Tackling the Minimal Superpermutation Problem

2014-08-22 § 5 Comments

What’s the shortest string that contains every possible permutation of ABCD somewhere inside it? As it happens, it’s 33 letters long: ABCDABCADBCABDCABACDBACBDACBADCBA. A string like this is called a minimal superpermutation.

So what’s the shortest string that contains every possible permutation of ABCDE? It was recently shown that 153 letters is the shortest possible, and that there are eight different superpermutations of this length.

Okay, what about ABCDEF? The answer is that nobody knows. Until this week the shortest known superpermutation of ABCDEF was 873 letters long:

ABCDEFABCDEAFBCDEABFCDEABCFDEABCDFEABCDAEFBCDAEBFCDAEBCFDAEBCDFAEB
CDAFEBCDABEFCDABECFDABECDFABECDAFBECDABFECDABCEFDABCEDFABCEDAFBCED
ABFCEDABCFEDABCADEFBCADEBFCADEBCFADEBCAFDEBCADFEBCADBEFCADBECFADBE
CAFDBECADFBECADBFECADBCEFADBCEAFDBCEADFBCEADBFCEADBCFEADBCAEFDBCAE
DFBCAEDBFCAEDBCFAEDBCAFEDBCABDEFCABDECFABDECAFBDECABFDECABDFECABDC
EFABDCEAFBDCEABFDCEABDFCEABDCFEABDCAEFBDCAEBFDCAEBDFCAEBDCFAEBDCAF
EBDCABEFDCABEDFCABEDCFABEDCAFBEDCABFEDCABACDEFBACDEBFACDEBAFCDEBAC
FDEBACDFEBACDBEFACDBEAFCDBEACFDBEACDFBEACDBFEACDBAEFCDBAECFDBAECDF
BAECDBFAECDBAFECDBACEFDBACEDFBACEDBFACEDBAFCEDBACFEDBACBDEFACBDEAF
CBDEACFBDEACBFDEACBDFEACBDAEFCBDAECFBDAECBFDAECBDFAECBDAFECBDACEFB
DACEBFDACEBDFACEBDAFCEBDACFEBDACBEFDACBEDFACBEDAFCBEDACFBEDACBFEDA
CBADEFCBADECFBADECBFADECBAFDECBADFECBADCEFBADCEBFADCEBAFDCEBADFCEB
ADCFEBADCBEFADCBEAFDCBEADFCBEADCFBEADCBFEADCBAEFDCBAEDFCBAEDCFBAED
CBFAEDCBAFEDCBA

and it was thought that might be the shortest possible.

But we now know it isn’t, because I found a shorter one:

ABCDEFABCDEAFBCDEABFCDEABCFDEACBFDEACFBDEACFDBEACFDEBACFDEABCDFEAB
CDAEFBCDAEBFCDAEBCFDAEBCDFAEBCDAFEBCDABEFCDABECFDABECDFABECDAFBECD
ABFECDABCEFDABCEDFABCEDAFBCEDABFCEDABCFEDACBFEDCABFDECAFBDCEAFBDCA
EFBDCAFEBDCAFBEDCAFBDECAFDBECADFBECADBFECADBEFCADBECFADBECAFDEBCAD
FEBCADEFBCADEBFCADEBCFADEBCAFDECBAFDECABFDCEABFDCAEBFDCABEFDCBAEFD
BCAEDFBCAEDBFCAEDBCFAEDBCAFEDBCAEFDBACEFDBAECFBDAECFBADECFBAEDCFBA
ECDFBACEDFBACDEFBACDFEBACDFBEACDFBAECFDBAEFCDBAFECDBAFCEDBAFCDEBAF
CDBEAFCDBAEFDCBEAFDCBEFADCBEFDACBEFDCABFEDCBAFEDCBFAECDBFACEDBFACD
EBFACDBEFACDBFEACDBFAECBDFEACBDFECABDFCEABDFCAEBDFCABEDFCBAEDFCBEA
DFCBEDAFCBEDFACBEDFCABDEFCBADEFCBDAEFCBDEAFCBDEFACBDEFCABDFECBADFE
CBDAFECBDFAECBFDAECBFADECBFAEDCBFEADCFBEADCFEBADCEFBADCEBFADCEBAFD
CEBADFCEBADCFEABDCFAEBDCFABEDCFABDECFABDCEFABDCFEADBCEFADBCEAFDBCE
ADFBCEADBFCEADBCFEADCBFEDACFBEDACFEBDACEFBDACEBFDACEBDFACEBDAFCEBD
ACFEDBACFEDABC

I’ve uploaded a short note about it to the arxiv.

Revisiting “On editing text”

2014-06-19 § 1 Comment

This document is an incomplete draft.

About two years ago I wrote about a category-theoretic treatment of collaborative text editing. That post is unique in the history of Bosker Blog in having been cited – twice so far that I know of – in the academic literature; so it’s a little embarrassing for me to have to explain that it is almost entirely wrong. The good news is that the core idea can be rescued, and the corrected story is quite interesting. Other writers on this subject seem to have made at least some of the same mistakes I did, so I hope this will be useful to at least a few other people too. « Read the rest of this entry »

Decoding the mysterious symmetry of the bicycle lock numbers

2014-02-25 § Leave a comment

Suppose you have a lock of this sort bicycle lock that has n dials and k numbers on each dial. Let m(nk) be the minimum number of turns that always suffice to open the lock from any starting position, where a turn consists of rotating any number of adjacent rings by one place.

In the previous post, we found an algorithm for computing these bicycle lock numbers, revealing a mysterious symmetry, « Read the rest of this entry »

The bicycle lock problem

2014-02-18 § 3 Comments

Giant bicycle lock

Photo: WickedVT


Don’t lock your bicycle with a combination lock. Someone will steal it: I learnt this the hard way. It’s quite easy to open a combination lock by feel, without knowing the combination. Try it: with a bit of practice, you can open the lock with your eyes shut. (It’s easier to do this with an old wobbly lock than a tight-fitting new one.)
« Read the rest of this entry »

Beyond Bézier curves

2013-11-13 § 4 Comments

There is a new feature of Pages and Keynote, not mentioned in any of Apple’s publicity nor in any press coverage I’ve seen, that is really very interesting. Perhaps it will even one day prove to have been revolutionary, in a quiet way. « Read the rest of this entry »

I hate the Pumping Lemma

2013-08-18 § 34 Comments

I hate the Pumping Lemma for regular languages. It’s a complicated way to express an idea that is fundamentally very simple, and it isn’t even a very good way to prove that a language is not regular.

Here it is, in all its awful majesty: for every regular language L, there exists a positive whole number p such that every string w∈L that has p characters or more can be broken down into three substrings xyz, where y is not the empty string and the total length of xy is at most p, and for every natural number i the string xyiz is also in L.
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The algebra of Unix command substitution

2013-08-16 § 6 Comments

Shadab Ahmed raised an interesting question. Open a Unix command shell, type : '!!' and press return. Then type : "!!" '!!' and press return. Now repeat the following a few times: press the up arrow, and press return.

« Read the rest of this entry »

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