https://wiki.changeringing.co.uk/index.php?title=Spliced_treble-dodging_minor_-_Introduction&feed=atom&action=historySpliced treble-dodging minor - Introduction - Revision history2024-03-29T00:40:34ZRevision history for this page on the wikiMediaWiki 1.34.0https://wiki.changeringing.co.uk/index.php?title=Spliced_treble-dodging_minor_-_Introduction&diff=1395&oldid=prevHolroyd at 06:54, 27 June 20112011-06-27T06:54:28Z<p></p>
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</table>Holroydhttps://wiki.changeringing.co.uk/index.php?title=Spliced_treble-dodging_minor_-_Introduction&diff=1235&oldid=prevHolroyd at 06:48, 28 October 20102010-10-28T06:48:40Z<p></p>
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</table>Holroydhttps://wiki.changeringing.co.uk/index.php?title=Spliced_treble-dodging_minor_-_Introduction&diff=1233&oldid=prevHolroyd at 06:44, 28 October 20102010-10-28T06:44:09Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Richard Smith richard at ex-parrot.com</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Richard Smith richard at ex-parrot.com</div></td></tr>
</table>Holroydhttps://wiki.changeringing.co.uk/index.php?title=Spliced_treble-dodging_minor_-_Introduction&diff=1232&oldid=prevHolroyd at 06:41, 28 October 20102010-10-28T06:41:53Z<p></p>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Tue Sep 28 04:19:50 BST 2010</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Tue Sep 28 04:19:50 BST 2010</div></td></tr>
</table>Holroydhttps://wiki.changeringing.co.uk/index.php?title=Spliced_treble-dodging_minor_-_Introduction&diff=1223&oldid=prevHolroyd: moved Spliced treble-dodging minor to Spliced treble-dodging minor - Introduction2010-10-24T22:55:45Z<p>moved <a href="/Spliced_treble-dodging_minor" class="mw-redirect" title="Spliced treble-dodging minor">Spliced treble-dodging minor</a> to <a href="/Spliced_treble-dodging_minor_-_Introduction" title="Spliced treble-dodging minor - Introduction">Spliced treble-dodging minor - Introduction</a></p>
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</td></tr></table>Holroydhttps://wiki.changeringing.co.uk/index.php?title=Spliced_treble-dodging_minor_-_Introduction&diff=1222&oldid=prevHolroyd: Created page with 'Richard Smith richard at ex-parrot.com Tue Sep 28 04:19:50 BST 2010 I've spent quite a lot of the last month looking at spliced extents of treble dodging minor. Thanks to a cu…'2010-10-24T22:50:08Z<p>Created page with 'Richard Smith richard at ex-parrot.com Tue Sep 28 04:19:50 BST 2010 I've spent quite a lot of the last month looking at spliced extents of treble dodging minor. Thanks to a cu…'</p>
<p><b>New page</b></p><div>Richard Smith richard at ex-parrot.com<br />
Tue Sep 28 04:19:50 BST 2010<br />
<br />
I've spent quite a lot of the last month looking at spliced <br />
extents of treble dodging minor.<br />
<br />
Thanks to a cunning algorithm (which I shall describe in a <br />
moment) designed by Ander which we've been fine-tuning it <br />
turns out to be possible to do exhaustive searches over <br />
search spaces that I had previously thought were impossibly <br />
large.<br />
<br />
As a demonstration, I have just done a search for all true <br />
extents of minor using just methods from the standard 147 <br />
treble dodging minor methods rung with 4ths place lead-end <br />
bobs. I will do some further verification of this result <br />
over the next few days, but I believe the number of extents <br />
of this form is<br />
<br />
5,862,727,200,079,423,275,554<br />
<br />
To put this number into perspective, if I were to produce a <br />
booklet listing these in a similar format to that used in <br />
the CC's spliced minor collection, then the resulting <br />
booklet would be about 5 light-years thick.<br />
<br />
<br />
THE ALGORITHM<br />
<br />
There are five main stages to the search algorithm.<br />
<br />
First we remove lead splices and lead-end variants from the <br />
list of methods. So, for example, we only want to include <br />
one of Beverley, Surfleet, Berwick and Hexham. This reduces <br />
the list of methods from 147 to 75.<br />
<br />
The second stage is to associate each lead end or lead head <br />
row with a method. Start with a list of the 60 in-course <br />
rows with the treble leading -- these will all appear as a <br />
l.e. or a l.h., and we need to choose a method for each one, <br />
and doing so will join a l.h. to the subsequent l.e.<br />
<br />
Suppose some l.e./l.h. rows already have methods chosen. Of <br />
the remaining rows, we call a method 'possible' if<br />
<br />
(i) the l.e. that would be reached by ringing a lead of<br />
the method starting at the given l.h. row is not<br />
associated with a method; and<br />
<br />
(ii) the lead would be true against all other chosen<br />
leads.<br />
<br />
Take the row that has the fewest possible methods and, in <br />
sequence, try each of its possible methods, recursing. <br />
This gives an exhaustive tree search. The result of this is <br />
a 'plan' -- a list of which method is rung from each lead, <br />
but with no information on how to join the leads up.<br />
<br />
Stage two can be speeded up significantly by implementing a <br />
form of rotational pruning. Put the methods in some <br />
arbitrary order. Any method (other than the first one <br />
chosen) must not be before the first one chosen in the <br />
ordering. This will remove some but not all rotations and <br />
reflections. If you want an accurate count, it's a good <br />
idea to check whether a plan is in its canonical rotation <br />
and only output it if it is.<br />
<br />
The third stage is to do an exhaustive search of ways to <br />
join the 30 leads in each plan using 12, 14 or 16 lead end <br />
changes. An normal tree search for compositions will do <br />
this fine. There's no need to check for truth beyond <br />
checking for repetition of lead heads and lead ends as this <br />
was dealt with in stage two. For each plan you then have a <br />
list of compositions that produce the extent.<br />
<br />
Fourth, we remove compositions that include 16 lead ends in <br />
London (3-3.4) or Hills (3-34.6) backworks. This is a <br />
little subtle for plans that include one of these backworks <br />
and another one -- as 16 lead ends are fine as long as they <br />
only occur in the non-London, non-Hills backworks.<br />
<br />
A further subtlety arises if rotational pruning was done in <br />
stage two. Because there is no clear distinction between <br />
rotation and reflection of a plan (because we don't yet know <br />
which rows will become a l.h. and which a l.e.), pruning <br />
removes both rotations and reflections. However, going from <br />
Carlisle-over to London-over with a 16 l.e. is fine; but <br />
going the other way is not.<br />
<br />
This gives the complete set of extents.<br />
<br />
Fifth, and assuming we want to count them, for each plan, <br />
the number of extents is the product of three terms: the <br />
number of distinct rotations / reflections (assuming <br />
rotational pruning); the number of lead splices (N^n where N <br />
is the number of methods in the lead splice set -- 2 or 4 <br />
for everything in the 147 -- and n the number of leads of <br />
it); and the number of compositions for each plan. Adding <br />
the values for each plan gives the overall total.<br />
<br />
For the 147, the five stages took: 4s, 4h 1m, 1h 7m, 16m 44, <br />
and 1m 18s. So the total search time was just under 6h. <br />
I've only made an effort to optimise stages two and three <br />
(stage five in particular is woefully suboptimal), but given <br />
that's where most of the time is spent, that seems <br />
reasonable. I reckon that without too much work the search <br />
could be reduced to under 4h -- maybe even under 3h.<br />
<br />
<br />
THE EXTENTS<br />
<br />
Because the search first finds plans, and the number of <br />
plans (modulo rotation) is a fairly managable 4614, it's <br />
fairly easy to get a good idea of what's there. And a quick <br />
scan through the list of plans shows that there are some <br />
interesting plans that are new (at least to me). I'll give <br />
a breakdown of what's there in a later email.<br />
<br />
RAS</div>Holroyd