
Computer Scientists Make Progress on Math Puzzle - J3L2404
http://www.utdallas.edu/news/2010/10/28-6621_Computer-Scientists-Make-Progress-on-Math-Puzzle_article.html
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dasht
That article appears to be partly wrong and partly misleading. The game is
described in a way that makes no sense and so the significance of the new
proof is not made clear. I think this is more accurate:

In topswops we start with a shuffled deck of N cards numbered 1..N. In each
move we reverse the order of the top K cards where K is the number initially
on the top card. (After that reversal, the original top card, K, is now the
Kth card down and the formerly Kth card down is now the top card. Any cards in
between have reversed order.)

The game terminates when the card number 1 comes to the top. (For reversing
the order of a single card has no effect.)

Here is what the article leaves out: If, when 1 comes to the top, the deck is
_sorted_ 1..N, you win. Otherwise, you lose. This is a very tedious form of
solitaire.

Note that if your shuffle produces an already sorted deck, you win trivially
with 0 moves: the deck is already sorted and 1 is on top! If the shuffle
produces a reverse sorted deck, N..1, you win trivially after 1 move!

If someone hands you a different shuffle, you might be playing for a very long
time, uncertain whether you will win or lose.

For every deck-size N, there is some number of moves - call it ENOUGH(N) -
such that if you haven't won in ENOUGH(N) moves, then you will not win with
that deck. For example, if the deck has 12 cards, and you've been playing for
64 moves ... why you might as well quit. We know for sure that any shuffle
that wins ends after at most 63 moves for a deck of size 12. ENOUGH(12) is 63.
[supposedly, I haven't double checked the brute-force proof]

What is the formula for ENOUGH(N)? or at least an approximate formula?

For values of N up to (the article says) size 18, ENOUGH(N) has been computed
by brute force.

Knuth gave a formula KNUTHTOPS(N) and proved that ENOUGH(N) <= KNUTHTOPS(N).
His proof was interesting because, asymptotically, KNUTHTOPS(N) is lowest
upper bound known. You can confidently stop a losing game after KNUTHTOPS(N)
moves -- but it is possible you could have stopped earlier if you had a better
formula.

These new guys gave a formula, HALnLINDAWOPS(N), and proved that
HALnLINDAWOPS(N) <= ENOUGH(N). HALnLINDAWOPS(N) is interesting for being the
highest lower bound known. Don't even think about stopping until you've made
at least HALnLINDAWOPS(N) moves or you are guaranteed, in some cases, to quit
part way through a winning shuffle.

The number at which you can quit safely, no fear of stopping a winning game,
is between HALnLINDAWOPS(N) and KNUTHTOPS(N) (inclusive). The main advance is
squeezing that range from below with a new proof.

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leftnode
Solving that puzzle is far beyond my intelligence, but I did have Dr.
Sudborough as a professor. He was one of the best professors I had at UTD,
really taught Automata Theory very well.

I'm glad he's getting more recognition.

~~~
ronnier
So did I. 2007 I believe. When did you?

~~~
leftnode
Hah, same here. I graduated Summer of 2007, so I probably had him in the
Spring 2007 class. As a tenured professor, he could've easily made me see a TA
when I had questions. Instead, he let me come to his office and worked with me
for a while on problems that were tough for me but rote for him. I really
appreciated it.

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jonafato
There seems to be a lot of disagreement about lower bounds here. The best case
lower bound is trivially 0 since you could start with a terminating
permutation. The article is talking about a worst case lower bound, which is
what one would use to assign the problem to a complexity class.

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jbillingsley
It's nice to see some noteworthy CS/Mathmatics accomplishments coming from
outside Stanford and MIT. But perhaps I'm biased being from the Dallas area.

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tonygonz
I must be misunderstanding the problem. Why is the lower bound on the number
of operations not 1, since you have a 1 in n chance of turning over the "1"
card as your first operation?

~~~
seles
The lower bound indeed is 1, however higher lower bounds can be found.

Here lower bound means that there exist a permutation that will take this many
operations. Their goal is to find what the maximum number of operations
assuming worst permutation. This is accomplished when the lower bound equals
the upper bound.

Proving lower bounds is easy they just need to give an example. Proving upper
bounds is hard, somehow they have a way besides running all possible cases.
(The article doesn't explain how and I haven't delven deeper).

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amichail
The article doesn't make sense. It implies that the lower bound proved is
exponential but much better than the conjectured lower bound (also
exponential):

"Knuth had previously proved an exponential upper bound on the number of
Topswops steps, and conjectured that one might also prove a matching lower
bound. What Dr. Hal Sudborough and Dr. Linda Morales did, however, was to
prove a lower bound that is much better than that proposed in Knuth’s
conjecture..."

However, the paper cited says “A Quadratic Lower Bound for Topswops”, so the
lower bound proved is much worse than the one conjectured by Knuth.

~~~
dasht
You are right. The article is a muddle.

I think they meant: the quadratic lower bound is the highest yet proved -- not
that it is "better" than Knuth's unproved conjecture.

~~~
amichail
It's more likely that the person who wrote this article doesn't even know what
a lower bound is.

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Brashman
ACM link for the paper: <http://portal.acm.org/citation.cfm?id=1866548>

~~~
michael_dorfman
Note that that is the ACM link to the citation; the paper itself is published
in an Elsevier journal, and is not available to ACM Digital Library
subscribers (like myself.)

~~~
carussell
[http://linkinghub.elsevier.com/retrieve/pii/S030439751000428...](http://linkinghub.elsevier.com/retrieve/pii/S0304397510004287)

You can see the article there if you have a subscription. If you don't have
one, you can buy a copy for $40. Otherwise, you can just go to the library.

It's also not too difficult to find a 2005 review copy online. Preprints for
the paper apparently date back to 1995.

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botptr
Vague title of the month award?

