Carl Love

Carl Love

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8 years, 245 days
Natick, Massachusetts, United States
My name was formerly Carl Devore.

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These are Posts that have been published by Carl Love

Yesterday, user @lcz , while responding as a third party to one of my Answers regarding GraphTheory, asked about breadth-first search. So, I decided to write a more-interesting example of it than the relatively simple example that was used in that Answer. I think that this is general enough to be worthy of a Post.

This application generates all maximal paths in a graph that begin with a given vertex. (I'm calling a path maximal if it cannot be extended and remain a path.) This code requires Maple 2019 or later and 1D input. This works for both directed and undirected graphs. Weights, if present. are ignored.


AllMaximalPaths:= proc(G::GRAPHLN, v)
    "All maximal paths of G starting at v by breadth-first search"
option `Author: Carl Love <> 2021-Mar-17`;
uses GT= GraphTheory;
    P:= [rtable([v])], R:= rtable(1..0),
    VL:= GT:-Vertices(G), V:= table(VL=~ [$1..nops(VL)]),
    Departures:= {op}~(GT:-Departures(G))
    while nops(P) <> 0 do
        P:= [
            for local p in P do
                local New:= Departures[V[p[-1]]] minus {seq}(p);
                if New={} then R,= [seq](p); next fi;                
                    for local u in New do 
                        local p1:= rtable(p); p1,= u
end proc
#large example:
GT:= GraphTheory:
K9:= GT:-CompleteGraph(9):
Pa:= CodeTools:-Usage(AllMaximalPaths(K9,1)):
memory used=212.56MiB, alloc change=32.00MiB, 
cpu time=937.00ms, real time=804.00ms, gc time=312.50ms

#fun example:
P:= GT:-SpecialGraphs:-PetersenGraph():
Pa:= CodeTools:-Usage(AllMaximalPaths(P,1)):
memory used=0.52MiB, alloc change=0 bytes, 
cpu time=0ns, real time=3.00ms, gc time=0ns


Pa[..9]; #sample paths
    {[1, 2, 3, 4, 10, 9, 8, 5], [1, 2, 3, 7, 8, 9, 10, 6], 
      [1, 2, 9, 8, 7, 3, 4, 5], [1, 2, 9, 10, 4, 3, 7, 6], 
      [1, 5, 4, 3, 7, 8, 9, 2], [1, 5, 4, 10, 9, 8, 7, 6], 
      [1, 5, 8, 7, 3, 4, 10, 6], [1, 5, 8, 9, 10, 4, 3, 2], 
      [1, 6, 7, 3, 4, 10, 9, 2]}

Notes on the procedure:

The two dynamic data structures are

  • P: a list of vectors of vertices. Each vector contains a path which we'll attempt to extend.
  • R: a vector of lists of vertices. Each list is a maximal path to be returned.

The static data structures are

  • V: a table mapping vertices (which may be named) to their index numbers.
  • Departures: a list of sets of vertices whose kth set is the possible next vertices from vertex number k.

On each iteration of the outer loop, P is completely reconstructed because each of its entries, a path p, is either determined to be maximal or it's extended. The set New is the vertices that can be appended to the (connected to vertex p[-1]). If New is empty, then p is maximal, and it gets moved to R

The following code constructs an array plot of all the maximal paths in the Petersen graph. I can't post the array plot, but you can see it in the attached worksheet:

#Do an array plot of each path embedded in the graph:
n:= nops(Pa):
c:= 9: 
    (PA:= rtable(
        (1..ceil(n/c), 1..c),
            if (local k:= (i-1)*ceil(n/c) + j) > n then 
                plot(axes= none)
                    GT:-HighlightTrail(P, Pa[k], inplace= false), 
                    stylesheet= "legacy", title= typeset(Pa[k])
    titlefont= [Times, Bold, 12]

#And recast that as an animation so that I can post it:
    [seq](`$`~(plots:-display~(PA), 5)),


When re-initializing a module (for example, while executing its ModuleApply), you'd likely want to re-initialize all of the remember tables and caches of its procedures. Unfortunately, forget(thismodule) (a direct self reference) doesn't work (I wonder why not?), and it doesn't even tell you that it didn't work. You could do forget(external name of module(an indirect self reference), but I consider that not robust because you'd need to change it if you change the external name. Even less robust is forget~([procedure 1, procedure 2, submodule 3, ...]). Here's something that works and is robust:

forget~([seq(x, x= thismodule)])[]

Or, using Maple 2019 or later syntax,

forget~([for local x in thismodule do x od])[]

Both of these handle submodules and ignore non-procedures. Both handle both locals and exports.

We need a check-off box for Maple Companion in the Products category of Question and Post headers.

While you're looking at that, there's also a bug that the Product indication gets stripped off when converting a Post to a Question, which is a common Moderator action.

I experienced a significant obstacle while trying to generate independent random samples with Statistics:-Sample on different nodes of a Grid multi-processing environment. After many hours of trial-and-error, I discovered an astonishing workaround, and I achieved excellent time and memory performance. Since this seems like a generally useful computation, I thought that it was worthy of a Post.

This Post is also worth reading to learn how to use Grid when you need to initialize a substantial environment on each node before using Grid:-Map or Grid:-Seq.

All remaining details are in the following worksheet.

How to use Statistics:-Sample in the `Grid` environment

Author: Carl Love <> 1 August 2019


I experienced a significant obstacle while trying to generate indenpendent random samples with Statistics:-Sample on the nodes of a multi-processor Grid (on a single computer). After several hours of trial-and-error, I discovered that two things are necessary to do this:


The random number generator needs to be seeded differently in each node. (The reason for this is easy to understand.)


The random variables generated by Statistics:-RandomVariable need to have different names in each node. This one is mind-boggling to me. Afterall, each node has its own kernel and so its own memory It's as if the names of random variables are stored in a disk file which all kernels access. And also the generator has been seeded differently in each node.


Once these things were done, the time and memory performance of the computation were excellent.


Digits:= 15

#Specify the size of the computation:
(n1,n2,n3):= (100, 100, 1000):
# n1 = size of each random sample;
# n2 = number of samples in a batch;
# n3 = number of batches.

#Procedure to initialize needed globals on each node:
Init:= proc(n::posint)
local node:= Grid:-MyNode();
   #This is wrapped in parse so that it'll act globally. Otherwise, an environment
   #variable would be reset when this procedure ends.
   parse("Digits:= 15;", 'statement');

   randomize(randomize()+node); #Initialize independent RNG for this node.
   #If repeatability of results is desired, remove the inner randomize().

   (:-X,:-Y):= Array(1..n, 'datatype'= 'hfloat') $ 2;

   #Perhaps due to some oversight in the design of Statistics, it seems necessary that
   #r.v.s in different nodes **need different names** in order to be independent:
   N||node:= Statistics:-RandomVariable('Normal'(0,1));
   :-TRS:= (X::rtable)-> Statistics:-Sample(N||node, X);
   #To verify that different names are needed, change N||node to N in both lines.
   #Doing so, each node will generate identical samples!

   #Perform some computation. For the pedagogical purpose of this worksheet, all that
   #matters is that it's some numeric computation on some Arrays of random Samples.
   :-GG:= (X::Array, Y::Array)->
         proc(X::Array, Y::Array, n::posint)
         local s, k, S:= 0, p:= 2*Pi;
            for k to n do
               s:= sin(p*X[k]);  
               S:= S + X[k]^2*cos(p*Y[k])/sqrt(2-sin(s)) + Y[k]^2*s
         end proc
         (X, Y, n)
   #Perform a batch of the above computations, and somehow numerically consolidate the
   #results. Once again, pedagogically it doesn't matter how they're consolidated.  
   :-TRX1:= (n::posint)-> add(GG(TRS(X), TRS(Y)), 1..n);
   #It doesn't matter much what's returned. Returning `node` lets us verify that we're
   #actually running this on a grid.
   return node
end proc

The procedure Init above uses the :- syntax to set variables globally for each node. The variables set are X, Y, N||node, TRS, GG, and TRX1. Names constructed by concatenation, such as N||node, are always global, so :- isn't needed for those.

#Time the initialization:
st:= time[real]():
   #Send Init to each node, but don't run it yet:
   #Run Init on each node:
   Nodes:= Grid:-Run(Init, [n1], 'wait');
time__init_Grid:= time[real]() - st;

Array(%id = 18446745861500764518)


The only purpose of array Nodes is that it lets us count the nodes, and it lets us verify that Grid:-MyNode() returned a different value on each node.

num_nodes:= numelems(Nodes);


#Time the actual execution:
st:= time[real]():
   R1:= [Grid:-Seq['tasksize'= iquo(n3, num_nodes)](TRX1(k), k= [n2 $ n3])]:
time__run_Grid:= time[real]() - st


#Just for comparison, run it sequentially:
st:= time[real]():
time__init_noGrid:= time[real]() - st;

st:= time[real]():
   R2:= [seq(TRX1(k), k= [n2 $ n3])]:
time__run_noGrid:= time[real]() - st;



R1 and R2 will be different because different random numbers were used, but they should have similar histograms.

      <R1 | R2>, #side-by-side plots
      'title'=~ <<"With Grid\n"> | <"Without Grid\n">>,
      'gridlines'= false

(Plot output deleted because MaplePrimes cannot handle side-by-side plots!)

They look similar enough to me!


Let's try to quantify the benefit of using Grid:

speedup_factor:= time__run_noGrid / time__run_Grid;


Express that as a fraction of the theoretical maximum speedup:

efficiency:= speedup_factor / num_nodes;


I think that that's really good!


The memory usage of this code is insignificant, which can be verified from an external memory monitor such as Winodws Task Manager. It's just a little bit more than that needed to start a kernel on each node. It's also possible to measure the memory usage programmatically. Doing so for a Grid:-Seq computation is a little bit beyond the scope of this worksheet.




Here are the histograms:

The procedure presented here does independence tests of a contingency table by four methods:

  1. Pearson's chi-squared (equivalent to Statistics:-ChiSquareIndependenceTest),
  2. Yates's continuity correction to Pearson's,
  3. G-chi-squared,
  4. Fisher's exact.

(All of these have Wikipedia pages. Links are given in the code below.) All computations are done in exact arithmetic. The coup de grace is Fisher's. The first three tests are relatively easy computations and give approximations to the p-value (the probability that the categories are independent), but Fisher's exact test, as its name says, computes it exactly. This requires the generation of all matrices of nonnegative integers that have the same row and column sums as the input matrix, and for each of these matrices computing the product of the factorials of its entries. So, there are relatively few implementations of it, and perhaps none that do it exactly. (Could some with access check Mathematica please?)

Our own Joe Riel's amazing and fast Iterator package makes this computation considerably easier and faster than it otherwise would've been, and I also found inspiration in his example of recursively counting contingency tables found at ?Iterator,BoundedComposition

ContingencyTableTests:= proc(
   O::Matrix(nonnegint), #contingency table of observed counts 
   {method::identical(Pearson, Yates, G, Fisher):= 'Pearson'}
   "Returns p-value for Pearson's (w/ or w/o Yates's continuity correction)" 
   " or G chi-squared or Fisher's exact test."
   " All computations are done in exact arithmetic."
   author= "Carl Love <>, 27-Oct-2018",
   references= (                                                           #Ref #s:
      "",         #*1
      "",  #*2
      "",                               #*3
      "",                #*4
      "Eric W Weisstein \"Fisher's Exact Test\" _MathWorld_--A Wolfram web resource:"
      ""                 #*5
uses AT= ArrayTools, St= Statistics, It= Iterator;
   #column & row sums: 
   C:= AT:-AddAlongDimension(O,1), R:= AT:-AddAlongDimension(O,2),
   r:= numelems(R), c:= numelems(C), #counts of rows & columns
   n:= add(R), #number of observations
   #matrix of expected values under null hypothesis (independence):
   #(A 0 entry would mean a 0 row or column in the original, which is not allowed.)
   E:= Matrix((r,c), (i,j)-> R[i]*C[j], datatype= 'positive') / n,
   #Pearson's, Yates's, and G all use a chi-sq statistic, each computed by 
   #slightly different formulae.
   Chi2:= add@~table([
       'Pearson'= (O-> (O-E)^~2 /~ E),                     #see *1
       'Yates'= (O-> (abs~(O - E) -~ 1/2)^~2 /~ E),        #see *2
       'G'= (O-> 2*O*~map(x-> `if`(x=0, 0, ln(x)), O/~E))  #see *3
   row, #alternative rows generated for Fisher's
   Cutoff:= mul(O!~), #cut-off value for less likely matrices
   #Generate recursively all contingency tables whose row and column sums match O.
   #Compute their probabilities under independence. Sum probabilities of all those
   #at most as probable as O. (see *5, *4)
   #   C = column sums remaining to be filled; 
   #   F = product of factorials of entries of contingency table being built;
   #   i = row to be chosen this iteration
   AllCTs:= (C, F, i)->
      if i = r then #Recursion ends; last row is C, the unused portion of column sums. 
         (P-> `if`(P >= Cutoff, 1/P, 0))(F*mul(C!~))
            thisproc(C - row[], F*mul(row[]!~), i+1), 
            row= It:-BoundedComposition(C, R[i])
   userinfo(1, ContingencyTableTests, "Table of expected values:", print(E));
   if method = 'Fisher' then AllCTs(C, 1, 1)*mul(R!~)*mul(C!~)/n!
   else 1 - St:-CDF(ChiSquare((r-1)*(c-1)), Chi2[method](O)) 
end proc:

The worksheet below contains the code above and one problem solved by the 4 methods



DrugTrial:= <
   20, 11, 19;
   4,  4,  17

infolevel[ContingencyTableTests]:= 1:

ContingencyTableTests(DrugTrial, method= Pearson):  % = evalf(%);

ContingencyTableTests: Table of expected values:

Matrix(2, 3, {(1, 1) = 16, (1, 2) = 10, (1, 3) = 24, (2, 1) = 8, (2, 2) = 5, (2, 3) = 12})

exp(-257/80) = 0.4025584775e-1

#Compare with:

hypothesis = false, criticalvalue = HFloat(5.991464547107979), distribution = ChiSquare(2), pvalue = HFloat(0.04025584774823787), statistic = 6.425000000

infolevel[ContingencyTableTests]:= 0:
ContingencyTableTests(DrugTrial, method= Yates):  % = evalf(%);

exp(-1569/640) = 0.8615885805e-1

ContingencyTableTests(DrugTrial, method= G):  % = evalf(%);

exp(-20*ln(5/4)+4*ln(2)-11*ln(11/10)-4*ln(4/5)-19*ln(19/24)-17*ln(17/12)) = 0.3584139703e-1

CodeTools:-Usage(ContingencyTableTests(DrugTrial, method= Fisher)):  % = evalf(%);

memory used=0.82MiB, alloc change=0 bytes, cpu time=0ns, real time=5.00ms, gc time=0ns

747139720973921/15707451356376611 = 0.4756594205e-1




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