Copyright (c) 2005 John Abbott
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
A copy of the licence is included in the file COPYING in this directory.
User documentation for files DivMask.H and DivMask.C
====================================================
The main reason for creating a [DivMask] is to permit a quick, coarse test
of divisibility between power products. The test is coarse because we
accept as responses "definitely not divisible" or "possibly". This is 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
and think you're interested, read on.
Note: currently [DivMask]s cannot be used to ascertain coprimality (see Bugs section).
To use [DivMasks]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 guiding principle:
if t1 divides t2 then (DivMask(t1) & DivMask(t2)) == DivMask(t1)
[i.e. DivMask(t1) is a "subset" of DivMask(t2)]
There are no other guarantees: in particular, the converse of the guiding
principle does not hold in general. Here is a code snippet to show how
[DivMask]s can be used (but it is easier to use [PPWithMask]):
DivMaskRule DMR = NewDivMaskEvenPowers();
PPMonoidElem PP1 ...
PPMonoidElem PP2 ...
...
const size_t nvars = NumIndets(owner(PP1));
vector<SmallExponent_t> expv(nvars);
DivMask dm1;
DivMask dm2;
exponents(expv, PP1);
DMR->myAssignFromExpv(dm1, &expv[0], nvars);
exponents(expv, PP2);
DMR->myAssignFromExpv(dm2, &expv[0], nvars);
if (!IsSubset(dm1, dm2)) { }
else { }
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].
There are a few comparison functions on [DivMask] objects:
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
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 [exps] must be of type [vector<SmallExponent_t>], but this may change.
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].
You can create five different sorts of rule:
DivMaskRule DMR = NewDivMaskNull();
no bit is ever set (relatively fast, but otherwise pretty useless).
DivMaskRule DMR = NewDivMaskSingleBit();
if the k-th indet appears in the PP then the k-th bit is set
(indets with index >= DivMask::MaskWidth are ignored completely).
DivMaskRule DMR = NewDivMaskSingleBitWrap();
if the k-th indet appears in the PP then the K-th bit is set where
K = k%DivMask::MaskWidth.
DivMaskRule DMR = NewDivMaskEvenPowers();
unlike the above rules, this rule may set several bits for a PP divisible
by a "high" power of an indeterminate. For instance, with a mask width of
32 and 4 indets, up to 8 bits can be set for each indet; the actual number
of bits set is ceiling(exponent/2). With many indets this rule behaves
like SingleBit.
DivMaskRule DMR = NewDivMaskHashing();
this rule uses a hashing scheme to allow many bits to be set for each indet
even when there are many indets. The number of bits set for an indet with
exponent exp is ceiling(sqrt(exp)).
Maintainer documentation for files DivMask.H and DivMask.C
==========================================================
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 [MaskWidth] to being a power of 2. There are no
member functions, and just one friend function:
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 durig debugging. The protected member
function [myBits] simply allows write access to the [bitset] held inside a
[DivMask]; I have to do it this way because friendship is not inherited.
The type [DivMaskRule] is just a reference counting smart pointer to 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 like this:
let var be the index of the indeterminate, and exp the exponent;
the first bit to be set for var will be in position (-5*var)%MaskWidth
and subsequent bits (for var) will be in positions separated by multiples
of step [where step is 2*(var/MaskWidth)+3].
Bugs, Shortcomings, and other ideas
===================================
Publicly visible use of [SmallExponent_t] is most unfortunate; how to fix it?
Is the restriction on [DivMask::MaskWidth] 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 too sarcastic.
=== Old logs ===
Revision 1.2 2006/08/07 21:23:25 cocoa
Removed almost all publicly visible references to SmallExponent_t;
changed to long in all PPMonoid functions and SparsePolyRing functions.
DivMask remains to sorted out.
Revision 1.3 2006/01/18 16:15:16 cocoa
Cleaned up DivMask considerably; everything still works,
so I'm checking in (and then going home).
Revision 1.2 2006/01/17 18:08:01 cocoa
Added new DivMask type: DivMaskHashingImpl.
Updated DivMask documentation.
Revision 1.2 2005/04/19 14:06:05 cocoa
Added GPL and GFDL licence stuff.