  The functions z=f(x) are listed below.  Unless otherwise specified,
  z and x have the same type.  Some functions have synonyms, as listed.
 
  For all 13 types:
    identity    z = x       also '+', 'uplus'
    ainv        z = -x      additive inverse, also '-', 'negate', 'uminus'
    minv        z = 1/x     multiplicative inverse
    one         z = 1       does not depend on x, also '1'
    abs         z = |x|     'abs.complex' returns a real result
 
  For all 11 real types:
    lnot        z = ~(x ~= 0)   logical negation (z is 1 or 0, with the
                                same type as x), also '~', 'not'.
 
  For 4 floating-point types (real & complex)x(single & double):
    sqrt        z = sqrt (x)    square root
    log         z = log (x)     base-e logarithm
    log2        z = log2 (x)    base-2 logarithm
    log10       z = log10 (x)   base-10 logarithm
    log1p       z = log1p (x)   log (x-1), base-e
    exp         z = exp (x)     base-e exponential, e^x
    pow2        z = pow2 (x)    base-2 exponential, 2^x
    expm1       z = exp1m (x)   e^x-1
    sin         z = sin (x)     sine
    cos         z = cos (x)     cosine
    tan         z = tan (x)     tangent
    acos        z = acos (x)    arc cosine
    asin        z = asin (x)    arc sine
    atan        z = atan (x)    arc tangent
    sinh        z = sinh (x)    hyperbolic sine
    cosh        z = cosh (x)    hyperbolic cosine
    tanh        z = tanh (x)    hyperbolic tangent
    asinh       z = asinh (x)   inverse hyperbolic sine
    acosh       z = acosh (x)   inverse hyperbolic cosine
    atanh       z = atanh (x)   inverse hyperbolic tangent
    signum      z = signum (x)  signum function, also 'sign'
    ceil        z = ceil (x)    ceiling
    floor       z = floor (x)   floor
    round       z = round (x)   round to nearest
    trunc       z = trunc (x)   truncate, also 'fix'
 
  For 'single complex' and 'double complex' only:
    creal       z = real (x)    real part of x (z is real), also 'real'
    cimag       z = imag (x)    imag. part of x (z is real), also 'imag'
    carg        z = carg (x)    phase angle (z is real), also 'angle'
    conj        z = conj (x)    complex conjugate (z is complex)
 
  For all 4 floating-point types (result is logical):
    isinf       z = isinf (x)       true if x is +Inf or -Inf
    isnan       z = isnan (x)       true if x is NaN
    isfinite    z = isfinite (x)    true if x is finite
 
  For single and double (result same type as input):
    lgamma      z = lgamma (x)  log of gamma function, also 'gammaln'
    tgamma      z = tgamma (x)  gamma function, also 'gamma'
    erf         z = erf (x)     error function
    erfc        z = erfc (x)    complementary error function
    frexpx      z = frexpx (x)  mantissa from ANSI C11 frexp function
    frexpe      z = frexpe (x)  exponent from ANSI C11 frexp function
                                The MATLAB [f,e]=log2(x) returns
                                f = frexpx (x) and e = frexpe (x).
 
  For integer types only (result is same type as input):
    bitcmp      z = ~(x)        bitwise complement, also 'bitnot'
 
  For int32 and int64 types, applied to an entry A(i,j)
    positioni0  z = i-1     also 'i0'
    positioni1  z = i       also 'i', 'i1', and 'positioni'
    positionj0  z = j-1     also 'j0'
    positionj1  z = j       also 'j', 'j1', and 'positionj'
 
  Example:
 
    % valid unary operators
    GrB.unopinfo ('+.double') ;     % also a valid binary operator
    GrB.unopinfo ('abs.double') ;
    GrB.unopinfo ('not.int32') ;
    GrB.unopinfo ('pow2.double') ;  % also a valid binary operator
    GrB.binopinfo ('pow2.double') ;
 
    % invalid unary operator (generates an error; this is a binary op):
    GrB.unopinfo ('*.double') ;
 
  See also GrB.binopinfo, GrB.descriptorinfo, GrB.monoidinfo,
  GrB.selectopinfo, GrB.semiringinfo.

help GrB.semiringinfo
 GRB.SEMIRINGINFO list the details of a GraphBLAS semiring.
 
    GrB.semiringinfo
    GrB.semiringinfo (semiring)
    GrB.semiringinfo (semiring, type)
 
  For GrB.semiring(semiring), the semiring must be a string of the form
  'add.mult.type', where 'add' and 'mult' are binary operators.  The
  second usage allows the type to be omitted from the first argument, as
  just 'add.mult'.  This is valid for all GraphBLAS operations, since the
  type defaults to the type of the input matrices.  However,
  GrB.semiringinfo does not have a default type and thus one must be
  provided, either in the semiring as GrB.semiringinfo ('+.*.double'), or
  in the second argument, GrB.semiringinfo ('+.*', 'double').
 
  The additive operator must be the binary operator of a valid monoid (see
  'help GrB.monoidinfo').  The multiplicative operator can be any binary
  operator z=f(x,y) listed by 'help GrB.binopinfo', but the type of z must
  match the operand type of the monoid.  The type in the string
  'add.mult.type' is the type of x for the multiply operator z=f(x,y), and
  the type of its z output defines the type of the monoid.
 
  Example:
 
    % valid semirings
    GrB.semiringinfo ('+.*.double') ;
    GrB.semiringinfo ('min.1st.int32') ;
 
    % invalid semiring (generates an error; since '<' is not a monoid)
    GrB.semiringinfo ('<.*.double') ;
 
  See also GrB.binopinfo, GrB.descriptorinfo, GrB.monoidinfo,
  GrB.selectopinfo, GrB.unopinfo.

Element-wise operations

Binary operators can be used in element-wise matrix operations, like C=A+B and C=A.*B. For the matrix addition C=A+B, the pattern of C is the set union of A and B, and the '+' operator is applied for entries in the intersection. Entries in A but not B, or in B but not A, are assigned to C without using the operator. The '+' operator is used for C=A+B but any operator can be used with GrB.eadd.

A = GrB (sprand (3, 3, 0.5)) ;
B = GrB (sprand (3, 3, 0.5)) ;
C1 = A + B
C2 = GrB.eadd ('+', A, B)
err = norm (C1-C2,1)
C1 =

  3x3 GraphBLAS double matrix, bitmap by col
  7 nonzeros, 7 entries

    (1,1)    0.666139
    (3,1)    0.735859
    (1,2)    1.47841
    (2,2)    0.146938
    (3,2)    0.566879
    (2,3)    0.248635
    (3,3)    0.104226


C2 =

  3x3 GraphBLAS double matrix, bitmap by col
  7 nonzeros, 7 entries

    (1,1)    0.666139
    (3,1)    0.735859
    (1,2)    1.47841
    (2,2)    0.146938
    (3,2)    0.566879
    (2,3)    0.248635
    (3,3)    0.104226

err =
     0

Subtracting two matrices

A-B and GrB.eadd ('-', A, B) are not the same thing, since the '-' operator is not applied to an entry that is in B but not A.

C1 = A-B
C2 = GrB.eadd ('-', A, B)
C1 =

  3x3 GraphBLAS double matrix, bitmap by col
  7 nonzeros, 7 entries

    (1,1)    -0.666139
    (3,1)    -0.735859
    (1,2)    -0.334348
    (2,2)    -0.146938
    (3,2)    0.566879
    (2,3)    0.248635
    (3,3)    0.104226


C2 =

  3x3 GraphBLAS double matrix, bitmap by col
  7 nonzeros, 7 entries

    (1,1)    0.666139
    (3,1)    0.735859
    (1,2)    -0.334348
    (2,2)    0.146938
    (3,2)    0.566879
    (2,3)    0.248635
    (3,3)    0.104226

But these give the same result

C1 = A-B
C2 = GrB.eadd ('+', A, GrB.apply ('-', B))
err = norm (C1-C2,1)
C1 =

  3x3 GraphBLAS double matrix, bitmap by col
  7 nonzeros, 7 entries

    (1,1)    -0.666139
    (3,1)    -0.735859
    (1,2)    -0.334348
    (2,2)    -0.146938
    (3,2)    0.566879
    (2,3)    0.248635
    (3,3)    0.104226


C2 =

  3x3 GraphBLAS double matrix, bitmap by col
  7 nonzeros, 7 entries

    (1,1)    -0.666139
    (3,1)    -0.735859
    (1,2)    -0.334348
    (2,2)    -0.146938
    (3,2)    0.566879
    (2,3)    0.248635
    (3,3)    0.104226

err =
     0

Element-wise 'multiplication'

For C = A.*B, the result C is the set intersection of the pattern of A and B. The operator is applied to entries in both A and B. Entries in A but not B, or B but not A, do not appear in the result C.

C1 = A.*B
C2 = GrB.emult ('*', A, B)
C3 = double (A) .* double (B)
C1 =

  3x3 GraphBLAS double matrix, bitmap by col
  1 nonzero, 1 entry

    (1,2)    0.518474


C2 =

  3x3 GraphBLAS double matrix, bitmap by col
  1 nonzero, 1 entry

    (1,2)    0.518474

C3 =
   (1,2)       0.5185

Just as in GrB.eadd, any operator can be used in GrB.emult:

A
B
C2 = GrB.emult ('max', A, B)
A =

  3x3 GraphBLAS double matrix, bitmap by col
  4 nonzeros, 4 entries

    (1,2)    0.572029
    (3,2)    0.566879
    (2,3)    0.248635
    (3,3)    0.104226


B =

  3x3 GraphBLAS double matrix, bitmap by col
  4 nonzeros, 4 entries

    (1,1)    0.666139
    (3,1)    0.735859
    (1,2)    0.906378
    (2,2)    0.146938


C2 =

  3x3 GraphBLAS double matrix, bitmap by col
  1 nonzero, 1 entry

    (1,2)    0.906378

Overloaded operators

The following operators all work as you would expect for any matrix. The matrices A and B can be GraphBLAS matrices, or MATLAB sparse or dense matrices, in any combination, or scalars where appropriate, The matrix M is logical (MATLAB or GraphBLAS):

  A+B   A-B  A*B   A.*B  A./B  A.\B  A.^b   A/b   C=A(I,J)  C(M)=A
  -A    +A   ~A    A'    A.'   A&B   A|B    b\A   C(I,J)=A  C=A(M)
  A~=B  A>B  A==B  A<=B  A>=B  A<B   [A,B]  [A;B] C(A)
  A(1:end,1:end)

For A^b, b must be a non-negative integer.

C1 = [A B] ;
C2 = [double(A) double(B)] ;
assert (isequal (double (C1), C2))
C1 = A^2
C2 = double (A)^2 ;
err = norm (C1 - C2, 1)
assert (err < 1e-12)
C1 =

  3x3 GraphBLAS double matrix, bitmap by col
  5 nonzeros, 5 entries

    (2,2)    0.140946
    (3,2)    0.0590838
    (1,3)    0.142227
    (2,3)    0.0259144
    (3,3)    0.151809

err =
     0
C1 = A (1:2,2:end)
A = double (A) ;
C2 = A (1:2,2:end) ;
assert (isequal (double (C1), C2))
C1 =

  2x2 GraphBLAS double matrix, bitmap by col
  2 nonzeros, 2 entries

    (1,1)    0.572029
    (2,2)    0.248635

Overloaded functions

Many MATLAB built-in functions can be used with GraphBLAS matrices:

A few differences with the built-in functions:

S = sparse (G)        % converts G to sparse/hypersparse
F = full (G)          % adds explicit zeros, so numel(F)==nnz(F)
F = full (G,type,id)  % adds explicit identity values to a GrB matrix
disp (G, level)       % display a GrB matrix G; level=2 is the default.

In the list below, the first set of Methods are overloaded built-in methods. They are used as-is on GraphBLAS matrices, such as C=abs(G). The Static methods are prefixed with "GrB.", as in C = GrB.apply ( ... ).

methods GrB
Methods for class GrB:

GrB             disp            ismatrix        real            
abs             display         isnan           repmat          
acos            dmperm          isnumeric       reshape         
acosh           double          isreal          round           
acot            eig             isscalar        sec             
acoth           end             issparse        sech            
acsc            eps             issymmetric     sign            
acsch           eq              istril          sin             
all             erf             istriu          single          
amd             erfc            isvector        sinh            
and             etree           kron            size            
angle           exp             ldivide         sparse          
any             expm1           le              spfun           
asec            false           length          spones          
asech           find            log             sprand          
asin            fix             log10           sprandn         
asinh           flip            log1p           sprandsym       
assert          floor           log2            sprintf         
atan            fprintf         logical         sqrt            
atan2           full            lt              struct          
atanh           gamma           mat2cell        subsasgn        
bandwidth       gammaln         max             subsindex       
bitand          ge              min             subsref         
bitcmp          graph           minus           sum             
bitget          gt              mldivide        symamd          
bitor           horzcat         mpower          symrcm          
bitset          hypot           mrdivide        tan             
bitshift        imag            mtimes          tanh            
bitxor          int16           ne              times           
cat             int32           nnz             transpose       
ceil            int64           nonzeros        tril            
colamd          int8            norm            triu            
complex         isa             not             true            
conj            isbanded        num2cell        uint16          
cos             isdiag          numel           uint32          
cosh            isempty         nzmax           uint64          
cot             isequal         ones            uint8           
coth            isfinite        or              uminus          
csc             isfloat         plus            uplus           
csch            ishermitian     pow2            vertcat         
ctranspose      isinf           power           xor             
diag            isinteger       prod            zeros           
digraph         islogical       rdivide         

Static methods:

MATLAB_vs_GrB   empty           kronecker       save            
apply           emult           ktruss          select          
apply2          entries         laplacian       selectopinfo    
assign          expand          load            semiringinfo    
bfs             extract         mis             speye           
binopinfo       extracttuples   monoidinfo      subassign       
build           eye             mxm             threads         
burble          finalize        nonz            trans           
cell2mat        format          normdiff        tricount        
chunk           incidence       offdiag         type            
clear           init            optype          unopinfo        
compact         isbycol         pagerank        ver             
descriptorinfo  isbyrow         prune           version         
dnn             isfull          random          vreduce         
eadd            issigned        reduce          

Zeros are handled differently

Explicit zeros cannot be automatically dropped from a GraphBLAS matrix, like they are in MATLAB sparse matrices. In a shortest-path problem, for example, an edge A(i,j) that is missing has an infinite weight, (the monoid identity of min(x,y) is +inf). A zero edge weight A(i,j)=0 is very different from an entry that is not present in A. However, if a GraphBLAS matrix is converted into a MATLAB sparse matrix, explicit zeros are dropped, which is the convention for a MATLAB sparse matrix. They can also be dropped from a GraphBLAS matrix using the GrB.select method.

G = GrB (magic (2)) ;
G (1,1) = 0      % G(1,1) still appears as an explicit entry
A = double (G)   % but it's dropped when converted to MATLAB sparse
H = GrB.select ('nonzero', G)  % drops the explicit zeros from G
fprintf ('nnz (G): %d  nnz (A): %g nnz (H): %g\n', ...
    nnz (G), nnz (A), nnz (H)) ;
fprintf ('num entries in G: %d\n', GrB.entries (G)) ;
G =

  2x2 GraphBLAS double matrix, full by col
  3 nonzeros, 4 entries

    (1,1)    0
    (2,1)    4
    (1,2)    3
    (2,2)    2

A =
     0     3
     4     2

H =

  2x2 GraphBLAS double matrix, bitmap by col
  3 nonzeros, 3 entries

    (2,1)    4
    (1,2)    3
    (2,2)    2

nnz (G): 3  nnz (A): 3 nnz (H): 3
num entries in G: 4

Displaying contents of a GraphBLAS matrix

Unlike MATLAB, the default is to display just a few entries of a GrB matrix. Here are all 100 entries of a 10-by-10 matrix, using a non-default disp(G,3):

G = GrB (rand (10)) ;
% display everything:
disp (G,3)
G =

  10x10 GraphBLAS double matrix, full by col
  100 nonzeros, 100 entries

    (1,1)    0.0342763
    (2,1)    0.17802
    (3,1)    0.887592
    (4,1)    0.889828
    (5,1)    0.769149
    (6,1)    0.00497062
    (7,1)    0.735693
    (8,1)    0.488349
    (9,1)    0.332817
    (10,1)    0.0273313
    (1,2)    0.467212
    (2,2)    0.796714
    (3,2)    0.849463
    (4,2)    0.965361
    (5,2)    0.902248
    (6,2)    0.0363252
    (7,2)    0.708068
    (8,2)    0.322919
    (9,2)    0.700716
    (10,2)    0.472957
    (1,3)    0.204363
    (2,3)    0.00931977
    (3,3)    0.565881
    (4,3)    0.183435
    (5,3)    0.00843818
    (6,3)    0.284938
    (7,3)    0.706156
    (8,3)    0.909475
    (9,3)    0.84868
    (10,3)    0.564605
    (1,4)    0.075183
    (2,4)    0.535293
    (3,4)    0.072324
    (4,4)    0.515373
    (5,4)    0.926149
    (6,4)    0.949252
    (7,4)    0.0478888
    (8,4)    0.523767
    (9,4)    0.167203
    (10,4)    0.28341
    (1,5)    0.122669
    (2,5)    0.441267
    (3,5)    0.157113
    (4,5)    0.302479
    (5,5)    0.758486
    (6,5)    0.910563
    (7,5)    0.0246916
    (8,5)    0.232421
    (9,5)    0.38018
    (10,5)    0.677531
    (1,6)    0.869074
    (2,6)    0.471459
    (3,6)    0.624929
    (4,6)    0.987186
    (5,6)    0.282885
    (6,6)    0.843833
    (7,6)    0.869597
    (8,6)    0.308209
    (9,6)    0.201332
    (10,6)    0.706603
    (1,7)    0.563222
    (2,7)    0.575795
    (3,7)    0.056376
    (4,7)    0.73412
    (5,7)    0.608022
    (6,7)    0.0400164
    (7,7)    0.540801
    (8,7)    0.023064
    (9,7)    0.165682
    (10,7)    0.250393
    (1,8)    0.23865
    (2,8)    0.232033
    (3,8)    0.303191
    (4,8)    0.579934
    (5,8)    0.267751
    (6,8)    0.916376
    (7,8)    0.833499
    (8,8)    0.978692
    (9,8)    0.734445
    (10,8)    0.102896
    (1,9)    0.353059
    (2,9)    0.738955
    (3,9)    0.57539
    (4,9)    0.751433
    (5,9)    0.93256
    (6,9)    0.281622
    (7,9)    0.51302
    (8,9)    0.24406
    (9,9)    0.950086
    (10,9)    0.303638
    (1,10)    0.563593
    (2,10)    0.705101
    (3,10)    0.0604146
    (4,10)    0.672065
    (5,10)    0.359793
    (6,10)    0.62931
    (7,10)    0.977758
    (8,10)    0.394328
    (9,10)    0.765651
    (10,10)    0.457809


That was disp(G,3), so every entry was printed. It's a little long, so the default is not to print everything.

With the default display (level = 2):

G
G =

  10x10 GraphBLAS double matrix, full by col
  100 nonzeros, 100 entries

    (1,1)    0.0342763
    (2,1)    0.17802
    (3,1)    0.887592
    (4,1)    0.889828
    (5,1)    0.769149
    (6,1)    0.00497062
    (7,1)    0.735693
    (8,1)    0.488349
    (9,1)    0.332817
    (10,1)    0.0273313
    (1,2)    0.467212
    (2,2)    0.796714
    (3,2)    0.849463
    (4,2)    0.965361
    (5,2)    0.902248
    (6,2)    0.0363252
    (7,2)    0.708068
    (8,2)    0.322919
    (9,2)    0.700716
    (10,2)    0.472957
    (1,3)    0.204363
    (2,3)    0.00931977
    (3,3)    0.565881
    (4,3)    0.183435
    (5,3)    0.00843818
    (6,3)    0.284938
    (7,3)    0.706156
    (8,3)    0.909475
    (9,3)    0.84868
    ...

That was disp(G,2) or just display(G), which is what is printed by a MATLAB statement that doesn't have a trailing semicolon. With level = 1, disp(G,1) gives just a terse summary:

disp (G,1)
G =

  10x10 GraphBLAS double matrix, full by col
  100 nonzeros, 100 entries


Storing a matrix by row or by column

MATLAB stores its sparse matrices by column, refered to as 'sparse by col' in SuiteSparse:GraphBLAS. In the 'sparse by col' format, each column of the matrix is stored as a list of entries, with their value and row index. In the 'sparse by row' format, each row is stored as a list of values and their column indices. GraphBLAS uses both 'by row' and 'by col', and the two formats can be intermixed arbitrarily. In its C interface, the default format is 'by row'. However, for better compatibility with MATLAB, the SuiteSparse:GraphBLAS MATLAB interface uses 'by col' by default instead.

rng ('default') ;
GrB.clear ;                      % clear prior GraphBLAS settings
fprintf ('the default format is: %s\n', GrB.format) ;
C = sparse (rand (2))
G = GrB (C)
GrB.format (G)
the default format is: by col
C =
   (1,1)       0.8147
   (2,1)       0.9058
   (1,2)       0.1270
   (2,2)       0.9134

G =

  2x2 GraphBLAS double matrix, full by col
  4 nonzeros, 4 entries

    (1,1)    0.814724
    (2,1)    0.905792
    (1,2)    0.126987
    (2,2)    0.913376

ans =
    'by col'

Many graph algorithms work better in 'by row' format, with matrices stored by row. For example, it is common to use A(i,j) for the edge (i,j), and many graph algorithms need to access the out-adjacencies of nodes, which is the row A(i,;) for node i. If the 'by row' format is desired, GrB.format ('by row') tells GraphBLAS to create all subsequent matrices in the 'by row' format. Converting from a MATLAB sparse matrix (in standard 'by col' format) takes a little more time (requiring a transpose), but subsequent graph algorithms can be faster.

G = GrB (C, 'by row')
fprintf ('the format of G is:    %s\n', GrB.format (G)) ;
H = GrB (C)
fprintf ('the format of H is:    %s\n', GrB.format (H)) ;
err = norm (H-G,1)
G =

  2x2 GraphBLAS double matrix, full by row
  4 nonzeros, 4 entries

    (1,1)    0.814724
    (1,2)    0.126987
    (2,1)    0.905792
    (2,2)    0.913376

the format of G is:    by row

H =

  2x2 GraphBLAS double matrix, full by col
  4 nonzeros, 4 entries

    (1,1)    0.814724
    (2,1)    0.905792
    (1,2)    0.126987
    (2,2)    0.913376

the format of H is:    by col
err =
     0

Hypersparse, sparse, bitmap, and full matrices

SuiteSparse:GraphBLAS can use four kinds of sparse matrix data structures: hypersparse, sparse, bitmap, and full, in both 'by col' and 'by row' formats, for a total of eight different combinations. In the 'sparse by col' that MATLAB uses for its sparse matrices, an m-by-n matrix A takes O(n+nnz(A)) space. MATLAB can create huge column vectors, but not huge matrices (when n is huge).

clear
[c, huge] = computer ;
C = sparse (huge, 1)    % MATLAB can create a huge-by-1 sparse column
try
    C = sparse (huge, huge)     % but this fails
catch me
    error_expected = me
end
C =
   All zero sparse: 281474976710655x1
error_expected = 
  MException with properties:

    identifier: 'MATLAB:array:SizeLimitExceeded'
       message: 'Requested 281474976710655x281474976710655 (2097152.0GB) array exceeds maximum array size preference. Creation of arrays greater than this limit may take a long time and cause MATLAB to become unresponsive.'
         cause: {}
         stack: [4x1 struct]
    Correction: []

In a GraphBLAS hypersparse matrix, an m-by-n matrix A takes only O(nnz(A)) space. The difference can be huge if nnz (A) << n.

clear
[c, huge] = computer ;
G = GrB (huge, 1)            % no problem for GraphBLAS
H = GrB (huge, huge)         % this works in GraphBLAS too
G =

  281474976710655x1 GraphBLAS double matrix, sparse by col
  no nonzeros, no entries


H =

  281474976710655x281474976710655 GraphBLAS double matrix, hypersparse by col
  no nonzeros, no entries

Operations on huge hypersparse matrices are very fast; no component of the time or space complexity is Omega(n).

I = randperm (huge, 2) ;
J = randperm (huge, 2) ;
H (I,J) = magic (2) ;        % add 4 nonzeros to random locations in H
H (I,I) = 10 * [1 2 ; 3 4] ; % so H^2 is not all zero
H = H^2 ;                    % square H
H = (H' * 2) ;               % transpose H and double the entries
K = pi * spones (H) ;
H = H + K                    % add pi to each entry in H
H =

  281474976710655x281474976710655 GraphBLAS double matrix, hypersparse by col
  8 nonzeros, 8 entries

    (27455183225557,27455183225557)    4403.14
    (78390279669562,27455183225557)    383.142
    (153933462881710,27455183225557)    343.142
    (177993304104065,27455183225557)    3003.14
    (27455183225557,177993304104065)    2003.14
    (78390279669562,177993304104065)    183.142
    (153933462881710,177993304104065)    143.142
    (177993304104065,177993304104065)    1403.14

numel uses vpa if the matrix is really huge

e1 = numel (G)               % this is huge, but still a flint
e2 = numel (H)               % this is huge^2, which needs vpa
whos e1 e2
e1 =
   2.8147e+14
e2 =
79228162514263774643590529025.0
  Name      Size            Bytes  Class     Attributes

  e1        1x1                 8  double              
  e2        1x1                 8  sym                 

All of these matrices take very little memory space:

whos C G H K
  Name                    Size                         Bytes  Class    Attributes

  G         281474976710655x1                            948  GrB                
  H         281474976710655x281474976710655             1268  GrB                
  K         281474976710655x281474976710655             1268  GrB                

The mask and accumulator

When not used in overloaded operators or built-in functions, many GraphBLAS methods of the form GrB.method ( ... ) can optionally use a mask and/or an accumulator operator. If the accumulator is '+' in GrB.mxm, for example, then C = C + A*B is computed. The mask acts much like logical indexing in MATLAB. With a logical mask matrix M, C<M>=A*B allows only part of C to be assigned. If M(i,j) is true, then C(i,j) can be modified. If false, then C(i,j) is not modified.

For example, to set all values in C that are greater than 0.5 to 3:

A = rand (3)
C = GrB.assign (A, A > 0.5, 3) ;     % in GraphBLAS
C1 = GrB (A) ; C1 (A > .5) = 3       % also in GraphBLAS
C2 = A       ; C2 (A > .5) = 3       % in MATLAB
err = norm (C - C1, 1)
err = norm (C - C2, 1)
A =
    0.9575    0.9706    0.8003
    0.9649    0.9572    0.1419
    0.1576    0.4854    0.4218

C1 =

  3x3 GraphBLAS double matrix, full by col
  9 nonzeros, 9 entries

    (1,1)    3
    (2,1)    3
    (3,1)    0.157613
    (1,2)    3
    (2,2)    3
    (3,2)    0.485376
    (1,3)    3
    (2,3)    0.141886
    (3,3)    0.421761

C2 =
    3.0000    3.0000    3.0000
    3.0000    3.0000    0.1419
    0.1576    0.4854    0.4218
err =
     0
err =
     0

The descriptor

Most GraphBLAS functions of the form GrB.method ( ... ) take an optional last argument, called the descriptor. It is a MATLAB struct that can modify the computations performed by the method. 'help GrB.descriptorinfo' gives all the details. The following is a short summary of the primary settings:

d.out = 'default' or 'replace', clears C after the accum op is used.

d.mask = 'default' or 'complement', to use M or ~M as the mask matrix; 'structural', or 'structural complement', to use the pattern of M or ~M.

d.in0 = 'default' or 'transpose', to transpose A for C=A*B, C=A+B, etc.

d.in1 = 'default' or 'transpose', to transpose B for C=A*B, C=A+B, etc.

d.kind = 'default', 'GrB', 'sparse', or 'full'; the output of GrB.method.

A = sparse (rand (2)) ;
B = sparse (rand (2)) ;
C1 = A'*B ;
C2 = GrB.mxm ('+.*', A, B, struct ('in0', 'transpose')) ;
err = norm (C1-C2,1)
err =
     0

Integer arithmetic is different in GraphBLAS

MATLAB supports integer arithmetic on its full matrices, using int8, int16, int32, int64, uint8, uint16, uint32, or uint64 data types. None of these integer data types can be used to construct a MATLAB sparse matrix, which can only be double, double complex, or logical. Furthermore, C=A*B is not defined for integer types in MATLAB, except when A and/or B are scalars.

GraphBLAS supports all of those types for all of its matrices (hyper, sparse, bitmap, or full). All operations are supported, including C=A*B when A or B are any integer type, in 1000s of semirings.

However, integer arithmetic differs in GraphBLAS and MATLAB. In MATLAB, integer values saturate if they exceed their maximum value. In GraphBLAS, integer operators act in a modular fashion. The latter is essential when computing C=A*B over a semiring. A saturating integer operator cannot be used as a monoid since it is not associative.

C = uint8 (magic (3)) ;
G = GrB (C) ;
C1 = C * 40
C2 = G * uint8 (40)
S = double (C1 < 255) ;
assert (isequal (double (C1).*S, double (C2).*S))
C1 =
  3x3 uint8 matrix
   255    40   240
   120   200   255
   160   255    80

C2 =

  3x3 GraphBLAS uint8_t matrix, full by col
  9 nonzeros, 9 entries

    (1,1)   64
    (2,1)   120
    (3,1)   160
    (1,2)   40
    (2,2)   200
    (3,2)   104
    (1,3)   240
    (2,3)   24
    (3,3)   80

An example graph algorithm: breadth-first search

The breadth-first search of a graph finds all nodes reachable from the source node, and their level, v. v=GrB.bfs(A,s) or v=bfs_matlab(A,s) compute the same thing, but GrB.bfs uses GraphBLAS matrices and operations, while bfs_matlab uses pure MATLAB operations. v is defined as v(s) = 1 for the source node, v(i) = 2 for nodes adjacent to the source, and so on.

clear
rng ('default') ;
n = 1e5 ;
A = logical (sprandn (n, n, 1e-3)) ;

tic
v1 = GrB.bfs (A, 1) ;
gb_time = toc ;

tic
v2 = bfs_matlab (A, 1) ;
matlab_time = toc ;

assert (isequal (double (v1'), v2))
fprintf ('\nnodes reached: %d of %d\n', nnz (v2), n) ;
fprintf ('GraphBLAS time: %g sec\n', gb_time) ;
fprintf ('MATLAB time:    %g sec\n', matlab_time) ;
fprintf ('Speedup of GraphBLAS over MATLAB: %g\n', ...
    matlab_time / gb_time) ;
fprintf ('\n# of threads used by GraphBLAS: %d\n', GrB.threads) ;
nodes reached: 100000 of 100000
GraphBLAS time: 0.325293 sec
MATLAB time:    0.470943 sec
Speedup of GraphBLAS over MATLAB: 1.44775

# of threads used by GraphBLAS: 4

Example graph algorithm: Luby's method in GraphBLAS

The GrB.mis function is variant of Luby's randomized algorithm [Luby 1985]. It is a parallel method for finding an maximal independent set of nodes, where no two nodes are adjacent. See the GraphBLAS/@GrB/mis.m function for details. The graph must be symmetric with a zero-free diagonal, so A is symmetrized first and any diagonal entries are removed.

A = GrB (A) ;
A = GrB.offdiag (A|A') ;

tic
s = GrB.mis (A) ;
toc
fprintf ('# nodes in the graph: %g\n', size (A,1)) ;
fprintf ('# edges: : %g\n', GrB.entries (A) / 2) ;
fprintf ('size of maximal independent set found: %g\n', ...
    full (double (sum (s)))) ;

% make sure it's independent
p = find (s) ;
S = A (p,p) ;
assert (GrB.entries (S) == 0)

% make sure it's maximal
notp = find (s == 0) ;
S = A (notp, p) ;
deg = GrB.vreduce ('+.int64', S) ;
assert (logical (all (deg > 0)))
Elapsed time is 0.282035 seconds.
# nodes in the graph: 100000
# edges: : 9.9899e+06
size of maximal independent set found: 2811

Sparse deep neural network

The 2019 MIT GraphChallenge (see http://graphchallenge.org) is to solve a set of large sparse deep neural network problems. In this demo, the MATLAB reference solution is compared with a solution using GraphBLAS, for a randomly constructed neural network. See the GrB.dnn and dnn_matlab.m functions for details.

clear
rng ('default') ;
nlayers = 16 ;
nneurons = 4096 ;
nfeatures = 30000 ;
fprintf ('# layers:   %d\n', nlayers) ;
fprintf ('# neurons:  %d\n', nneurons) ;
fprintf ('# features: %d\n', nfeatures) ;
fprintf ('# of threads used: %d\n', GrB.threads) ;

tic
Y0 = sprand (nfeatures, nneurons, 0.1) ;
for layer = 1:nlayers
    W {layer} = sprand (nneurons, nneurons, 0.01) * 0.2 ;
    bias {layer} = -0.2 * ones (1, nneurons) ;
end
t_setup = toc ;
fprintf ('construct problem time: %g sec\n', t_setup) ;

% convert the problem from MATLAB to GraphBLAS
t = tic ;
[W_gb, bias_gb, Y0_gb] = dnn_mat2gb (W, bias, Y0) ;
t = toc (t) ;
fprintf ('setup time: %g sec\n', t) ;
# layers:   16
# neurons:  4096
# features: 30000
# of threads used: 4
construct problem time: 4.68352 sec
setup time: 0.282884 sec

Solving the sparse deep neural network problem with GraphbLAS

Please wait ...

tic
Y1 = GrB.dnn (W_gb, bias_gb, Y0_gb) ;
gb_time = toc ;
fprintf ('total time in GraphBLAS: %g sec\n', gb_time) ;
total time in GraphBLAS: 10.9011 sec

Solving the sparse deep neural network problem with MATLAB

Please wait ...

tic
Y2 = dnn_matlab (W, bias, Y0) ;
matlab_time = toc ;
fprintf ('total time in MATLAB:    %g sec\n', matlab_time) ;
fprintf ('Speedup of GraphBLAS over MATLAB: %g\n', ...
    matlab_time / gb_time) ;
fprintf ('\n# of threads used by GraphBLAS: %d\n', GrB.threads) ;

err = norm (Y1-Y2,1)
total time in MATLAB:    91.4765 sec
Speedup of GraphBLAS over MATLAB: 8.39146

# of threads used by GraphBLAS: 4
err =
     0

For objects, GraphBLAS has better colon notation than MATLAB

The MATLAB notation C = A (start:inc:fini) is very handy, and it works great if A is a MATLAB matrix. But for objects like the GraphBLAS matrix, MATLAB starts by creating the explicit index vector I = start:inc:fini. That's fine if the matrix is modest in size, but GraphBLAS can construct huge matrices. The problem is that 1:n cannot be explicitly constructed when n is huge.

The C API for GraphBLAS can represent the colon notation start:inc:fini in an implicit manner, so it can do the indexing without actually forming the explicit list I = start:inc:fini. But there is no access to this method using the MATLAB notation start:inc:fini.

Thus, to compute C = A (start:inc:fini) for very huge matrices, you need to use use a cell array to represent the colon notation, as { start, inc, fini }, instead of start:inc:fini. See 'help GrB.extract', 'help GrB.assign' for the functional form. For the overloaded syntax C(I,J)=A and C=A(I,J), see 'help GrB/subsasgn' and 'help GrB/subsref'. The cell array syntax isn't conventional, but it is far faster than the MATLAB colon notation for objects, and takes far less memory when I is huge.

n = 1e14 ;
H = GrB (n, n) ;            % a huge empty matrix
I = [1 1e9 1e12 1e14] ;
M = magic (4)
H (I,I) = M ;
J = {1, 1e13} ;            % represents 1:1e13 colon notation
C1 = H (J, J)              % computes C1 = H (1:e13,1:1e13)
c = nonzeros (C1) ;
m = nonzeros (M (1:3, 1:3)) ;
assert (isequal (c, m)) ;
M =
    16     2     3    13
     5    11    10     8
     9     7     6    12
     4    14    15     1

C1 =

  10000000000000x10000000000000 GraphBLAS double matrix, hypersparse by col
  9 nonzeros, 9 entries

    (1,1)    16
    (1000000000,1)    5
    (1000000000000,1)    9
    (1,1000000000)    2
    (1000000000,1000000000)    11
    (1000000000000,1000000000)    7
    (1,1000000000000)    3
    (1000000000,1000000000000)    10
    (1000000000000,1000000000000)    6

try
    % try to compute the same thing with colon
    % notation (1:1e13), but this fails:
    C2 = H (1:1e13, 1:1e13)
catch me
    error_expected = me
end
error_expected = 
  MException with properties:

    identifier: 'MATLAB:array:SizeLimitExceeded'
       message: 'Requested 10000000000000x1 (74505.8GB) array exceeds maximum array size preference. Creation of arrays greater than this limit may take a long time and cause MATLAB to become unresponsive.'
         cause: {}
         stack: [4x1 struct]
    Correction: []

Iterative solvers work as-is

Many built-in functions work with GraphBLAS matrices unmodified.

A = sparse (rand (4)) ;
b = sparse (rand (4,1)) ;
x = gmres (A,b)
norm (A*x-b)
x = gmres (GrB(A), GrB(b))
norm (A*x-b)
gmres converged at iteration 4 to a solution with relative residual 0.
x =
    0.9105
    3.8949
   -0.5695
   -1.3867
ans =
   8.6711e-16
gmres converged at iteration 4 to a solution with relative residual 0.
x =
    0.9105
    3.8949
   -0.5695
   -1.3867
ans =
   7.2802e-16

... even in single precision

x = gmres (GrB(A,'single'), GrB(b,'single'))
norm (A*x-b)
gmres converged at iteration 4 to a solution with relative residual 0.
x =
    0.9105
    3.8949
   -0.5695
   -1.3867
ans =
   8.3369e-08

Both of the following uses of minres (A,b) fail to converge because A is not symmetric, as the method requires. Both failures are correctly reported, and both the MATLAB version and the GraphBLAS version return the same incorrect vector x.

x = minres (A, b)
x = minres (GrB(A), GrB(b))
minres stopped at iteration 4 without converging to the desired tolerance 1e-06
because the maximum number of iterations was reached.
The iterate returned (number 4) has relative residual 0.21.
x =
    0.2489
    0.2081
    0.0700
    0.3928
minres stopped at iteration 4 without converging to the desired tolerance 1e-06
because the maximum number of iterations was reached.
The iterate returned (number 4) has relative residual 0.21.

x =

  4x1 GraphBLAS double matrix, full by col
  4 nonzeros, 4 entries

    (1,1)    0.248942
    (2,1)    0.208128
    (3,1)    0.0699707
    (4,1)    0.392812

With a proper symmetric matrix

A = A+A' ;
x = minres (A, b)
norm (A*x-b)
x = minres (GrB(A), GrB(b))
norm (A*x-b)
minres converged at iteration 4 to a solution with relative residual 1.3e-11.
x =
 -114.0616
   -1.4211
  134.8227
    2.0694
ans =
   1.3650e-11
minres converged at iteration 4 to a solution with relative residual 1.3e-11.

x =

  4x1 GraphBLAS double matrix, full by col
  4 nonzeros, 4 entries

    (1,1)    -114.062
    (2,1)    -1.4211
    (3,1)    134.823
    (4,1)    2.0694

ans =
   1.3650e-11

Extreme performance differences between GraphBLAS and MATLAB.

The GraphBLAS operations used so far are perhaps 2x to 50x faster than the corresponding MATLAB operations, depending on how many cores your computer has. To run a demo illustrating a 500x or more speedup versus MATLAB, run this demo:

  gbdemo2

It will illustrate an assignment C(I,J)=A that can take under a second in GraphBLAS but several minutes in MATLAB. To make the comparsion even more dramatic, try:

  gbdemo2 (20000)

assuming you have enough memory.

Sparse logical indexing is much, much faster in GraphBLAS

The mask in GraphBLAS acts much like logical indexing in MATLAB, but it is not quite the same. MATLAB logical indexing takes the form:

     C (M) = A (M)

which computes the same thing as the GraphBLAS statement:

     C = GrB.assign (C, M, A)

The GrB.assign statement computes C(M)=A(M), and it is vastly faster than C(M)=A(M) for MATLAB sparse matrices, even if the time to convert the GrB matrix back to a MATLAB sparse matrix is included.

GraphBLAS can also compute C(M)=A(M) using overloaded operators for subsref and subsasgn, but C = GrB.assign (C, M, A) is a bit faster.

Here are both methods in GraphBLAS (both are very fast). Setting up:

clear
n = 4000 ;
tic
C = sprand (n, n, 0.1) ;
A = 100 * sprand (n, n, 0.1) ;
M = (C > 0.5) ;
t_setup = toc ;
fprintf ('nnz(C): %g, nnz(M): %g, nnz(A): %g\n', ...
    nnz(C), nnz(M), nnz(A)) ;
fprintf ('\nsetup time:     %g sec\n', t_setup) ;
nnz(C): 1.5226e+06, nnz(M): 761163, nnz(A): 1.52245e+06

setup time:     0.664685 sec

First method in GraphBLAS, with GrB.assign

Including the time to convert C1 from a GraphBLAS matrix to a MATLAB sparse matrix:

tic
C1 = GrB.assign (C, M, A) ;
C1 = double (C1) ;
gb_time = toc ;
fprintf ('\nGraphBLAS time: %g sec for GrB.assign\n', gb_time) ;
GraphBLAS time: 0.023441 sec for GrB.assign

Second method in GraphBLAS, with C(M)=A(M)

now using overloaded operators, also include the time to convert back to a MATLAB sparse matrix, for good measure:

A2 = GrB (A) ;
C2 = GrB (C) ;
tic
C2 (M) = A2 (M) ;
C2 = double (C2) ;
gb_time2 = toc ;
fprintf ('\nGraphBLAS time: %g sec for C(M)=A(M)\n', gb_time2) ;
GraphBLAS time: 0.132812 sec for C(M)=A(M)

Now with MATLAB matrices, with C(M)=A(M)

Please wait, this will take about 10 minutes or so ...

tic
C (M) = A (M) ;
matlab_time = toc ;

fprintf ('\nGraphBLAS time: %g sec (GrB.assign)\n', gb_time) ;
fprintf ('GraphBLAS time: %g sec (overloading)\n', gb_time2) ;
fprintf ('MATLAB time:    %g sec\n', matlab_time) ;
fprintf ('Speedup of GraphBLAS (overloading) over MATLAB: %g\n', ...
    matlab_time / gb_time2) ;
fprintf ('Speedup of GraphBLAS (GrB.assign)  over MATLAB: %g\n', ...
    matlab_time / gb_time) ;
fprintf ('\n# of threads used by GraphBLAS: %d\n', GrB.threads) ;

assert (isequal (C1, C))
assert (isequal (C2, C))
fprintf ('Results of GrB and MATLAB match perfectly.\n')
GraphBLAS time: 0.023441 sec (GrB.assign)
GraphBLAS time: 0.132812 sec (overloading)
MATLAB time:    632.374 sec
Speedup of GraphBLAS (overloading) over MATLAB: 4761.42
Speedup of GraphBLAS (GrB.assign)  over MATLAB: 26977.3

# of threads used by GraphBLAS: 4
Results of GrB and MATLAB match perfectly.

Limitations and their future solutions

The MATLAB interface for SuiteSparse:GraphBLAS is a work-in-progress. It has some limitations, most of which will be resolved over time.

(1) Nonblocking mode:

GraphBLAS has a 'non-blocking' mode, in which operations can be left pending and completed later. SuiteSparse:GraphBLAS uses the non-blocking mode to speed up a sequence of assignment operations, such as C(I,J)=A. However, in its MATLAB interface, this would require a MATLAB mexFunction to modify its inputs. That breaks the MATLAB API standard, so it cannot be safely done. As a result, using GraphBLAS via its MATLAB interface can be slower than when using its C API.

(2) Integer element-wise operations:

Integer operations in MATLAB saturate, so that uint8(255)+1 is 255. To allow for integer monoids, GraphBLAS uses modular arithmetic instead. This is the only way that C=A*B can be defined for integer semirings. However, saturating integer operators could be added in the future, so that element- wise integer operations on GraphBLAS sparse integer matrices could work just the same as their MATLAB counterparts.

So in the future, you could perhaps write this, for both sparse and dense integer matrices A and B:

     C = GrB.eadd ('+saturate.int8', A, B)

to compute the same thing as C=A+B in MATLAB for its full int8 matrices. Note that MATLAB can do this only for dense integer matrices, since it doesn't support sparse integer matrices.

(3) Faster methods:

Most methods in this MATLAB interface are based on efficient parallel C functions in GraphBLAS itself, and are typically as fast or faster than the equivalent built-in operators and functions in MATLAB.

There are few notable exceptions; these will be addressed in the future. These include bandwidth, istriu, istril, isdiag, reshape, issymmetric, and ishermitian, all of which should be faster in a future release.

Here is an example that illustrates the performance of istril.

A = sparse (rand (2000)) ;
tic
c1 = istril (A) ;
matlab_time = toc ;
A = GrB (A) ;
tic
c2 = istril (A) ;
gb_time = toc ;
fprintf ('\nMATLAB: %g sec, GraphBLAS: %g sec\n', ...
    matlab_time, gb_time) ;
if (gb_time > matlab_time)
    fprintf ('GraphBLAS is slower by a factor of %g\n', ...
        gb_time / matlab_time) ;
end
MATLAB: 0.000142 sec, GraphBLAS: 0.011394 sec
GraphBLAS is slower by a factor of 80.2394

(4) Linear indexing:

If A is an m-by-n 2D MATLAB matrix, with n > 1, A(:) is a column vector of length m*n. The index operation A(i) accesses the ith entry in the vector A(:). This is called linear indexing in MATLAB. It is not yet available for GraphBLAS matrices in this MATLAB interface to GraphBLAS, but will be added in the future.

(5) Implicit singleton dimension expansion

In MATLAB C=A+B where A is m-by-n and B is a 1-by-n row vector implicitly expands B to a matrix, computing C(i,j)=A(i,j)+B(j). This implicit expansion is not yet suported in GraphBLAS with C=A+B. However, it can be done with C = GrB.mxm ('+.+', A, diag(GrB(B))). That's a nice example of the power of semirings, but it's not immediately obvious, and not as clear a syntax as C=A+B. The GraphBLAS/@GrB/dnn.m function uses this 'plus.plus' semiring to apply the bias to each neuron.

A = magic (3)
B = 1000:1000:3000
C1 = A + B
C2 = GrB.mxm ('+.+', A, diag (GrB (B)))
err = norm (C1-C2,1)
A =
     8     1     6
     3     5     7
     4     9     2
B =
        1000        2000        3000
C1 =
        1008        2001        3006
        1003        2005        3007
        1004        2009        3002

C2 =

  3x3 GraphBLAS double matrix, bitmap by col
  9 nonzeros, 9 entries

    (1,1)    1008
    (2,1)    1003
    (3,1)    1004
    (1,2)    2001
    (2,2)    2005
    (3,2)    2009
    (1,3)    3006
    (2,3)    3007
    (3,3)    3002

err =
     0

(6) MATLAB object overhead.

The GrB matrix is a MATLAB object, and there are some cases where performance issues can arise as a result. Extracting the contents of a MATLAB object (G.field) takes much more time than for a MATLAB struct with % the same syntax, and building an object has similar issues. The difference is small, and it does not affect large problems. But if you have many calls to GrB operations with a small amount of work, then the time can be dominated by the MATLAB object-oriented overhead.

There is no solution or workaround to this issue.

A = rand (3,4) ;
G = GrB (A) ;
tic
for k = 1:100000
    [m, n] = size (A) ;
end
toc
tic
for k = 1:100000
    [m, n] = size (G) ;
end
toc
Elapsed time is 0.036292 seconds.
Elapsed time is 0.614848 seconds.

GraphBLAS operations

In addition to the overloaded operators (such as C=A*B) and overloaded functions (such as L=tril(A)), GraphBLAS also has methods of the form GrB.method. Most of them take an optional input matrix Cin, which is the initial value of the matrix C for the expression below, an optional mask matrix M, and an optional accumulator operator.

in GrB syntax:  C<#M,replace> = accum (C, A*B)
in @GrB MATLAB: C = GrB.mxm (Cin, M, accum, semiring, A, B, desc) ;

In the above expression, #M is either empty (no mask), M (with a mask matrix) or ~M (with a complemented mask matrix), as determined by the descriptor (desc). 'replace' can be used to clear C after it is used in accum(C,T) but before it is assigned with C<...> = Z, where Z=accum(C,T). The matrix T is the result of some operation, such as T=A*B for GrB.mxm, or T=op(A,B) for GrB.eadd.

For a complete list of GraphBLAS overloaded operators and methods, type:

help GrB

Thanks for watching!

Tim Davis, Texas A&M University, http://faculty.cse.tamu.edu/davis, https://twitter.com/DocSparse