[r-t] Bobs only Erin Triples (was: Composition search)
Alexander Holroyd
holroyd at math.ubc.ca
Tue Dec 1 10:48:26 UTC 2015
Richard, can you clarify this point, please?
On Sat, 28 Nov 2015, Alexander Holroyd wrote:
> But I'm a bit confused by your assertion that none of these searches even
> produced a set of mutually true blocks. There are a bunch of sets of
> mutually true blocks for various part end groups. For example, composition
> number 5 (by PABS) in PABS's Collection (http://www.ringing.info/stedman.pdf)
> is based on a single bobs-only 10-part block, and about 20 years ago I came
> up with several sets of 7-part block sets in which the extent is divided into
> an _odd_ number of bobs-only blocks - not joinable, alas).
>
> Ander
>
> On Sat, 28 Nov 2015, Richard Smith wrote:
>
>> Philip Earis wrote:
>>
>>> I think Richard has ruled out a perfect 7-part composition, and possibly
>>> further - I'm sure he'll be able to confirm.
>>
>> As I mentioned the other day, I'm not sure I did. I looked at the rather
>> meagre notes I made when I last investigated this, and there's no
>> indication that I did eliminate it and I made several comments about it
>> perhaps being beyond the scope of a reasonable search.
>>
>> So I decided to take another look at it. The code I wrote on Wednesday is
>> considerably better than any of the previous generations of search code
>> I've used, in a significant part because I cracked the problem of how to do
>> efficient rotation pruning in a multi-part search and in doing so came up
>> with a good way of parallelising the search.
>>
>> As I speculated on Wednesday, this is done by applying the normal one-part
>> rotation pruning algorithm to a multi-part search, and repeating for each
>> conjugate part-end group. In the case of the 7-part search, there are 120
>> conjugate part-end groups which naturally divides the search space into 120
>> roughly equal chunks. This is much more even division than dividing by
>> search prefix, especially in the presence of rotation pruning, and there
>> was less than a factor of 2.5 difference between the largest and smallest
>> of the 120 searches. As no communication is required between 120 search
>> processes, there's no need for multi-threading, and they can be run
>> independently on separate machines.
>>
>> Increased computer speeds have helped too. Ten years ago, which is the
>> last time I seriously studied this, I used a single-core 800 MHz processor;
>> now even my laptop has a dual-core 1.9 GHz processor, and the machines I
>> actually used had 2.66 GHz processors. But the clock speed (and extra
>> core) is not representative of the actual speed increase, and in fact my
>> laptop (with an 1.9 Ghz Intel i3-3227U processor) ran the code about 1.95
>> times as fast as the machines I actually used (with a 2.66 GHz Intel P8800
>> processor). This is because the Ivy Bridge microarchitecture in my laptop
>> is that much faster than the Penryn microarchitecture.
>>
>> On average each of the 120 search processes took 17 minutes to complete and
>> none exceeded 40 minutes. No blocks were found.
>>
>> I also checked all larger part-end groups. The part-end group cannot
>> contain any elements conjugate to the in-course falseness (i.e. a simple
>> 3-cycle) which severely limits the available part structures. If a
>> multi-part bobs-only extent of Erin exists, then the number of parts
>> must be one of 168, 60, 24, 21, 20, 12, 10, 8, 7, 6, 5, 4, 3, 2 or 1. I
>> have now eliminated the 7-part and all larger options. None even produced
>> a set of mutually true blocks.
>>
>> In fact, I think we can rule out several of these structures on Q-set
>> grounds: specifically the 168, 60, 24 and 8 part structures. But I'd need
>> to think a bit harder about it to be certain. The point is moot anyway.
>>
>> RAS
>>
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