Re: Re: Help to remove equivalent (redundant) solutions

*To*: mathgroup at smc.vnet.net*Subject*: [mg91544] Re: [mg91500] Re: Help to remove equivalent (redundant) solutions*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Tue, 26 Aug 2008 03:29:58 -0400 (EDT)*References*: <g8o858$l6a$1@smc.vnet.net> <200808241105.HAA15487@smc.vnet.net>

mark mcclure wrote: > On Aug 23, 1:45 am, Bob Hanlon <hanl... at cox.net> wrote: >> Use SameTest option in Union > > I've had to think about this question at some length in my > work with fractal geometry, where I deal with large numbers > of points. In this context, there are some issues with the > use of SameTest. In particular, Union[S] first sorts its > elements in n*log(n) time and then compares in linear time. > Union[S, SameTest -> f] cannot assume any ordering so it > must compare pairwise, running in quadratic time. If at all > possible, it is much faster to express each element in S in > some canonical form, then apply Union without the SameTest. > I've got some examples below but would be very curious to > know if anyone has improvements. > > Here's a list of 400 pairs of points with each pair very > close to another. Note that Union works very quickly, but > distinguishes between close pairs of points. > > ------ > points = RandomReal[{0, 1}, {200, 2}]; > points = Join[points, points + 10^(-15)]; > Union[points] // Length // Timing > > {0.000159, 400} > ------ > > > We can apply Bob's solution which, correctly, treats close > points the same but runs much longer. > > ------ > Union[points, SameTest -> (Rationalize[#1, 10^(-10)] === > Rationalize[#2, 10^(-10)] &)] // Length // Timing > > {4.36683, 200} > ------ > > > Note that SameTest -> (Norm[#1 - #2] < 10^(-10) &) is > faster, but we can get the best of both worlds by > rationalizing first and then applying the Union. > > ----- > Union[Rationalize[points, 10^(-10)]] // Length // Timing > > {0.04629, 200} > ----- > > > Of course, there are issues with Rationalize, or Round, as > well. First, you lose some precision in the output, which > might or might not be important for your application. Also, > while it seems to work well with randomly generated data, > it's easy to devise situations where points close together > Rationalize to different results. Thus, I'd be leary of > using Rationalize in this manner for serious work. What > is needed is a reliable function that places things in a > canonical form. Here's one such function I developed based > on clustering. > > ----- > Needs["HierarchicalClustering`"]; > canonicalFunction[nonCanonicalValues_List] := Module[ > {toCanonical}, > Quiet[heirarchy = Agglomerate[N[nonCanonicalValues], > DistanceFunction -> EuclideanDistance, > Linkage -> "Average"]]; > segregate[Cluster[cl1_, cl2_, d_, _, _], tol_] := > > MyClusters[cl1, cl2] /; d > tol; > segregate[mine_MyClusters, tol_] := segregate[#, tol] & /@ mine; > segregate[x_, _] := x; > cf[cl_Cluster] := ClusterFlatten[cl]; > cf[x_] := {x}; > clusters = > cf /@ List @@ Flatten[FixedPoint[segregate[#, 10^-12] &, > MyClusters[heirarchy]]]; > canonicalValues = Chop[First /@ clusters]; > toCanonical[x_] := First[Nearest[canonicalValues][x]]; > toCanonical]; > ----- > > We use it as follows. > > ----- > toCan = canonicalFunction[points]; > Union[toCan /@ points] // Length // Timing > > {0.138291, 200} > ----- > > Again, I'd love to see alternatives. > > Mark McClure Here is a way to get a representative from an "equivalence" class, using Nearest to remove from consideration all elements within a given radius of that representative. It's fuzzy, and in a serious way: not only will the result depend on the order of the input points, but even the size of the result might depend on it. This is due to nontransitivity of the "equivalence" relation (so of course it is not a true equivalence relation). Anyway, the up side to this is in the speed. It will tend to be O(n log(n)), provided Nearest behaves itself. removeDuplicatesInRange[points_, eps_] := Module[ {keepers, len=Length[points], nearfunc, unusedlist, nearpts}, nearfunc = Nearest[points]; keepers = ConstantArray[False,len]; Map [(unusedlist[#]=True)&, points]; Do [ If [TrueQ[unusedlist[points[[j]]]], keepers[[j]] = True; nearpts = Rest[nearfunc[points[[j]],{len,N[eps]}]]; Map [(unusedlist[#]=False)&, nearpts]; ], {j,len}]; Clear[usedlist]; Pick[points, keepers] ] Example: len = 10^5; disp = 10^(-4); eps = 10^(-3); points = RandomReal[1, {len,2}]; points = Join[points, points + RandomReal[disp, {len,2}], points - 3*RandomReal[disp, {len,2}]]; Timing[Length[np = removeDuplicatesInRange[points,eps]]] This should run in ten seconds or so. Daniel Lichtblau Wolfram Research

**References**:**Re: Help to remove equivalent (redundant) solutions from***From:*mark mcclure <mcmcclur@unca.edu>