# lebesgue.py # tested with python 2.6 and 3.1 on a linux platform. # source file: algebre.tm # license: MIT license (http://en.wikipedia.org/wiki/MIT_License) # copyright (c) 2010 Gribouillis at www.daniweb.com """ Module lebesgue This module defines a class LebesgueSet which instances represent finite unions of intervals of real numbers. Set operations on these finite unions of intervals include intersection, union, set difference, symetric difference (xor), set complement. Topological operations like testing wether a point is interior or on the boundary, or adherent to the set are provided as well as access to certain quantities, like the set's upper and lower bound and the Lebesgue measure of the set. Of course, the class is improperly named LebesgueSet because a Lebesgue Set in mathematics is a wild set which can be much more complicated than a finite union of intervals. However, conceptually, it's best to consider that the intervals are nor opened nor closed, and that an instance represents a union of intervals up to a set of measure 0. In this class, infinite values are represented by using the python value Float("infinity"). Infinite floats can be tested with math.isinf(). """ import unittest import itertools from bisect import bisect import operator import sys if sys .version_info >=(3 ,): xrange =range class LebesgueSet (object ): _inf =float ('infinity')# can be tested with math.isinf() _minf =-_inf UNION ,INTER ,XOR =range (3 ) def __init__ (self ,points ,left_infinite =False ,fast =False ): """Create a new LebesgueSet object. @ points: a sequence of real numbers @ left_infinite : a boolean, True if the first interval is semi-infinite ]-infinity, .[ @ fast : can be set to True if the sequence of points is already sorted and has no duplicates. """ if not fast : points =sorted (set (points )) self .points =list (points ) self .left_infinite =bool (left_infinite ) def intervals (self ): start =0 p =self .points n =len (p ) if self .left_infinite : if n : yield (self ._minf ,p [0 ]) start =1 else : yield (self ._minf ,self ._inf ) return while start +1 <n : yield (p [start ],p [start +1 ]) start +=2 if start <n : yield (p [start ],self ._inf ) def __str__ (self ): return str (list (self .intervals ())) def __nonzero__ (self ): return self .left_infinite or self .points def __invert__ (self ): """Compute the set's complement""" return LebesgueSet (self .points ,not self .left_infinite ,fast =True ) def is_bounded (self ): return not (self .left_infinite or (len (self .points )&1 )) def lower_bound (self ): if self .left_infinite : return self ._minf elif not self .points : return self ._inf else : return self .points [0 ] def upper_bound (self ): if self .left_infinite ^(len (self .points )&1 ): return self ._inf elif not self .points : return self ._minf else : return self .points [-1 ] def zoom (self ,center =0.0 ,factor =1.0 ): if factor ==0.0 : points =() left_infinite =False else : points =(center +factor *(x -center )for x in self .points ) if factor >0.0 : left_infinite =self .left_infinite else : left_infinite =(self .upper_bound ()==self ._inf ) points =reversed (points ) return LebesgueSet (points ,left_infinite ,fast =True ) def measure (self ): """self.measure() -> a finite or infinite float. Compute the Lebesgue measure of the set self. If this measure is infinite, return float('infinity').""" if self .is_bounded (): p =self .points return sum (p [i +1 ]-p [i ]for i in xrange (0 ,len (p )/2 ,2 )) else : return self ._inf def status (self ,x_real ): """self.status(x) returns -1, 0, 1 depending on x being outside, on the boundary, or inside the set self (x is a real number)""" i =bisect (self .points ,x_real ) if (i >0 )and x_real ==self .points [i -1 ]: return 0 return 1 if self .left_infinite ^(i &1 )else -1 def is_interior (self ,x ): """True if x is in the topological interior of the set self.""" return self .status (x )==1 def is_exterior (self ,x ): """True if x is in the topological interior of the complement.""" return self .status (x )==-1 def is_boundary (self ,x ): """True if x is one end of one of the intervals""" return self .status (x )==0 def is_adherent (self ,x ): """True if x is in one of the closed intervals defining self.""" return self .status (x )>=0 def deltas (self ,negated =False ): infinite =self .left_infinite ^(1 if negated else 0 ) assert infinite in (0 ,1 ) for i ,x in enumerate (self .points ): yield (x ,-1 if (infinite ^(i &1 ))else 1 ) @classmethod def operate (cls ,op ,family ): """Compute the union, intersection or xor of a family of sets. @ op : one of LebesgueSet.UNION or .INTER or .XOR @ family : a non empty sequence of LebesgueSet's. A real number x belongs to the xor of the family if it belongs to an odd number of members of the family. """ family =tuple (family ) value =sum (u .left_infinite for u in family ) P = if op ==cls .XOR : left_infinite =bool (value &1 ) L =sorted (itertools .chain (*(u .points for u in family ))) for k ,group in itertools .groupby (L ): if len (tuple (group ))&1 : P .append (k ) else : inter =(op ==cls .INTER ) if inter : value =len (family )-value left_infinite =not (value )if inter else bool (value ) L =sorted (itertools .chain (*(u .deltas (inter )for u in family ))) for k ,group in itertools .groupby (L ,key =operator .itemgetter (0 )): group =tuple (group ) new_value =value +sum (delta [1 ]for delta in group ) if not (value and new_value ): P .append (k ) value =new_value return LebesgueSet (P ,left_infinite ,fast =True ) def __and__ (self ,other ): """self & other -> intersection of self and other.""" return self .operate (self .INTER ,(self ,other )) def __or__ (self ,other ): """self | other, self + other -> union of self and other.""" return self .operate (self .UNION ,(self ,other )) __add__ =__or__ def __xor__ (self ,other ): """self ^ other -> symetric difference of self and other.""" return self .operate (self .XOR ,(self ,other )) def __sub__ (self ,other ): """self - other -> set difference.""" return self &~other @classmethod def union (cls ,family ): """union of a family of Lebesgue sets.""" return cls .operate (cls .UNION ,family ) @classmethod def inter (cls ,family ): """intersection of a family of Lebesgue sets""" return cls .operate (cls .INTER ,family ) @classmethod def xor (cls ,family ): """xor of a family of Lebesgue sets.""" return cls .operate (cls .XOR ,family ) class LebesgueTestCase (unittest .TestCase ): """Test class for LebesgueSets.""" def setUp (self ): import random self .gauss =random .gauss self .randint =random .randint self .nbr_families =100 self .nbr_test_points =1000 def random_point (self ): return self .gauss (0 ,10 ) def random_union (self ): points =(self .random_point ()for i in range (20 )) left_infinite =self .randint (0 ,1 ) return LebesgueSet (points ,left_infinite ) def random_family (self ): return [self .random_union ()for i in range (3 )] def runTest (self ): itt =itertools for i in xrange (self .nbr_families ): family =self .random_family () points =set (itt .chain (*(u .points for u in family ))) union_res =LebesgueSet .union (family ) inter_res =LebesgueSet .inter (family ) xor_res =LebesgueSet .xor (family ) for j in xrange (self .nbr_test_points ): while True : x =self .random_point () if not x in points : break status_list =list (u .status (x )for u in family ) us_ ,is_ ,xs_ =(u .status (x ) for u in (union_res ,inter_res ,xor_res )) u_expected =1 if any (s ==1 for s in status_list )else -1 i_expected =1 if all (s ==1 for s in status_list )else -1 x_expected =1 if sum (s ==1 for s in status_list )&1 else -1 self .assertEqual (us_ ,u_expected ,"Union failed") self .assertEqual (is_ ,i_expected ,"Intersection failed") self .assertEqual (xs_ ,x_expected ,"Xor failed") if __name__ =="__main__": unittest .main ()
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