# Source code for sympy.combinatorics.util

from __future__ import print_function, division

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

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 range(base_len): ordering[base[i]] = i current = base_len for i in range(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 range(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 [(0 1 2), (0 2), (1 2)] >>> D.base [0, 1] >>> _distribute_gens_by_base(D.base, D.strong_gens) [[(0 1 2), (0 2), (1 2)], [(1 2)]] See Also ======== _strong_gens_from_distr, _orbits_transversals_from_bsgs, _handle_precomputed_bsgs """ base_len = len(base) degree = gens[0].size stabs = [[] for _ in range(base_len)] max_stab_index = 0 for gen in gens: j = 0 while j < base_len - 1 and gen._array_form[base[j]] == base[j]: j += 1 if j > max_stab_index: max_stab_index = j for k in range(j + 1): stabs[k].append(gen) for i in range(max_stab_index + 1, base_len): stabs[i].append(_af_new(list(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: (2), 1: (0 1 2), 2: (0 2)}, {1: (2), 2: (1 2)}], [[0, 1, 2], [1, 2]], [[(0 1 2), (0 2), (1 2)], [(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 range(base_len): basic_orbits[i] = list(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 switching 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: (2), 1: (0 1 2), 2: (0 2 1)}, {1: (2), 2: (1 2)}]) See Also ======== _distribute_gens_by_base, _handle_precomputed_bsgs """ from sympy.combinatorics.perm_groups import _orbit_transversal base_len = len(base) degree = strong_gens_distr[0][0].size transversals = [None]*base_len if transversals_only is False: basic_orbits = [None]*base_len for i in range(base_len): transversals[i] = dict(_orbit_transversal(degree, strong_gens_distr[i], base[i], pairs=True)) if transversals_only is False: basic_orbits[i] = list(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() >>> 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 _orbit base_len = len(base) degree = strong_gens[0].size if strong_gens_distr is None: strong_gens_distr = _distribute_gens_by_base(base, strong_gens) temp = strong_gens_distr[:] if basic_orbits is None: basic_orbits = [] for i in range(base_len): basic_orbit = _orbit(degree, strong_gens_distr[i], 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_orbit = _orbit(degree, temp_gens, 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 information 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) ((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
def _strip_af(h, base, orbits, transversals, j): """ optimized _strip, with h, transversals and result in array form if the stripped elements is the identity, it returns False, base_len + 1 j h[base[i]] == base[i] for i <= j """ base_len = len(base) for i in range(j+1, base_len): beta = h[base[i]] if beta == base[i]: continue if beta not in orbits[i]: return h, i + 1 u = transversals[i][beta] if h == u: return False, base_len + 1 h = _af_rmul(_af_invert(u), h) return 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 [(0 1 2), (2)(0 1), (1 2)] >>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens) >>> _strong_gens_from_distr(strong_gens_distr) [(0 1 2), (2)(0 1), (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