The main reason for creating a DivMask
is to permit a quick, coarse
test of divisibility between power products -- but before you read on, you
might like to consider using PPWithMask
instead, which offers
essentially the same advantages with a *much more convenient interface*.
We say that DivMask
s permit a "coarse" test because we accept as
responses definitely not divisible or possibly divisible (but
further checks must be conducted to decide for certain).
For example the radical of a PP .... (WORK-IN-PROGRESS)
DivMask
s are
a fairly low-level concept, and probably of little use to most normal
CoCoALib users. If you need to do conduct a great many divisibility tests
(between power products) and think you're interested, read on (assuming you
have already decided that PPWithMask
does not fulfill your needs).
Note: currently DivMask
s cannot be used to ascertain coprimality (see Bugs section).
To use DivMask
s you must master two concepts. Firstly, the DivMask
itself is simply a bitset wrapped up in a class. The size of the bitset is
determined at compile time. There are various rules for how to set the
bits in the bitset, but they all satisfy the following guiding principle:
ift1
dividest2
then(DivMask(t1) & DivMask(t2)) == DivMask(t1)
i.e.DivMask(t1)
is a "subset" ofDivMask(t2)
There are no other guarantees: in particular, the converse of the guiding principle does not hold in general.
You can create five different sorts of DivMaskRule
:
WORK-IN-PROGRESS: explain what a DivMaskRule is
NewDivMaskNull();
DivMaskRule
is required and you know you
won't use it)
NewDivMaskSingleBit();
k
-th exponent in the PP is strictly positive then the
k
-th bit is set: at most a single bit is used for each indeterminate,
indets with index >= DivMask::ourMaskWidth
are ignored completely.
NewDivMaskSingleBitWrap();
k
-th exponent in the PP is strictly positive then the
k%DivMask::ourMaskWidth
-th bit is set: all indets are taken into
account, and each bit is used for a set of indeterminates.
This implementation is good when we have many indeterminates in
supposedly sparse PPs. (So far I don't have good examples with
more than 2*ourMaskWidth indeterminates)
NewDivMaskEvenPowers();
ceiling(exponent/2)
.
This implementation is good when we have few indeterminates with
high exponents (e.g. Buchberger's algorithm).
It is equivalent to SingleBitWrapImpl
if the number of indets is
bigger than ourMaskWidth
.
NewDivMaskHashing();
ceiling(sqrt(exponent))
.
Supposedly the implementation works well in all cases (e.g. few
indets and high degrees, or many indets and low degrees)
For indet x the first bit set has index x%ourMaskWidth
, and in
general the k-th bit set has index (x + k*hash2)%ourMaskWidth
.
(See code for definition of hash2)
The type DivMaskRule
is used to set the bits in a DivMask
object.
The possible function calls are:
DMR->myAssignFromExpv(mask, exps, NumIndets)
-- sets mask according to PP with exponent vector exps.
Currently the parameter exps
must be of type
vector<SmallExponent_t>
, but this may change.
This function might be quite expensive and its cost depends on the
DivMaskRule
, but this is not a problem if it is called much more rarely
than IsSubset
.
DMR->myOutputSelf(out)
The value of a DivMask
object may be set any number of times (even using
different DivMaskRule
s on each occasion). Any two DivMask
s may be
compared, but the result is meaningful only if both values were created
using the same DivMaskRule
.
There are a few comparison functions on DivMask
objects -- these are
guaranteed to be very fast and independent of the DivMaskRule
,
unlike myAssignFromExpv
dm1 == dm2
-- true iff the bitsets are equal
dm1 != dm2
-- false iff the bitsets are equal
IsSubset(dm1, dm2)
-- true if every bit set in dm1 is set in dm2
You can read the bits held inside a DivMask
object using this function:
bits(dm)
-- gives read-only access to the bitset inside the DivMask
,
the type of the result is DivMask::mask_t
which is a
typedef for a std::bitset
.
The class DivMask
is pretty simple: we don't use a naked
bitset
to ensure that only a DivMaskRule
can set the value.
Use of bitwise-and for modular reduction restricts ourMaskWidth
to
being a power of 2. There are no member functions, and just one
friend function (giving read access to the bitset):
friend const mask_t bits(const DivMask& dm);
The class DivMaskRuleBase
is an abstract base class with an intrusive
reference count: every concrete divmask rule must be derived from this
class. The virtual member function myAssignFromExpv
must be defined in
each concrete divmask rule class: it should set the bits in the DivMask
argument according to the exponents specified in the other two arguments.
The virtual member function myOutput
simply prints the name of the
divmask rule -- it might be useful during debugging. The protected member
function DivMaskRuleBase::myBits
simply allows write access to the
bitset
held inside a DivMask
value; I have to do it this way
because friendship is not inherited.
The type DivMaskRule
is just a reference counting smart pointer to an
instance of a concrete divmask rule class.
The entire declarations and definitions of the concrete classes are in the .C file. There is no need for them to be visible in the .H file.
The class DivMaskNullImpl
is quite simple.
The class DivMaskSingleBitImpl
is also very simple.
The class DivMaskSingleBitWrapImpl
is implemented assuming that the mask
width is a power of 2. It is quite simple.
The class DivMaskEvenPowersImpl
was (half) written by Anna while under the
influence of mind-altering drugs, I reckon.
The class DivMaskHashingImpl
is a bit involved, especially regarding the
choice of bits to set. I'm sure the heuristic can be improved (e.g. by actually
trying it on some real cases :-) Currently the heuristic works as follows.
We consider each indeterminate in turn:
let var
be the index of the indeterminate, and exp
the exponent, then
the total number of bits to be set is ceil(sqrt(exp))
, and
the first bit to be set will be in position var%ourMaskWidth
and subsequent bits will be in positions separated by multiples
of step (where step is 24*floor(var/ourMaskWidth)+13
-- this was chosen because
it happened to make DivMaskHashingImpl
perform well in the CoCoALib tests).
Publicly visible use of SmallExponent_t
is most unfortunate; how to fix it?
Define operator<=
for DivMasks, to do the same as IsSubset??
Should default ourMaskWidth
be 32 or 64?
Surely most current processors are 64 bit now?
Is the restriction that DivMask::ourMaskWidth
be a power of 2 reasonable? Would we really
lose that much speed if any value were allowed? Chances are that the
only interesting values are 32, 64 or 128 (which are indeed all powers
of 2).
COPRIMALITY: Do we want DivMask
s to permit a swift coprimality check?
Presumably the idea would be that two disjoint DivMask values would
imply that the corresponding PPs must be coprime. Another possibility
is that the DivMask values are disjoint iff the PPs are coprime; this
second possibility would exclude some ideas for implementing DivMasks
(for instance DivMaskSingleBitWrap
and DivMaskHashing
would be excluded).
Documentation is too sarcastic.
2006