Source code for sympy.combinatorics.util

from sympy.ntheory import isprime
from sympy.combinatorics.permutations import Permutation, _af_invert, _af_rmul

rmul = Permutation.rmul
_af_new = Permutation._af_new

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### Utilities for computational group theory
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[docs]def _base_ordering(base, degree): r""" Order `\{0, 1, ..., n-1\}` so that base points come first and in order. Parameters ========== ``base`` - the base ``degree`` - the degree of the associated permutation group Returns ======= A list ``base_ordering`` such that ``base_ordering[point]`` is the number of ``point`` in the ordering. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.util import _base_ordering >>> S = SymmetricGroup(4) >>> S.schreier_sims() >>> _base_ordering(S.base, S.degree) [0, 1, 2, 3] Notes ===== This is used in backtrack searches, when we define a relation `<<` on the underlying set for a permutation group of degree `n`, `\{0, 1, ..., n-1\}`, so that if `(b_1, b_2, ..., b_k)` is a base we have `b_i << b_j` whenever `i<j` and `b_i << a` for all `i\in\{1,2, ..., k\}` and `a` is not in the base. The idea is developed and applied to backtracking algorithms in [1], pp.108-132. The points that are not in the base are taken in increasing order. References ========== [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" """ base_len = len(base) ordering = [0]*degree for i in xrange(base_len): ordering[base[i]] = i current = base_len for i in xrange(degree): if i not in base: ordering[i] = current current += 1 return ordering
[docs]def _check_cycles_alt_sym(perm): """ Checks for cycles of prime length p with n/2 < p < n-2. Here `n` is the degree of the permutation. This is a helper function for the function is_alt_sym from sympy.combinatorics.perm_groups. Examples ======== >>> from sympy.combinatorics.util import _check_cycles_alt_sym >>> from sympy.combinatorics.permutations import Permutation >>> a = Permutation([[0,1,2,3,4,5,6,7,8,9,10], [11, 12]]) >>> _check_cycles_alt_sym(a) False >>> b = Permutation([[0,1,2,3,4,5,6], [7,8,9,10]]) >>> _check_cycles_alt_sym(b) True See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym """ n = perm.size af = perm.array_form current_len = 0 total_len = 0 used = set() for i in xrange(n//2): if not i in used and i < n//2 - total_len: current_len = 1 used.add(i) j = i while(af[j] != i): current_len += 1 j = af[j] used.add(j) total_len += current_len if current_len > n//2 and current_len < n-2 and isprime(current_len): return True return False
[docs]def _distribute_gens_by_base(base, gens): """ Distribute the group elements ``gens`` by membership in basic stabilizers. Notice that for a base `(b_1, b_2, ..., b_k)`, the basic stabilizers are defined as `G^{(i)} = G_{b_1, ..., b_{i-1}}` for `i \in\{1, 2, ..., k\}`. Parameters ========== ``base`` - a sequence of points in `\{0, 1, ..., n-1\}` ``gens`` - a list of elements of a permutation group of degree `n`. Returns ======= List of length `k`, where `k` is the length of ``base``. The `i`-th entry contains those elements in ``gens`` which fix the first `i` elements of ``base`` (so that the `0`-th entry is equal to ``gens`` itself). If no element fixes the first `i` elements of ``base``, the `i`-th element is set to a list containing the identity element. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.named_groups import DihedralGroup >>> from sympy.combinatorics.util import _distribute_gens_by_base >>> D = DihedralGroup(3) >>> D.schreier_sims() >>> D.strong_gens [Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)] >>> D.base [0, 1] >>> _distribute_gens_by_base(D.base, D.strong_gens) [[Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)], [Permutation(1, 2)]] See Also ======== _strong_gens_from_distr, _orbits_transversals_from_bsgs, _handle_precomputed_bsgs """ base_len = len(base) stabs = [] degree = gens[0].size for i in xrange(base_len): stabs.append([]) num_gens = len(gens) max_stab_index = 0 for i in xrange(num_gens): j = 0 while j < base_len - 1 and gens[i](base[j]) == base[j]: j += 1 if j > max_stab_index: max_stab_index = j for k in xrange(j + 1): stabs[k].append(gens[i]) for i in range(max_stab_index + 1, base_len): stabs[i].append(_af_new(range(degree))) return stabs
[docs]def _handle_precomputed_bsgs(base, strong_gens, transversals=None,\ basic_orbits=None, strong_gens_distr=None): """ Calculate BSGS-related structures from those present. The base and strong generating set must be provided; if any of the transversals, basic orbits or distributed strong generators are not provided, they will be calculated from the base and strong generating set. Parameters ========== ``base`` - the base ``strong_gens`` - the strong generators ``transversals`` - basic transversals ``basic_orbits`` - basic orbits ``strong_gens_distr`` - strong generators distributed by membership in basic stabilizers Returns ======= ``(transversals, basic_orbits, strong_gens_distr)`` where ``transversals`` are the basic transversals, ``basic_orbits`` are the basic orbits, and ``strong_gens_distr`` are the strong generators distributed by membership in basic stabilizers. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.named_groups import DihedralGroup >>> from sympy.combinatorics.util import _handle_precomputed_bsgs >>> D = DihedralGroup(3) >>> D.schreier_sims() >>> _handle_precomputed_bsgs(D.base, D.strong_gens, ... basic_orbits=D.basic_orbits) ([{0: Permutation(2), 1: Permutation(0, 1, 2), 2: Permutation(0, 2)}, {1: Permutation(2), 2: Permutation(1, 2)}], [[0, 1, 2], [1, 2]], [[Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)], [Permutation(1, 2)]]) See Also ======== _orbits_transversals_from_bsgs, distribute_gens_by_base """ if strong_gens_distr is None: strong_gens_distr = _distribute_gens_by_base(base, strong_gens) if transversals is None: if basic_orbits is None: basic_orbits, transversals =\ _orbits_transversals_from_bsgs(base, strong_gens_distr) else: transversals =\ _orbits_transversals_from_bsgs(base, strong_gens_distr, transversals_only=True) else: if basic_orbits is None: base_len = len(base) basic_orbits = [None]*base_len for i in xrange(base_len): basic_orbits[i] = transversals[i].keys() return transversals, basic_orbits, strong_gens_distr
[docs]def _orbits_transversals_from_bsgs(base, strong_gens_distr,\ transversals_only=False): """ Compute basic orbits and transversals from a base and strong generating set. The generators are provided as distributed across the basic stabilizers. If the optional argument ``transversals_only`` is set to True, only the transversals are returned. Parameters ========== ``base`` - the base ``strong_gens_distr`` - strong generators distributed by membership in basic stabilizers ``transversals_only`` - a flag swithing between returning only the transversals/ both orbits and transversals Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.util import _orbits_transversals_from_bsgs >>> from sympy.combinatorics.util import (_orbits_transversals_from_bsgs, ... _distribute_gens_by_base) >>> S = SymmetricGroup(3) >>> S.schreier_sims() >>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens) >>> _orbits_transversals_from_bsgs(S.base, strong_gens_distr) ([[0, 1, 2], [1, 2]], [{0: Permutation(2), 1: Permutation(0, 1, 2), 2: Permutation(0, 2, 1)}, {1: Permutation(2), 2: Permutation(1, 2)}]) See Also ======== _distribute_gens_by_base, _handle_precomputed_bsgs """ from sympy.combinatorics.perm_groups import PermutationGroup base_len = len(base) transversals = [None]*base_len if transversals_only is False: basic_orbits = [None]*base_len for i in xrange(base_len): group = PermutationGroup(strong_gens_distr[i]) transversals[i] = dict(group.orbit_transversal(base[i], pairs=True)) if transversals_only is False: basic_orbits[i] = transversals[i].keys() if transversals_only: return transversals else: return basic_orbits, transversals
[docs]def _remove_gens(base, strong_gens, basic_orbits=None, strong_gens_distr=None): """ Remove redundant generators from a strong generating set. Parameters ========== ``base`` - a base ``strong_gens`` - a strong generating set relative to ``base`` ``basic_orbits`` - basic orbits ``strong_gens_distr`` - strong generators distributed by membership in basic stabilizers Returns ======= A strong generating set with respect to ``base`` which is a subset of ``strong_gens``. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.util import _remove_gens >>> from sympy.combinatorics.testutil import _verify_bsgs >>> S = SymmetricGroup(15) >>> base, strong_gens = S.schreier_sims_incremental() >>> len(strong_gens) 26 >>> new_gens = _remove_gens(base, strong_gens) >>> len(new_gens) 14 >>> _verify_bsgs(S, base, new_gens) True Notes ===== This procedure is outlined in [1],p.95. References ========== [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" """ from sympy.combinatorics.perm_groups import PermutationGroup base_len = len(base) if strong_gens_distr is None: strong_gens_distr = _distribute_gens_by_base(base, strong_gens) if basic_orbits is None: basic_orbits = [] for i in range(base_len): stab = PermutationGroup(strong_gens_distr[i]) basic_orbit = stab.orbit(base[i]) basic_orbits.append(basic_orbit) strong_gens_distr.append([]) res = strong_gens[:] for i in range(base_len - 1, -1, -1): gens_copy = strong_gens_distr[i][:] for gen in strong_gens_distr[i]: if gen not in strong_gens_distr[i + 1]: temp_gens = gens_copy[:] temp_gens.remove(gen) if temp_gens == []: continue temp_group = PermutationGroup(temp_gens) temp_orbit = temp_group.orbit(base[i]) if temp_orbit == basic_orbits[i]: gens_copy.remove(gen) res.remove(gen) return res
[docs]def _strip(g, base, orbits, transversals): """ Attempt to decompose a permutation using a (possibly partial) BSGS structure. This is done by treating the sequence ``base`` as an actual base, and the orbits ``orbits`` and transversals ``transversals`` as basic orbits and transversals relative to it. This process is called "sifting". A sift is unsuccessful when a certain orbit element is not found or when after the sift the decomposition doesn't end with the identity element. The argument ``transversals`` is a list of dictionaries that provides transversal elements for the orbits ``orbits``. Parameters ========== ``g`` - permutation to be decomposed ``base`` - sequence of points ``orbits`` - a list in which the ``i``-th entry is an orbit of ``base[i]`` under some subgroup of the pointwise stabilizer of ` `base[0], base[1], ..., base[i - 1]``. The groups themselves are implicit in this function since the only infromation we need is encoded in the orbits and transversals ``transversals`` - a list of orbit transversals associated with the orbits ``orbits``. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.util import _strip >>> S = SymmetricGroup(5) >>> S.schreier_sims() >>> g = Permutation([0, 2, 3, 1, 4]) >>> _strip(g, S.base, S.basic_orbits, S.basic_transversals) (Permutation(4), 5) Notes ===== The algorithm is described in [1],pp.89-90. The reason for returning both the current state of the element being decomposed and the level at which the sifting ends is that they provide important information for the randomized version of the Schreier-Sims algorithm. References ========== [1] Holt, D., Eick, B., O'Brien, E. "Handbook of computational group theory" See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random """ h = g.array_form base_len = len(base) for i in range(base_len): beta = h[base[i]] if beta == base[i]: continue if beta not in orbits[i]: return _af_new(h), i + 1 u = transversals[i][beta].array_form h = _af_rmul(_af_invert(u), h) return _af_new(h), base_len + 1
[docs]def _strong_gens_from_distr(strong_gens_distr): """ Retrieve strong generating set from generators of basic stabilizers. This is just the union of the generators of the first and second basic stabilizers. Parameters ========== ``strong_gens_distr`` - strong generators distributed by membership in basic stabilizers Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.util import (_strong_gens_from_distr, ... _distribute_gens_by_base) >>> S = SymmetricGroup(3) >>> S.schreier_sims() >>> S.strong_gens [Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)] >>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens) >>> _strong_gens_from_distr(strong_gens_distr) [Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)] See Also ======== _distribute_gens_by_base """ if len(strong_gens_distr) == 1: return strong_gens_distr[0][:] else: result = strong_gens_distr[0] for gen in strong_gens_distr[1]: if gen not in result: result.append(gen) return result