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Re: Simple n-tuple problem - with no simple solution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116000] Re: Simple n-tuple problem - with no simple solution
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 28 Jan 2011 06:17:25 -0500 (EST)

It seems to me that he did. He writes (below) that the problem involves optimization of a function over the space of n-tupes and there is no reason at all to assume that the function is invariant with respect to the action of the group of permutations of the coordinates.

In fact, finding the "arrangements", if any, under which a problem is invariant is one of the most important techniques in mathematics. In particular, the symmetric group is a very interesting object even though all its elements are "only arrangements" of {1,2,...,n}.

Andrzej Kozlowski


On 27 Jan 2011, at 09:39, DrMajorBob wrote:

> "Therefore, the time it takes to generate all possible states
> is prohibitive for large n (n > 10) and searches through
> such a voluminous state space is not feasible."
>
> Not true. The number of distinct solutions never exceeds 627. Those 
> millions of others are only rearrangements.
>
> Unless there's some reason the order matters? You haven't mentioned a 
> reason.
>
> Bobby
>
> On Wed, 26 Jan 2011 04:06:07 -0600, Don <donabc at comcast.net> wrote:
>
>> On Jan 24, 2:22 am, DrMajorBob <btre... at austin.rr.com> wrote:
>>> Brilliant!
>>>
>>> Bobby
>>>
>>> On Sun, 23 Jan 2011 16:32:03 -0600, Dana DeLouis <dana.... at gmail.com>
>>> wrote:
>>>
>>>
>>>
>>>
>>>
>>>> On Jan 22, 3:23 am, DrMajorBob <btre... at austin.rr.com> wrote:
>>>>> Here are some timings.
>>>
>>>>> n = 5; addends = Rationalize@Range[0, 1.0, 0.05];
>>>>> Length@t1[addends, n] // Timing
>>>
>>>>> {1.44372, 192}
>>>>> {23.6318, 192}
>>>
>>>>> n = 7;
>>>>> Length@t1[addends, n] // Timing
>>>>> {1.85135, 364}
>>>
>>>>> n = 9;
>>>>> Length@t1[addends, n] // Timing
>>>>> {2.04216, 488}
>>>
>>>>> The Compositions method rendered Mathematica unresponsive at n = 10,
>>> but
>>>>> here's n = 25 for my approach:
>>>>> n = 25;
>>>>> Length@t1[addends, n] // Timing
>>>
>>>>> {2.57619, 627}
>>>
>>>>> I'd say Solve (which must use Integer Linear Programming for this) is
>>>>> the
>>>>> way to go.
>>>
>>>> Hi.  Just some thoughts.   With the smallest unit 1, the most n can be
>>>> is 20.
>>>
>>>> If we assume that solution sizes smaller than n have implied 0's, 
>>> then.  ..
>>>
>>>> IntegerPartitions[20,5]//Length //Timing
>>>> {0.000085,192}
>>>
>>>> IntegerPartitions[20,20]//Length //Timing
>>>> {0.000244,627}
>>>
>>>> If we want to look at each exact n, then the function uses {n}
>>>
>>>> Table[Length[IntegerPartitions[20,{j}]],{j,20}]
>>>> {1,10,33,64,84,90,82,70,54,42,30,22,15,11,7,5,3,2,1,1}
>>>
>>>> Accumulate[%]
>>>> =
>>>>
>>> {1,11,44,108,192,282,364,434,488,530,560,582,597,608,615,620,623,625,626,62==AD7}
>>>
>>>> By adding them up, we see that for n==5, we get the 192.  For n==7, 364,
>>>> etc.
>>>
>>>> If the question was "Count all possible combinations" as in the last
>>>> total above, then one additional way...
>>>
>>>> Timing[SeriesCoefficient[
>>>>   1/Times @@ (1 - n^#1 & ) /@
>>>>      Range[20], {n, 0, 20}]]
>>>
>>>> {0.0003, 627}
>>>
>>>> = = = = = = = = = =
>>>> HTH  : >)
>>>> Dana DeLouis
>>>
>>> --
>>> DrMajor... at yahoo.com- Hide quoted text -
>>>
>>> - Show quoted text -
>>
>> Thank you all for the in-depth analysis
>> of this problem.
>> The proposed solutions do solve the problem of
>> memory overflow as opposed to the simple
>> solution of generating all possible tuples and only then Selecting out
>> those whose elements sum to 1.
>>
>> I should add perhaps a little context to the original problem.
>> It is part of an optimization problem.
>> All the n-tuples (with the requirement
>> that all elements of any given tuple  add to 1.0 - as explained in
>> the original problem definition)  constitute
>> essentially the state space
>> that has to be searched ( in some clever way)  to find
>> the maximum value of an objective function subject
>> to well defined independent constraints.
>>
>> The hope was that
>> the requirement of summing to one would inhibit the
>> state space from growing too fast for "reasonable"
>> sizes of n ("reasonable" being about a max of 15 for
>> the current application).   Many of the solutions
>> proposed show that that is not feasible: the state space
>> grows very rapidly from about a 5-tuple to larger values of n up to
>> 15.
>> (A person more knowledgeable than I in combinatorics might have seen
>> this coming.)
>> Using the NumberOfCompositons function
>> from the Combinatorica package (special thanks to those
>> who introduced me to this package) shows how quickly
>> the state space grows.
>>
>> Needs["Combinatorica`"] // Quiet
>> max = 15;
>> For[n = 5, n <= max, n++,
>> Print["n = ",n, " " ,  AbsoluteTiming[c2b = 0.05 Compositions[20,
>> n] ;] ]
>> ]
>>
>> Therefore, the time it takes to generate all possible states
>> is prohibitive for large n (n > 10) and searches through
>> such a voluminous state space is not feasible.
>> A heuristic is being worked on that will, hopefully, reduce
>> the size of the state space while not sacrificing an optimum
>> solution for the objective function.  Thank you again for all your
>> input into this.
>>
>
>
> --
> DrMajorBob at yahoo.com
>


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