ferari-1.0.0/AUTHORS0000644000000000000000000000031511667671022012165 0ustar 00000000000000Main author: Robert C. Kirby email: robert.c.kirby@ttu.edu www: http://www.math.ttu.edu/~kirby/ Contributors: Anders Logg email: logg@simula.no www: http://home.simula.no/~logg/ ferari-1.0.0/COPYING0000644000000000000000000010437411667671022012162 0ustar 00000000000000 GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The GNU General Public License is a free, copyleft license for software and other kinds of works. The licenses for most software and other practical works are designed to take away your freedom to share and change the works. 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Copyright (C) This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . Also add information on how to contact you by electronic and paper mail. If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode: Copyright (C) This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an "about box". You should also get your employer (if you work as a programmer) or school, if any, to sign a "copyright disclaimer" for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see . The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read . ferari-1.0.0/COPYING.LESSER0000644000000000000000000001672711667671022013162 0ustar 00000000000000 GNU LESSER GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. This version of the GNU Lesser General Public License incorporates the terms and conditions of version 3 of the GNU General Public License, supplemented by the additional permissions listed below. 0. Additional Definitions. As used herein, "this License" refers to version 3 of the GNU Lesser General Public License, and the "GNU GPL" refers to version 3 of the GNU General Public License. "The Library" refers to a covered work governed by this License, other than an Application or a Combined Work as defined below. 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If the Library as you received it specifies that a proxy can decide whether future versions of the GNU Lesser General Public License shall apply, that proxy's public statement of acceptance of any version is permanent authorization for you to choose that version for the Library. ferari-1.0.0/ChangeLog0000644000000000000000000000053611667671022012674 0ustar 000000000000001.0.0 - Change license to LGPL v3 or later 0.2.0 - Bug fixes 0.1.0: - Now have roughly O(n log n) process for building graph instead of the brute force O(n^2) process 0.0.2: - Minor update to remove zeros from abstract syntax and to use unary relations (number of nonzeros) as well as binary relations 0.0.1: - First edition of FEarari ferari-1.0.0/README0000644000000000000000000000170411667671022012000 0ustar 00000000000000FErari: Finite Element rearrangement to automatically reduce instructions ------------------------------------------------------------------------- FErari generates optimized code for evaluation of the element tensor (element stiffness matrix) and functions as an optimizing backend for FFC. License: -------- This program is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this program. If not, see . ferari-1.0.0/TODO0000644000000000000000000000020711667671022011605 0ustar 00000000000000[TODO] Clean up and generate (abstract) code for geometric optimization [TODO] Pay attention to get sign correct for negative equality ferari-1.0.0/ferari/0000755000000000000000000000000011667671022012366 5ustar 00000000000000ferari-1.0.0/release.conf0000644000000000000000000000017411667671022013407 0ustar 00000000000000# Configuration file for fenics-release PACKAGE="ferari" LP_PACKAGE="ferari" FILES="setup.py ChangeLog ferari/__init__.py" ferari-1.0.0/setup.py0000644000000000000000000000054311667671022012632 0ustar 00000000000000#!/usr/bin/env python from distutils.core import setup setup(name="ferari", \ version="1.0.0", \ description="Optimizer for finite element code", \ author="Robert C. Kirby", \ author_email="kirby@uchicago.edu", \ url="http://people.cs.uchicago.edu/~kirby", \ license="LGPL v3 or later", \ packages=['ferari'] ) ferari-1.0.0/ferari/__init__.py0000644000000000000000000000002211667671022014471 0ustar 00000000000000VERSION = "1.0.0" ferari-1.0.0/ferari/binary.py0000644000000000000000000003054611667671022014234 0ustar 00000000000000# Copyright (C) 2006 Robert C. Kirby # # This file is part of FErari. # # FErari is free software: you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # FErari is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with FErari. If not, see . # # First added: 2005-04-01 # Last changed: 2006-04-05 import sigdig, numpy, string, graph, pg, util from xpermutations import xuniqueCombinations eps = util.eps # This are distance measures used in the (hopefully) # now obsolete brute-force O(n^2) method of forming # the graph def edplus( u , v ): """Returns the Hamming distance (number of nonzero vectors) between u and v, after rounding both vectors to 5 significant digits""" ur = sigdig.vec_round_sig( u , 5 ) vr = sigdig.vec_round_sig( v , 5 ) return util.nnz( ur - vr ) def edminus( u , v ): """Returns the Hamming distance (number of nonzero vectors) between u and -v, after rounding both vectors to 5 significant digits""" ur = sigdig.vec_round_sig( u , 5 ) vr = sigdig.vec_round_sig( v , 5 ) return util.nnz( ur + vr ) def colinear( u , v ): """Returns 0 if both vectors are 0, 1 if they are colinear, and the length of the vectors if they are not.""" d = len( u ) z = numpy.zeros( u.shape , "d" ) uzhuh = numpy.alltrue( numpy.allclose( u , z ) ) vzhuh = numpy.alltrue( numpy.allclose( v , z ) ) if uzhuh and vzhuh: return 0 elif uzhuh or vzhuh: return d else: uhat = util.unit_vector( u ) vhat = util.unit_vector( v ) if numpy.alltrue( numpy.allclose( uhat , vhat ) ) \ or numpy.alltrue( numpy.allclose( uhat , -vhat ) ): return 1 else: return d def rho( u , v ): """Amalgamates several complexity-reducing relations into a single function for the purposes of the brute-force graph construction""" ep = edplus( u , v ) em = edminus( u , v ) c = colinear( u , v ) m = min( em , ep , c ) if ep == m: l = "edp" elif em == m: l = "edm" else: l = "c" return (m,l) def get_graph( vecs , dist ): """Brute force construction of the CRR graph.""" G = dict( [ (v,{}) for v in vecs ] ) for (u,v) in xuniqueCombinations( vecs.keys() , 2 ): (w,l) = dist( vecs[u] , vecs[v] ) G[u][v] = (w,l) G[v][u] = (w,l) return G def get_graph_clever( vecs ): """This builds a (possibly unconnected) graph based on Hamming distance and colinearity, but it typically works in O(n log n) time. Lack of connectedness is not a problem, as we can apply Prim's algorithm to each connected component.""" digits = 11 n = len( vecs ) d = len( vecs.itervalues().next() ) vecs = dict( [ (i,sigdig.vec_round_sig(v,digits)) \ for (i,v) in vecs.iteritems() ] ) G = dict( [ (i,{}) for i in vecs ] ) # compute positive/negative Hamming distance between all the vectors # need to set up array of hash tables of unique entries in # position i to keys of vectors with that value in position i Gbig = dict( [ ((1,i),{}) for i in vecs ] + \ [ ((-1,i),{}) for i in vecs ] ) tables = [ {} for i in range(d)] for i in vecs: for j in range(d): vj = vecs[i][j] if vj in tables[j]: tables[j][vj].append( (1,i) ) else: tables[j][vj] = [(1,i)] mvj = -vecs[i][j] if mvj in tables[j]: tables[j][mvj].append( (-1,i) ) else: tables[j][mvj] = [(-1,i)] sgncode = { 1:"edp" , -1:"edm" } # create the graph mapping \pm each vector to \pm each other # vector, *if* they share a common entry for j in range(d): for vj in tables[j]: for (tup1,tup2) in \ xuniqueCombinations( tables[j][vj] , 2 ): if tup2 in Gbig[tup1]: Gbig[tup1][tup2] -= 1 Gbig[tup2][tup1] -= 1 else: Gbig[tup1][tup2] = d-1 Gbig[tup2][tup1] = d-1 # this extracts the minimum Hamming distance between # +v1,+v2 and +v1,-v2 and writes it into the graph for i1 in vecs: nbs = [ i2 for (sgn,i2) in Gbig[(1,i1)] if i2 > i1 ] for i2 in nbs: wp = Gbig[(1,i1)].get((1,i2),d) wm = Gbig[(1,i1)].get((-1,i2),d) if wp <= wm: G[i1][i2] = (wp,"edp") G[i2][i1] = (wp,"edp") else: G[i1][i2] = (wm,"edm") G[i2][i1] = (wm,"edm") # now I need to write in colinear vectors. # first, filter out zeros z = numpy.zeros( (d,),"d" ) remaining = dict( [ x for x in vecs.iteritems() \ if not numpy.alltrue( \ numpy.allclose( x[1] , z , eps ) ) ] ) # then, call pg to get colinear terms Ls = pg.rp_line_finder( remaining , 1 ) for L in Ls: for (i1,i2) in xuniqueCombinations( list(L) , 2 ): if i2 in G[i1]: if G[i1][i2][0] >= 1: G[i1][i2] = (1,"c") G[i2][i1] = (1,"c") # this is a sanity check to make sure I didn't screw up for i in G: for j in G[i]: if G[i][j][0] != rho(vecs[i],vecs[j])[0]: print "foo" print G[i][j],rho(vecs[i],vecs[j]) print vecs[i], vecs[j] print return G def process_bf( vecs ): """Takes a dictionary mapping vector labels to the numpy.array objects, and returns the minimum spanning tree of the associated CRR graph. This runs in something like O(n^2) plus the cost of forming the minimum spanning tree. Officically, it's O(n^3) because I'm doing a suboptimal MST implementation, but the cost seems dominated by building the graph rather than the MST in the regime I've considered. Eventually I should implement a better MST algorithm if this is a bottleneck""" return graph.prim( get_graph( vecs , rho ) ) def process( vecs ): """Uses efficient graph construction algorithm then returns the minimum spanning forest.""" # G = get_graph_clever( vecs ) G = get_graph( vecs , rho ) return reduce( lambda a,b: graph.merge_disjoint( a , b ) , \ map( graph.prim , \ graph.connectedComponents( G ) ) ) def snip( mst , Adict ): """Takes the mst as input. For each node, if the cost of doing the dot product by brute force is less than or equal to the cost of using the parent, remove that dependency. This transforms the tree to a forest, which is actually good for data dependency and locality""" forest = {} for v in mst: if not mst[v]: forest[v] = {} else: num_nz = util.nnz( Adict[v] ) mst_cost = mst[v].values()[0][0] if num_nz <= mst_cost: forest[v] = {} else: forest[v] = mst[v] return forest def cost( g , vecs ): """Takes a MST graph constructed in process and returns the sum of edge weights plus the number of nonzeros in the root; this is the total MAPs needed to form an element stiffness matrix.""" w = 0 for u in g: if g[u]: v = g[u].keys()[0] w += g[u][v][0] else: w += util.nnz( vecs[u] ) return w def nodep_code( u ): """Abstract code for a dot product performed by brute force""" return [ (u[i],1,i) for i in range(len(u)) \ if abs(u[i]) > 1.e-6 ] # returns the abstract code # associated with various dependencies # u^t g is assumed known, v^t is to be computed. # v^t g = u^t g + ( v - u )^t g def ep_code( Adict , i1 , i2 ): """Returns abstract code for computing the dot product of Adict[i2] from Adict[i1] using positive Hamming distance""" u = Adict[i1] v = Adict[i2] diff = v - u diffinds = [] # which indices matter? for i in range( len( diff ) ): if abs( diff[i] ) >= 1.e-4: diffinds.append(i) oplist = [ (1.0,0,i1) ] for i in diffinds: oplist.append( (diff[i],1,i) ) return oplist # returns the abstract code # associated with various dependencies # u^t g is assumed known, v^t is to be computed. # v^t g = -u^t g + ( v + u )^t g def em_code( Adict , i1 , i2 ): """Returns abstract code for computing the dot product of Adict[i2] from Adict[i1] using negative Hamming distance""" u = Adict[i1] v = Adict[i2] diff = v + u diffinds = [] # which indices matter? for i in range( len( diff ) ): if abs( diff[i] ) >= 1.e-4: diffinds.append(i) oplist = [ (-1.0,0,i1) ] for i in diffinds: oplist.append( (diff[i],1,i) ) return oplist # v^t g = v[0]/u[0] ( u^t g ) def c_code( A , i1 , i2 ): """Returns abstract code for computing Adict[i2]^t g from Adict[i1]^t g using colinearity.""" u = numpy.reshape( A[i1] , (-1,) ) v = numpy.reshape( A[i2] , (-1,) ) i=0 while abs(u[i]) < 1.e-6 and i. # # First added: 2005-04-01 # Last changed: 2006-04-01 #from ffc.compiler.compiler import * import sigdig def symmetric_g( a ): if numpy.rank( a ) != 2 or a.shape[0] != a.shape[1]: print a raise RuntimeError, "Illegal matrix" n = a.shape[ 0 ] new_len = n * (n+1) / 2 result = numpy.zeros( (new_len,) , "d" ) cur = 0 for i in range( n ): result[ cur ] = a[ i , i ] cur += 1 for j in range( i+1 , n ): result[ cur ] = a[ i , j ] + a [ j , i ] cur += 1 return result def laplacianform( shape , degree ): el = FiniteElement( "Lagrange" , shape , degree ) k = Index() dx = Integral( "interior" ) v = BasisFunction( el ) u = BasisFunction( el ) a = v.dx( k ) * u.dx( k ) * dx ac = build( a , "fooform" , "raw" ) return ac.reference_tensor() ################################################################ # The following functions take a shape and degree and return # the reference tensor as a dictionary mapping indices to # the appropriately processed tensors ################################################################ def laplacian( shape , degree ): el = FiniteElement( "Lagrange" , shape , degree ) k = Index() dx = Integral( "interior" ) v = BasisFunction( el ) u = BasisFunction( el ) a = v.dx( k ) * u.dx( k ) * dx ac = build( a , "fooform" , "raw" ) A0 = sigdig.tensor_round_sig( ac.reference_tensor() , 10 ) # put in symmetric part, symmetry transform applied Adict = {} for i in range(A0.shape[0]): for j in range(i,A0.shape[1]): Adict[i,j] = sigdig.vec_round_sig( symmetric_g( A0[i,j] ) , 10 ) return Adict def laplacian_matvec( shape , degree ): el = FiniteElement( "Lagrange" , shape , degree ) k = Index() dx = Integral( "interior" ) v = BasisFunction( el ) u = BasisFunction( el ) a = v.dx( k ) * u.dx( k ) * dx ac = build( a , "fooform" , "raw" ) A0 = sigdig.tensor_round_sig( ac.reference_tensor() , 10 ) Adict = {} for i in range(A0.shape[0]): Adict[i] = numpy.reshape( A0[i] , (-1,) ) return Adict def weighted_laplacian( shape , degree ): el = FiniteElement( "Lagrange" , shape , degree ) k = Index() dx = Integral( "interior" ) w = Function( el ) v = BasisFunction( el ) u = BasisFunction( el ) a = w * v.dx( k ) * u.dx( k ) * dx ac = build( a , "fooform" , "raw" ) A0 = sigdig.tensor_round_sig( ac.reference_tensor() , 10 ) Adict = {} for i in range(A0.shape[0]): for j in range(i,A0.shape[1]): tmp = numpy.array( map( symmetric_g , A0[i,j] ) ) tmp = numpy.reshape( tmp , (-1,) ) Adict[i,j] = sigdig.vec_round_sig( tmp , 10 ) return Adict def weighted_laplacian_g_first( shape , degree ): el = FiniteElement( "Lagrange" , shape , degree ) k = Index() dx = Integral( "interior" ) w = Function( el ) v = BasisFunction( el ) u = BasisFunction( el ) a = w * v.dx( k ) * u.dx( k ) * dx ac = build( a , "fooform" , "raw" ) A0 = sigdig.tensor_round_sig( ac.reference_tensor() , 10 ) Adict = {} for i in range(A0.shape[0]): for j in range(i,A0.shape[1]): for k in range(0,A0.shape[2]): Adict[i,j,k] = symmetric_g(A0[i,j,k]) return Adict def weighted_laplacian_coeff_first( shape , degree ): el = FiniteElement( "Lagrange" , shape , degree ) k = Index() dx = Integral( "interior" ) w = Function( el ) v = BasisFunction( el ) u = BasisFunction( el ) a = w * v.dx( k ) * u.dx( k ) * dx ac = build( a , "fooform" , "raw" ) A0 = sigdig.tensor_round_sig( ac.reference_tensor() , 10 ) Adict = {} for i in range(A0.shape[0]): for j in range(i,A0.shape[1]): tmp = numpy.array( map( symmetric_g, A0[i,j] ) ) for k in range(tmp.shape[1]): Adict[i,j,k] = tmp[:,k] return Adict ferari-1.0.0/ferari/combined.py0000644000000000000000000001062711667671022014526 0ustar 00000000000000# Copyright (C) 2006 Robert C. Kirby # # This file is part of FErari. # # FErari is free software: you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # FErari is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with FErari. If not, see . # # First added: 2005-04-01 # Last changed: 2006-04-01 import binary, pg, build_tensors, graph from xpermutations import xuniqueCombinations def process_prime( vecs ): n = len( vecs ) d = len( vecs.iteritems().next() ) # get graph of CRR weights G = binary.get_graph( vecs , binary.rho ) # do partial geometric processing up to colinearity p = pg.process( vecs , 1 ) # "reverse arrows" in the dependency graph # to get a mapping from remaining vectors to their children # this will let us enumerate children = dict( [ (v,[]) for v in vecs ] ) for (v,parent) in p.iteritems(): if parent: pcur = parent.iterkeys().next() children[pcur].append( v ) # grab the things that didn't have a linear dependency remaining = dict( [ a for a in vecs.iteritems() if not p[a[0]] ] ) # search for second order linear dependencies Ls = pg.rp_line_finder( remaining , 2 ) vecs_to_lines = dict( [ (v,set()) for v in vecs ] ) for L in Ls: for v in L: vecs_to_lines[v].add( L ) def process( vecs ): n = len( vecs ) d = len( vecs.iteritems().next() ) # get graph of CRR weights G = binary.get_graph( vecs , binary.rho ) # do partial geometric processing up to colinearity p = pg.process( vecs , 1 ) # "reverse arrows" in the dependency graph # to get a mapping from remaining vectors to their children # this will let us enumerate children = dict( [ (v,[]) for v in vecs ] ) for (v,parent) in p.iteritems(): if parent: pcur = parent.iterkeys().next() children[pcur].append( v ) # grab the things that didn't have a linear dependency remaining = dict( [ a for a in vecs.iteritems() if not p[a[0]] ]) # search for second order linear dependencies Ls = pg.rp_line_finder( remaining , 2 ) vecs_to_triples = dict([ (v,set()) for v in vecs ]) for L in Ls: for trip in xuniqueCombinations( list(L) , 3 ): for c1 in [trip[0]]+ children[trip[0]]: for c2 in [trip[1]] + children[trip[1]]: for c3 in [trip[2]] + children[trip[2]]: newtrip = (c1,c2,c3) for c in newtrip: vecs_to_triples[c].add(newtrip) # modified Prim's algorithm start = G.iterkeys().next() weights = { start: 0 } parents = { start: None } done = set() while weights: u = graph.argmin( weights ) weights.pop( u ) done.add( u ) for v in G[u]: if v not in done: if v not in weights \ or G[u][v][0] < weights[v]: weights[v] = G[u][v][0] parents[v] = (u,G[u][v][0],G[u][v][1]) for trip in vecs_to_triples[u]: undone_in_trip = set(trip).difference( done ) if len( undone_in_trip ) == 1: vnew = undone_in_trip.pop() if vnew not in weights or 2 < weights[vnew]: weights[vnew] = 2 newparent = tuple(set(trip).difference((vnew,))) parents[vnew] = (newparent,2,"lc") return parents def graph_cost( p , A0dict ): s = 0 for u in p: if p[u]: s += p[u][1] else: s += binary.nnz(A0dict[u]) return s def main(): data = [] for shape in ("tetrahedron",): for degree in range(2,4): A0dict = build_tensors.laplacian( shape , degree ) p = process( A0dict ) data.append( (len(A0dict),len(A0dict.itervalues().next()),\ graph_cost( p , A0dict ) ) ) print data if __name__ == "__main__": main() ferari-1.0.0/ferari/graph.py0000644000000000000000000001617611667671022014054 0ustar 00000000000000# Copyright (C) 2006 Robert C. Kirby # # This file is part of FErari. # # FErari is free software: you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # FErari is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with FErari. If not, see . # # First added: 2005-04-01 # Last changed: 2006-04-01 import copy, os def argmin( a ): """a is a dictionary, returns the key for which the value is minimal""" am = a.keys()[0] minval = a[a.keys()[0]] for (k,ak) in a.iteritems(): if ak < minval: am = k minval = ak return am def prim( G ): """G is a dictionary whose keys are nodes in a graph. The values are dictionaries mapping neighbors to (weight,label) pairs/ Returns a directed graph rooted at the first node in G""" start = G.iterkeys().next() weights = { start: 0 } parents = { start: None } done = set() while weights: u = argmin( weights ) weights.pop( u ) done.add( u ) for v in G[u]: if v not in done: if v not in weights or \ G[u][v][0] < weights[v]: weights[v] = G[u][v][0] parents[v] = (u,G[u][v][0],G[u][v][1]) pgraph = dict( [ (u,{}) for u in parents ] ) for u in parents: if parents[u]: #not a root (p,w,l) = parents[u] pgraph[u][p] = (w,l) return pgraph def topsort( p ): """p is a digraph with nodes pointing from u to v if u depends on v. We produce an ordering such that u comes after v. This is the reverse of a typical topological sort.""" g = copy.deepcopy( p ) order = [] while g: found_u = False for u in g: if len( g[u] ) == 0: found_u = True break if not found_u: raise RuntimeError, "not acyclic" order.append( u ) for v in g: if u in g[v]: g[v].pop( u ) g.pop( u ) return order def bfs( g ): order = [] found = set() Q = set() u = g.iterkeys().next() def bfs( g , start ): """returns a list that is a breadth-first ordering of the nodes of g reachable from start""" Q = [ start ] found = set( (start,) ) done = set() order = [] while Q: u = Q.pop( 0 ) done.add( u ) order.append( u ) for v in g[u]: if v not in found: found.add( v ) Q.append( v ) return order def connectedComponents( g ): """takes an undirected graph g and returns a list of graphs that are the connected components.""" components = [] found = set() while len( found ) != len( g ): # pick a member of g that is not found yet for u in g: if u not in found: break order = bfs( g , u ) component_cur = dict( [ (v , g[v] ) for v in order ] ) found.update( order ) components.append( component_cur ) return components def merge_disjoint( g1 , g2 ): g3 = copy.deepcopy( g1 ) for u in g2: if u in g3: raise RuntimeError, "graphs not disjoint" g3[u] = copy.deepcopy( g2[u] ) return g3 # mooched/modified from pygraphlib. class Dot: ''' A class that creates a B{graphviz} (dot language) representation of the graph. To make use of the image generation features If the C{dot} and C{dotty} programs must be either be in the system path or their location needs to be specified in the L{constructor<__init__>}. Download and install the graphviz programs then either set the See the L{pydot} module level documentation for usage examples. ''' def __init__(self, graph, name="G", dot='dot', dotty='dotty', neato='neato'): self.graph = graph self.temp_file = 'pydot_temp.dot' self.name, self.style = name, {} self.dot , self.dotty, self.neato = dot, dotty, neato self.node_style, self.edge_style = {}, {} def set_style(self, **kwargs): 'Changes the overall style' self.style = kwargs def set_node_style(self, node, **kwargs): 'Sets the style for a node.' self.node_style[node] = kwargs def set_all_node_style(self, **kwargs): 'Sets the styles for all nodes' for node in self.graph: self.set_node_style(node, **kwargs) def set_edge_style(self, head, tail, **kwargs): 'Sets the stye for a single edge' key1, key2 = (head, tail), (tail, head) self.edge_style.setdefault(key1, kwargs).update(kwargs) def set_all_edge_style(self, **kwargs): 'Sets the styles for all edges' for u in self.graph: for v in self.graph[u]: self.set_edge_style(u,v, **kwargs) def save_dot(self, file_name=None): 'Saves the current graph represenation as a C{dot} file ' if not file_name: file_name = self.temp_file fp = open(file_name, "w") header = "digraph %s {\n" % self.name fp.write(header) # write overall graph style for attr_name, attr_value in self.style.items(): fp.write('%s="%s"; ' % (attr_name, attr_value)) fp.write("\n") # shortcuts to some reusable patterns beg_patt = '\t"%s" [' # to begin attributes mid_patt = '"%s"="%s",' # to write attributes end_patt = '];\n' # to end attributes edg_patt = '\t"%s" -> "%s" [' # to begin edges # write the node attributes for node in self.graph: fp.write( beg_patt % (node,)) if self.node_style.has_key(node): for attr_name, attr_value in self.node_style[node].items(): fp.write(mid_patt % (attr_name, attr_value)) fp.write(end_patt) seen = {} # write edge attributes for u in self.graph: for v in self.graph[u]: edge = (u, v) fp.write(edg_patt % edge ) if self.edge_style.has_key(edge): for attr_name, attr_value in self.edge_style[edge].items(): fp.write(mid_patt % (attr_name, attr_value)) fp.write(end_patt) fp.write("}\n") fp.close() def show(self): 'Displays the current graph via dotty' self.save_dot(self.temp_file) show_cmd = "%s %s&" % (self.dotty, self.temp_file) os.system(show_cmd) def save_image(self, file_name="out", mode="gif"): 'Saves the dot file as an image file' self.save_dot(self.temp_file) save_cmd = "%s -T%s %s -o %s" % (self.dot, mode, self.temp_file, file_name) os.system(save_cmd) ferari-1.0.0/ferari/migrate0000644000000000000000000000600311667671022013740 0ustar 00000000000000To migrate from Python to C++: ublas: They give us containers and some basic liner algebra operators. This is part of boost; we should use it. Make a single function "optimize" - Graph optimize( int num_vec , int vec_length , double *vec ); Why the low-level input interface? That is because we will have the vectors in contiguous storage from Numerical Python. Don't assume a dictionary input. Maybe we can add a layer that wraps the Numeric array into a ublas array inside of Python? - to find nonzeros: make list of zero indices, then add to graph? - to find equality: - go ahead and normalize sign of each vector rather than sort, define a comparison function that fudges potential floating-point roundoff. Use a map to store data -- this is something like an AVL tree underneath and will guarantee O(log(n)) insertion. Probably will be faster and easier to use than a hash table. normalized cross product: - call lapack to get the full svd? This should be more efficient than computing determinants. bf_line_finder: Need to figure out C++ equivalent of xuniqueCombinations when checking for linear dependence, doing connected components should be easy in boost -- there is boost/graph/connected_components.hpp "remaining" -- this can just be an stl list of the labels of vectors that are still active. This means not using filterdict, but something else. gen_graph: - We should be able to use a map to define the priority of each line in the line graph. This will allow efficient updates of priorities, and we can find/remove an extremal element in O(log(n)) time. This keeps us from having to write any funny mutable heaps. - Is it best to use STL sets in computing the gen_graph? For that matter, is it best to use sets in C++ everywhere we use them in Python? At this point, it will be interesting to see how fast bf_line_finder is in C++ versus Python (or versus the rp_line_finder in Python, for that matter). rp_line_finder: - random projection should require calling gesvd. Examples of putting in prototypes to FORTRAN LAPACK/BLAS functions: // LAPACK interface prototypes extern "C" { void dgetrf_( int *m , int *n , double *a, int *lda , int *ipiv , int *info ); void dgetri_( int *n , double *a , int *lda , int *ipiv , double *work , int *lwork , int *info ); void dgemm_( char *ta , char *tb , int *m , int *n, int *k , double *alpha , double *a , int *lda , double *b , int *ldb , double *beta , double *c , int *ldc ); }; Then you call it just like a regular function. Google the online references for BLAS and LAPACK for more information. You will want dgesvd for the svd, I think. Also, you will probably want to use the ublas overloaded operators instead of dgemm (matrix-matrix multiply) since it will be easier to code and not too much slower. Under no circumstances should you write your own matrix-matrix multiply. ferari-1.0.0/ferari/pg.py0000644000000000000000000003703411667671022013355 0ustar 00000000000000# Copyright (C) 2006 Robert C. Kirby # # This file is part of FErari. # # FErari is free software: you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # FErari is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with FErari. If not, see . # # First added: 2005-04-01 # Last changed: 2006-04-01 import numpy,build_tensors,graph from xpermutations import xuniqueCombinations from numpy.random import rand from numpy.linalg import svd, det precision = 5 eps = .1**precision def randproj( m , n ): """creates a random matrix in R^{m,n}""" # maps from R^n to R^m if m >= n: A = rand( m , n ) [u,s,vt] = svd( A ) return u else: A = rand( n , m ) [u,s,vt] = svd(A) return numpy.transpose(u) def nnz( u ): """Returns the number of items in u that are farther away from zero than eps.""" nnz_cur = 0 for ui in u: if abs( ui ) > eps: nnz_cur += 1 return nnz_cur def nnz_cmp( x , y ): """Allows sorting according to the sparsity of the vectors""" nnzx = nnz(x) nnzy = nnz(y) if nnzx < nnzy: return -1 elif nnzx == nnzy: return 0 else: return 1 def compare( u , v ): """Lexicographic ordering that respects floating-point fuzziness""" if len(u) != len(v): raise RuntimeError, "can't compare" diff = u-v for d in diff: if abs(d) > eps: if d < 0: return -1 else: return 1 return 0 def kvp_cmp( x , y ): return compare( x[1] , y[1] ) def normalize_sign( u ): """Returns u or -u such that the first nonzero entry (up to eps) is positive""" for ui in u: if abs(ui) > eps: if ui < 0.0: return -u else: return u return u def unit_vector( u ): mau = max(abs(u)) if mau < eps: raise RuntimeError, "barf: divide by zero" return normalize_sign( u / mau ) def filterequal( vs , compare ): """vs is a list of key/value pairs such that equal values must be adjacent in the list. Returns a list of lists, each of which is the set of key/value pairs corresponding to equal items.""" outer_index = 0 equal_items = [] while outer_index < len( vs ): cur_items = [ ] inner_index = outer_index while inner_index< len(vs) \ and compare( vs[inner_index] , \ vs[outer_index] ) == 0: cur_items.append( vs[inner_index] ) inner_index += 1 equal_items.append( cur_items ) outer_index = inner_index return equal_items def rank( A ): """Returns the numerical rank of an array.""" u,s,vt = svd( A ) return len( [ si for si in s if abs( si ) > eps ] ) def cohyperplanar(bs,n): """Predicate indicating whether the members of bs span an n-dimensional space""" mat = numpy.array( bs ) return rank( numpy.array( mat ) ) == n def strike_col( A , j ): m,n = A.shape return numpy.take( A , [ k for k in range(0,n) if k != j ] , 1 ) def normalized_cross( vecs ): n,d = len( vecs ) , len( vecs[ 0 ] ) mat = numpy.array( vecs ) if n != d - 1: for v in vecs: print v raise RuntimeError, "barf" # normalize the cross product to have unit value to avoid # rounding to zero in the cross product routine. foo = numpy.array( \ [ (-1)**i * det(strike_col(mat,i)) \ for i in range( d ) ] ) foo /= max( abs( foo ) ) return normalize_sign( foo ) def bf_line_finder( vecs , m ): tups = [ frozenset( t ) \ for t in xuniqueCombinations( vecs.keys() , m + 1 ) \ if cohyperplanar( [ vecs[i] for i in t ] , m ) ] gr = dict( [ ( tup , {}) for tup in tups ] ) for (tup1,tup2) in xuniqueCombinations( tups , 2 ): if len( tup1.intersection( tup2 ) ) == m: gr[tup1][tup2] = (m,None) gr[tup2][tup1] = (m,None) ccs = graph.connectedComponents( gr ) Ls = [] for cc in ccs: Lcur = set() for k in cc: Lcur.update( k ) Ls.append( Lcur ) return [ frozenset( L ) for L in Ls ] def rp_line_finder( vecs , m ): d = len( vecs.itervalues().next() ) udude = randproj(m+1,d) pi_vecs = dict( [ (i,numpy.dot(udude,v ) ) \ for (i,v) in vecs.iteritems() ] ) pi_normals = [ (tup,normalized_cross( [ pi_vecs[t] for t in tup ]))\ for tup in xuniqueCombinations( pi_vecs.keys() , \ m ) ] pi_normals.sort( kvp_cmp ) ## for i,n in pi_normals: ## print i,n eitems = filterequal( pi_normals , kvp_cmp ) Ls = [] for e in eitems: if len( e ) > 1: inds = set() for (indlist,a) in e: inds.update( indlist ) vs = numpy.array( [ vecs[i] for i in inds ] ) if rank( vs ) == m: Ls.append( frozenset( inds ) ) else: mini_vecs = dict( [ (i,vecs[i]) for i in inds ] ) Ls.extend( bf_line_finder( mini_vecs , m ) ) return Ls def line_graph( Ls , m ): """Returns a graph whose elements are the members of Ls and whose edges have weights that are the size of the intersection.""" gr = dict( [ (L,{}) for L in Ls ] ) for (L1,L2) in xuniqueCombinations(Ls,2): n = len( L1.intersection(L2) ) if n > 0: gr[L1][L2] = (m - n,None) gr[L2][L1] = (m - n,None) return gr def gen_graph ( vecs , Ls , m ): """vecs is a dictionary of label/vector and Ls is a list of sets indicating (hyper) lines in a partial geometry based on dependency between m+1 things.""" lg = line_graph( Ls , m ) gg = {} weights = dict( [ (L,m) for L in Ls ] ) while weights: # pick a line with minimal weight L = graph.argmin( weights ) w = weights[L] weights.pop( L ) Ldone = L.intersection( gg ) # sanity check: # if there are k items in Ldone and k <= m # then the weight I get had better # be m - k lldone = len( Ldone ) if (lldone <= m and w != m - lldone) \ or (lldone > m and w != 0): print ct print L print Ldone print w raise RuntimeError, "barf" Ltodo = L.difference( Ldone ) roots = set() for r in Ldone: if len( roots ) == m: break else: roots.add( r ) # pick the sparsest new elements as new generator members Ltodo_list = list( Ltodo ) Ltodo_list.sort(nnz_cmp) for r in Ltodo_list: if len( roots ) == m: break else: roots.add( r ) done_roots = roots.intersection( gg ) new_roots = roots.intersection( Ltodo ) # sanity check: length of new_roots # ought to be equal to w if len( new_roots ) != w: raise RuntimeError, "barf" # add new items to gg for u in new_roots: if u in gg: raise RuntimeError, "barf" gg[u] = {} # new roots go into graph have no out-edges # rest of Ltodo go into graph with out-edges to # each member of the roots. for u in Ltodo.difference( new_roots ): if u in gg: raise RuntimeError, "barf" gg[u] = {} for v in roots: gg[u][v] = (m,None) # update neighboring lines' weight for Lnb in lg[L]: if Lnb in weights: weights[Lnb] = max( 0 , m - len( Lnb.intersection( gg ) ) ) return gg def gen_graph_cost( gg , A0dict ): """Evaluates the total floating-point cost of the dot product algorithm associated with a generation graph.""" cost = 0 for u in gg: if len( gg[u] ) == 0: cost += nnz( A0dict[u] ) elif len( gg[u] ) == 1: cost += gg[u].values()[0][0] else: cost += len( gg[u] ) return cost def process( vecs , mmax ): first_index = vecs.iterkeys().next() n = len( vecs ) d = len( vecs[first_index] ) z = numpy.zeros( (d,) , "d" ) def filterdict( source , filt ): return dict( [ a for a in source.iteritems() \ if a[0] not in filt ] ) parents = {} print "finding zeros..." zeros = [ x for x in vecs.iteritems() \ if numpy.alltrue( numpy.allclose( x[1],z,eps ) ) ] for z in zeros: parents[z[0]] = {} remaining = filterdict( vecs , parents ) print "down to %d vecs" % ( len( remaining ) , ) print "filtering equal vectors" # create the list of projections of vectors, each with positive # first nonzero entry projected = [ (x[0],normalize_sign(x[1])) \ for x in remaining.iteritems() ] projected.sort( kvp_cmp ) eitems = filterequal( projected , kvp_cmp ) for eit in eitems: item_cur = eit[0] for item in eit[1:]: parents[item[0]] = { item_cur[0] : (0,"e") } remaining = filterdict( remaining , parents ) print "done filtering equal vectors" for m in range(1,mmax+1): if len( remaining ) <= m: break print "filtering linear dependence of order " , m Ls = rp_line_finder( remaining , m ) gg = gen_graph( remaining , Ls , m ) for u in gg: if gg[u]: parents[u] = dict( [ (v,(m,"lc")) for v in gg[u] ] ) remaining = filterdict( remaining , parents ) print "%s vectors remaining" % ( len(remaining,) ) for r in remaining: parents[r] = {} return parents def process2( vecs , max_level ): first_index = vecs.iterkeys().next() n = len( vecs ) d = len( vecs[first_index] ) z = numpy.zeros( (d,) , "d" ) def filterdict( source , filt ): return dict( [ a for a in source.iteritems() \ if a[0] not in filt ] ) parents = {} nv = len( vecs ) zeros = [ x for x in vecs.iteritems() \ if numpy.alltrue( numpy.allclose( x[1],z,eps ) ) ] nz = len( zeros ) nr = nv - nz for z in zeros: parents[z[0]] = {} remaining = filterdict( vecs , parents ) projected = [ (x[0],normalize_sign(x[1])) \ for x in remaining.iteritems() ] projected.sort( kvp_cmp ) eitems = filterequal( projected , kvp_cmp ) for eit in eitems: item_cur = eit[0] for item in eit[1:]: parents[item[0]] = { item_cur[0] : (0,"e") } remaining = filterdict( remaining , parents ) ne = nr - len( remaining ) nr = len( remaining ) print "done filtering equal vectors" nactive = [] nleft = [nr] curcost = [] for i in range(1,max_level+1): print "filtering linear dependence of order " , i print "%s vectors remain" % (nleft[-1],) Ls = rp_line_finder( remaining , i ) # figure out who lives in linear relations, who doesn't active = set() for L in Ls: for u in L: active.add( u ) nactive.append( len( active ) ) print "%s elements are active" % (len(active),) gg = gen_graph( remaining , Ls , i ) for u in gg: if gg[u]: parents[u] = dict( [(v,(i,"lc")) for v in gg[u] ] ) remaining = filterdict( remaining , parents ) nr = len( remaining ) nleft.append( nr ) print "%s remain now" % (nr,) # temporarily add in everything remaining by brute force to the # generation graph, compute the cost, and then remove them for r in remaining: parents[r] = {} cc = gen_graph_cost( parents , vecs ) curcost.append( cc ) print "current cost is down to %s" % (cc,) for r in remaining: parents.pop( r ) for r in remaining: parents[r] = {} return parents,n,d,nz,ne,nleft,nactive,curcost def process_gen_graph( vecs ): """Returns the generation graph for vectors active at 2-dependencies""" first_index = vecs.iterkeys().next() n = len( vecs ) d = len( vecs.itervalues().next() ) z = numpy.zeros( (d,) , "d" ) def filterdict( source , filt ): return dict( [ a for a in source.iteritems() \ if a[0] not in filt ] ) parents = {} print "finding zeros..." zeros = [ x for x in vecs.iteritems() \ if numpy.alltrue( numpy.allclose( x[1],z,eps ) ) ] for z in zeros: parents[z[0]] = {} remaining = filterdict( vecs , parents ) print "down to %d vecs" % ( len( remaining ) , ) print "filtering equal vectors" projected = [ (x[0],normalize_sign(x[1])) \ for x in remaining.iteritems() ] projected.sort( kvp_cmp ) eitems = filterequal( projected , kvp_cmp ) for eit in eitems: item_cur = eit[0] for item in eit[1:]: parents[item[0]] = { item_cur[0] : (0,"e") } remaining = filterdict( remaining , parents ) print "done filtering equal vectors" for m in range(1,3): if len( remaining ) <= m: break print "filtering linear dependence of order " , m Ls = rp_line_finder( remaining , m ) gg = gen_graph( remaining , Ls , m ) for u in gg: if gg[u]: parents[u] = dict( [ (v,(m,"lc")) for v in gg[u] ] ) remaining = filterdict( remaining , parents ) print "%s vectors remaining" % ( len(remaining,) ) ggdot = graph.Dot( gg ) ggdot.save_image( "foo%d.png" % (m,) , "png") for r in remaining: parents[r] = {} return parents def main_gg(): shape="triangle" degree=3 A0dict=build_tensors.laplacian(shape,degree) process_gen_graph(A0dict) def main(): import string results = {} shape = "triangle" degrees = range(2,4) max_level = 4 for degree in degrees: A0dict = \ build_tensors.weighted_laplacian_coeff_first(shape,degree) results[degree] = process2(A0dict,max_level) # now let's print the table # first row is heading result_str = """ \\begin{tabular}{%s} %s \\\\ \\hline %s \\\\ %s \\\\ %s \\\\ %s \\\\ %s \\\\ \\hline """ % ( string.join( ["c"]*(len(degrees)+1) , "|" ) , \ string.join( [""] + map(str,degrees) , " & ") , \ string.join( ["$n$"] + [str(a[1]) \ for a in results.values()] , " & ") , \ string.join( ["$d$"] + [str(a[2]) \ for a in results.values()] , " & ") , \ string.join( ["num zero"] + [str(a[3]) \ for a in results.values()] , " & ") , \ string.join( ["num equal"] + [str(a[4]) \ for a in results.values()] , " & ") , \ string.join( ["num colinear"] + [str(a[6][0]) \ for a in results.values()] , " & " ) ) for level in range(2,max_level+1): result_str += """%s \\\\ %s \\\\ %s \\\\ %s \\\\ \\hline """ % ( string.join( ["size for %d-search"%(level,)] + \ [ str(a[5][level-1]) \ for a in results.values()] , " & " ) , \ string.join( ["num with %d-dependency" % (level,) ] + \ [ str(a[6][level-1]) \ for a in results.values()] , " & " ) , \ string.join( ["generator size"] + \ [ str(a[5][level]) \ for a in results.values()] , " & " ) , \ string.join( ["MAPs"] + \ [ str(a[7][level-1]) \ for a in results.values()] , " & " ) ) result_str += "\\end{tabular}" print result_str if __name__ == "__main__": main_gg() ferari-1.0.0/ferari/sigdig.py0000644000000000000000000000253411667671022014212 0ustar 00000000000000# Copyright (C) 2006 Robert C. Kirby # # This file is part of FErari. # # FErari is free software: you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # FErari is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with FErari. If not, see . # # First added: 2005-04-01 # Last changed: 2006-04-01 import numpy def round_sig( a , m ): if abs( a ) < 10**(-m): return 0.0 else: m_str , e_str = ( "%.12e" % (a,) ).split( "e" ) return round( float( m_str ) , m ) * 10**int(e_str) def vec_round_sig( a , m ): return numpy.array( [ round_sig( x , m ) for x in a ] ) def tensor_round_sig( a , m ): asize = reduce( lambda a,b:a*b , a.shape ) flat = numpy.reshape( a , (asize,1) ) around = vec_round_sig( flat , m ) return numpy.reshape( around , a.shape ) if __name__ == "__main__": a = 1.66666666666666666666 print "%.16e" % (a,) print round_sig( a , 8 )ferari-1.0.0/ferari/util.py0000644000000000000000000000104711667671022013717 0ustar 00000000000000eps = 1.e-10 precision = 5 def nnz( u ): nnz_cur = 0 for ui in u: if abs( ui ) > eps: nnz_cur += 1 return nnz_cur def normalize_sign( u ): """Returns u or -u such that the first nonzero entry (up to eps) is positive""" for ui in u: if abs(ui) > eps: if ui < 0.0: return -u else: return u return u def unit_vector( u ): mau = max(abs(u)) if mau < eps: print u raise RuntimeError, "divide by zero" return normalize_sign( u / mau ) ferari-1.0.0/ferari/xpermutations.py0000644000000000000000000000346211667671022015667 0ustar 00000000000000#!/usr/bin/env python #from __future__ import generators __version__ = "1.0" """xpermutations.py Generators for calculating a) the permutations of a sequence and b) the combinations and selections of a number of elements from a sequence. Uses Python 2.2 generators. Similar solutions found also in comp.lang.python Keywords: generator, combination, permutation, selection See also: http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/105962 See also: http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/66463 See also: http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/66465 """ def xcombinations(items, n): if n==0: yield [] else: for i in xrange(len(items)): for cc in xcombinations(items[:i]+items[i+1:],n-1): yield [items[i]]+cc def xuniqueCombinations(items, n): if n==0: yield [] else: for i in xrange(len(items)): for cc in xuniqueCombinations(items[i+1:],n-1): yield [items[i]]+cc def xselections(items, n): if n==0: yield [] else: for i in xrange(len(items)): for ss in xselections(items, n-1): yield [items[i]]+ss def xpermutations(items): return xcombinations(items, len(items)) if __name__=="__main__": print "Permutations of 'love'" for p in xpermutations(['l','o','v','e']): print ''.join(p) print print "Combinations of 2 letters from 'love'" for c in xcombinations(['l','o','v','e'],2): print ''.join(c) print print "Unique Combinations of 2 letters from 'love'" for uc in xuniqueCombinations(['l','o','v','e'],2): print ''.join(uc) print print "Selections of 2 letters from 'love'" for s in xselections(['l','o','v','e'],2): print ''.join(s) print print map(''.join, list(xpermutations('done')))