Coverage for local/lib/python2.7/site-packages/sage/combinat/designs/evenly_distributed_sets.pyx : 92%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
# -*- coding: utf-8 -*- r""" Evenly distributed sets in finite fields
This module consists of a simple class :class:`EvenlyDistributedSetsBacktracker`. Its main purpose is to iterate through the evenly distributed sets of a given finite field.
The naive backtracker implemented here is not directly used to generate difference family as even for small parameters it already takes time to run. Instead, its output has been stored into a database :mod:`sage.combinat.designs.database`. If the backtracker is improved, then one might want to update this database with more values.
Classes and methods ------------------- """ from __future__ import print_function, absolute_import
cimport cython
from libc.limits cimport UINT_MAX from libc.string cimport memset, memcpy
from cysignals.memory cimport check_malloc, check_calloc, sig_free
from sage.rings.integer cimport Integer,smallInteger
cdef class EvenlyDistributedSetsBacktracker: r""" Set of evenly distributed subsets in finite fields.
**Definition:** Let `K` be a finite field of cardinality `q` and `k` an integer so that `k(k-1)` divides `q-1`. Let `H = K^*` be the multiplicative group of invertible elements in `K`. A `k`-*evenly distributed set* in `K` is a set `B = \{b_1, b_2, \ldots, b_k\}` of `k` elements of `K` so that the `k(k-1)` differences `\Delta B = \{b_i - b_j; i \not= j\}` hit each coset modulo `H^{2(q-1)/(k(k-1))}` exactly twice.
Evenly distributed sets were introduced by Wilson [Wi72]_ (see also [BJL99-1]_, Chapter VII.6). He proved that for any `k`, and for any prime power `q` large enough such that `k(k-1)` divides `k` there exists a `k`-evenly distributed set in the field of cardinality `q`. This existence result based on a counting argument (using Dirichlet theorem) does not provide a simple method to generate them.
From a `k`-evenly distributed set, it is straightforward to build a `(q,k,1)`-difference family (see :meth:`to_difference_family`). Another approach to generate a difference family, somehow dual to this one, is via radical difference family (see in particular :func:`~sage.combinat.designs.difference_family.radical_difference_family` from the module :mod:`~sage.combinat.designs.difference_family`).
By default, this backtracker only considers evenly distributed sets up to affine automorphisms, i.e. `B` is considered equivalent to `s B + t` for any invertible element `s` and any element `t` in the field `K`. Note that the set of differences is just multiplicatively translated by `s` as `\Delta (s B + t) = s (\Delta B)`, and so that `B` is an evenly distributed set if and only if `sB` is one too. This behaviour can be modified via the argument ``up_to_isomorphism`` (see the input section and the examples below).
INPUT:
- ``K`` -- a finite field of cardinality `q`
- ``k`` -- a positive integer such that `k(k-1)` divides `q-1`
- ``up_to_isomorphism`` - (boolean, default ``True``) whether only consider evenly distributed sets up to automorphisms of the field of the form `x \mapsto ax + b`. If set to ``False`` then the iteration is over all evenly distributed sets that contain ``0`` and ``1``.
- ``check`` -- boolean (default is ``False``). Whether you want to check intermediate steps of the iterator. This is mainly intended for debugging purpose. Setting it to ``True`` will considerably slow the iteration.
EXAMPLES:
The main part of the code is contained in the iterator. To get one evenly distributed set just do::
sage: from sage.combinat.designs.evenly_distributed_sets import EvenlyDistributedSetsBacktracker sage: E = EvenlyDistributedSetsBacktracker(Zmod(151),6) sage: B = E.an_element() sage: B [0, 1, 69, 36, 57, 89]
The class has a method to convert it to a difference family::
sage: E.to_difference_family(B) [[0, 1, 69, 36, 57, 89], [0, 132, 48, 71, 125, 121], [0, 59, 145, 10, 41, 117], [0, 87, 114, 112, 127, 42], [0, 8, 99, 137, 3, 108]]
It is also possible to run over all evenly distributed sets::
sage: E = EvenlyDistributedSetsBacktracker(Zmod(13), 4, up_to_isomorphism=False) sage: for B in E: print(B) [0, 1, 11, 5] [0, 1, 4, 6] [0, 1, 9, 3] [0, 1, 8, 10]
sage: E = EvenlyDistributedSetsBacktracker(Zmod(13), 4, up_to_isomorphism=True) sage: for B in E: print(B) [0, 1, 11, 5]
Or only count them::
sage: for k in range(13, 200, 12): ....: if is_prime_power(k): ....: K = GF(k,'a') ....: E1 = EvenlyDistributedSetsBacktracker(K, 4, False) ....: E2 = EvenlyDistributedSetsBacktracker(K, 4, True) ....: print("{:3} {:3} {:3}".format(k, E1.cardinality(), E2.cardinality())) 13 4 1 25 40 4 37 12 1 49 24 2 61 12 1 73 48 4 97 64 6 109 72 6 121 240 20 157 96 8 169 240 20 181 204 17 193 336 28
Note that by definition, the number of evenly distributed sets up to isomorphisms is at most `k(k-1)` times smaller than without isomorphisms. But it might not be exactly `k(k-1)` as some of them might have symmetries::
sage: B = EvenlyDistributedSetsBacktracker(Zmod(13), 4).an_element() sage: B [0, 1, 11, 5] sage: [9*x + 5 for x in B] [5, 1, 0, 11] sage: [3*x + 11 for x in B] [11, 1, 5, 0] """ # PYTHON DATA cdef K # the underlying field cdef list list_K # the elements of K (i -> x) cdef dict K_to_int # inverse of list_K (x -> i)
# FLAGS cdef int count # do we count or do we iterate cdef int check # do we need to check (debug) cdef int up_to_isom # do we care only about isomorphisms
# STATIC DATA cdef unsigned int q # cardinality of the field cdef unsigned int k # size of the subsets cdef unsigned int e # k(k-1)/2 cdef unsigned int m # (q-1) / e cdef unsigned int ** diff # qxq array: diff[x][y] = x - y cdef unsigned int ** ratio # qxq array: ratio[x][y] = x / y cdef unsigned int * min_orb # q array : min_orb[x] = min {x, 1-x, 1/x, # 1/(1-x), (x-1)/x, x/(x-1)}
# DYNAMIC DATA cdef unsigned int * B # current stack of elements of {0,...,q-1} cdef unsigned int * cosets # e array: cosets of differences of elts in B cdef unsigned int * t # e array: temporary variable for updates
def __dealloc__(self):
def __init__(self, K, k, up_to_isomorphism=True, check=False): r""" TESTS::
sage: from sage.combinat.designs.evenly_distributed_sets import EvenlyDistributedSetsBacktracker
sage: EvenlyDistributedSetsBacktracker(Zmod(4),2) Traceback (most recent call last): ... ValueError: Ring of integers modulo 4 is not a field
sage: EvenlyDistributedSetsBacktracker(Zmod(71),7) Traceback (most recent call last): ... ValueError: k(k-1)=42 does not divide q-1=70
For `q=421` which is congruent to 1 modulo `12`, `20`, `30` and `42` we run backtracker with the ``check`` argument set to ``True``::
sage: for _ in EvenlyDistributedSetsBacktracker(Zmod(421), 4, check=True): ....: pass sage: for _ in EvenlyDistributedSetsBacktracker(Zmod(421), 5, check=True): ....: pass sage: for _ in EvenlyDistributedSetsBacktracker(Zmod(421), 6, check=True): ....: pass sage: for _ in EvenlyDistributedSetsBacktracker(Zmod(421), 7, check=True): ....: pass """
cdef unsigned int i,j,ell
else:
def to_difference_family(self, B, check=True): r""" Given an evenly distributed set ``B`` convert it to a difference family.
As for any `x\in K^*=H` we have `|\Delta B \cap x H^{2(q-1)/(k(k-1))}|=2` (see :class:`EvenlyDistributedSetsBacktracker`), the difference family is produced as `\{xB:x\in H^{2(q-1)/(k(k-1))}\}`
This method is useful if you want to obtain the difference family from the output of the iterator.
INPUT:
- ``B`` -- an evenly distributed set
- ``check`` -- (boolean, default ``True``) whether to check the result
EXAMPLES::
sage: from sage.combinat.designs.evenly_distributed_sets import EvenlyDistributedSetsBacktracker sage: E = EvenlyDistributedSetsBacktracker(Zmod(41),5) sage: B = E.an_element(); B [0, 1, 13, 38, 31] sage: D = E.to_difference_family(B); D [[0, 1, 13, 38, 31], [0, 32, 6, 27, 8]]
sage: from sage.combinat.designs.difference_family import is_difference_family sage: is_difference_family(Zmod(41),D,41,5,1) True
Setting ``check`` to ``False`` is much faster::
sage: timeit("df = E.to_difference_family(B, check=True)") # random 625 loops, best of 3: 117 µs per loop
sage: timeit("df = E.to_difference_family(B, check=False)") # random 625 loops, best of 3: 1.83 µs per loop """ raise RuntimeError("a wrong evenly distributed set was " "produced by the Sage library for the parameters:\n" " q={} k={}\n" "Please send an e-mail to " "sage-devel@googlegroups.com".format(self.q, self.k))
def an_element(self): r""" Return an evenly distributed set.
If there is no such subset raise an :class:`~sage.categories.sets_cat.EmptySetError`.
EXAMPLES::
sage: from sage.combinat.designs.evenly_distributed_sets import EvenlyDistributedSetsBacktracker
sage: E = EvenlyDistributedSetsBacktracker(Zmod(41),5) sage: E.an_element() [0, 1, 13, 38, 31]
sage: E = EvenlyDistributedSetsBacktracker(Zmod(61),6) sage: E.an_element() Traceback (most recent call last): ... EmptySetError: no 6-evenly distributed set in Ring of integers modulo 61 """
def __repr__(self): r""" A string representative.
EXAMPLES::
sage: from sage.combinat.designs.evenly_distributed_sets import EvenlyDistributedSetsBacktracker
sage: EvenlyDistributedSetsBacktracker(GF(25,'a'), 4) 4-evenly distributed sets (up to isomorphism) in Finite Field in a of size 5^2 sage: EvenlyDistributedSetsBacktracker(GF(25,'a'), 4, up_to_isomorphism=False) 4-evenly distributed sets in Finite Field in a of size 5^2 """
def cardinality(self): r""" Return the number of evenly distributed sets.
Use with precaution as there can be a lot of such sets and this method might be very long to answer!
EXAMPLES::
sage: from sage.combinat.designs.evenly_distributed_sets import EvenlyDistributedSetsBacktracker
sage: E = EvenlyDistributedSetsBacktracker(GF(25,'a'),4) sage: E 4-evenly distributed sets (up to isomorphism) in Finite Field in a of size 5^2 sage: E.cardinality() 4
sage: E = EvenlyDistributedSetsBacktracker(GF(25,'a'), 4, up_to_isomorphism=False) sage: E.cardinality() 40 """
def _B_relabelled_copies(self): r""" Check whether ``self.B`` is minimal among its relabelization.
If `B=\{b_1,...,b_k\}` is an evenly distributed set and contains `0` and `1`, then for any two distinct `i,j` we define `f_{ij} : x \mapsto (x-b_j)/(b_i-b_j)` which maps `B` on another evenly distributed set of size `k` containing `0` and `1`. For each pair `i,j` we consider check whether the set `f_{ij}(B)` is smaller than `B`.
This is an internal function and should only be call by the backtracker implemented in the method `__iter__`.
OUTPUT:
- ``False`` if ``self.B`` is not minimal
- the list of evenly distributed sets isomorphs to ``self.B`` given as a list of tuples if ``self.up_to_isom=0`` or list containing only ``self.B`` as a tuple if ``self.up_to_isom=1``.
TESTS::
sage: from sage.combinat.designs.evenly_distributed_sets import \ ....: EvenlyDistributedSetsBacktracker sage: E = EvenlyDistributedSetsBacktracker(Zmod(13), 4, up_to_isomorphism=True) sage: E.cardinality() # indirect doctest 1 sage: E = EvenlyDistributedSetsBacktracker(Zmod(13), 4, up_to_isomorphism=False) sage: E.cardinality() # indirect doctest 4
.. NOTE::
this method is not seriously optimized. The main goal of this backtracker is to generate one evenly distributed set. In that case, this method will be called only once. """ cdef unsigned int i,j,k,tmp1,tmp2,verify
# z -> (z - B[i]) / (B[j] - B[i])
else: # the backtracker should never build a set which by # relabelling is strictly smaller than B[:3] raise RuntimeError("there is a problem got tmp2={}".format(tmp2,self.B[2]))
@cython.cdivision(True) @cython.boundscheck(False) @cython.wraparound(False) def __iter__(self): r""" Iterator through all evenly distributed sets that start with `[0,1]`.
EXAMPLES::
sage: from sage.combinat.designs.evenly_distributed_sets import EvenlyDistributedSetsBacktracker
sage: E = EvenlyDistributedSetsBacktracker(Zmod(13),4) sage: for B in E: ....: print(B) [0, 1, 11, 5] """
# in the list B we store the candidate for being an e.d.s. # we always have B[0] = 0 and B[1] = 1 # because 0 is in B, the cosets of the elements of B must be # disjoint.
raise RuntimeError("got x < m or x > q_m_1 (x={})".format(x)) raise RuntimeError("got x={} in an already occupied coset".format(x))
# try to append x
pass else:
# remove the differences created by x and increment else: else:
# now we determine the next element x to be tested # remove the differences created by x and increment else: else: else:
@cython.cdivision(True) cdef inline int _check_last_element(self, unsigned int kk) except -1: r""" Add the element ``x`` to ``B`` in position kk if the resulting set is still evenly distributed.
OUTPUT:
1 if the element was added, and 0 otherwise. """ cdef unsigned int i, j, x_m_i, x_m_j
# We check two things: # 1. that the newly created differences x-B[i] will not be in a coset # already occuppied # # 2. that by applying some automorphisms we will not get an # element smaller than B[2]. # # We should test all linear functions that send a subset of the form # {x, B[i], B[j]} to some {0, 1, ?}. # # Note that if {x, B[i], B[j]} can be mapped to {0, 1, z} by some # function, then it can also be mapped to all {0, 1, z'} where z'= # 1/z, 1-z, 1/(1-z), (z-1)/z and z/(z-1). The attribute # 'min_orbit[z]' is exactly the minimum among these values. # # So, it is enough to test one of these functions. We choose t -> (x # - t)/ (x - B[j]) (that maps x to 0 and B[j] to 1). Its value at # B[i] is just z = (x - B[i]) / (x - B[j]). # # In the special case when kk=2, or equivalently when we are testing if x # fits as a new B[2], then we just check that x is the minimum among # {x, 1/x, 1-x, 1/(1-x), (x-1)/x and x/(x-1)}.
# 1. check that the difference x-B[i] was not already in an # occuppied coset
# 2. check relabeling
# Now check that the x-B[i] belongs to distinct cosets
@cython.cdivision(True) cdef int _check_cosets(self, unsigned int kk) except -1: r""" Sanity check (only for debug purposes). """ cdef unsigned int i,j cdef unsigned int c
# count the number of elements in self.cosets raise RuntimeError("the number of elements in cosets is wrong! Got {} instead of {}.".format(c, (kk*(kk-1))/2))
raise RuntimeError("self.cosets misses the difference B[{}]-B[{}]".format(i,j))
|