[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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