Talk:Criss-cross algorithm

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Did You Know?

Old nomination: Inaccessible

I nominated the following hook: Kiefer.Wolfowitz (Discussion) 00:57, 22 March 2011 (UTC)[reply]

Did you know ... that the criss-cross algorithm' and the simplex algorithm are not polynomial-time algorithms because they visit all 2^D corners of a cube in dimension D?

That sounds a bit difficult for a DYK, don't you think? I mean, is there any way it could be worded to be more accessible? CRGreathouse (t | c) 04:25, 22 March 2011 (UTC)[reply]

In particular, I don't think many readers—even mathematically sophisticated ones—will know of the Klee–Minty cube, nor the significance of a vector space's dimension. If I were writing it (and I'm not!) I might say something about the fact that it's an exponential-time algorithm that performs far better in practice. Surely there are other things that educated but nonspecialist readers could appreciate? CRGreathouse (t | c) 19:50, 22 March 2011 (UTC)[reply]

That's helpful. At least the phrase "Klee-Minty" should go!

Caveat: I have not yet provided a reference to the expected behavior of the criss cross algorithm (which is D on cubes of dimension D, I believe). The expected number of steps of the simplex algorithm is also linear under a wide class of models (which are incompatible with reality, alas); the expected running time is usually thought to be about 3D on practical problems.

I simplified it a bit. Kiefer.Wolfowitz (Discussion) 19:59, 22 March 2011 (UTC)[reply]

New hook: Accessible

Maybe this would work better?

- Did you know ... that the criss-cross algorithm and the simplex algorithm can visit all 2^D corners of a cube in dimension D but visit only D corners on average?

If you like this, then I should find a reference. Best regards, Kiefer.Wolfowitz (Discussion) 20:04, 22 March 2011 (UTC)[reply]

I added some preliminary referencing about average behavior. The expected behavior of the simplex method (on problems from the unit sphere) is O(D) by Borgwardt and Smale. I don't have Klee & Minty's paper handy, but it seems trivial that a cube drawn from the sphere should have D steps on average. (I don't have time to reference properly today: Iterating the bibliography operator initialized with Fukuda & Terlaky should suffice. Sincerely, Kiefer.Wolfowitz (Discussion) 20:25, 22 March 2011 (UTC)[reply]

I nominated the alternative. Thanks for the suggestions. Kiefer.Wolfowitz (Discussion) 21:06, 22 March 2011 (UTC)[reply]

Looks good to me. CRGreathouse (t | c) 23:25, 22 March 2011 (UTC)[reply]

I submitted it, along with a graphic. Please consider vetting it at DYK. Thanks again for your feedback. (I evaluated the D and 2^D expressions with D=3, computing 3 and 8 respectively: I trust this does not violate Original Research policy!) Cheers, Kiefer.Wolfowitz (Discussion) 00:40, 23 March 2011 (UTC)[reply]