Comments on: A Small Constant http://magic.aladdin.cs.cmu.edu/2006/02/11/a-small-constant/ Algorithms are everywhere. Sun, 30 Aug 2009 23:42:14 -0400 http://wordpress.org/?v=2.9-rare hourly 1 By: Maverick Woo http://magic.aladdin.cs.cmu.edu/2006/02/11/a-small-constant/comment-page-1/#comment-184 Maverick Woo Mon, 13 Feb 2006 03:10:04 +0000 http://magic.aladdin.cs.cmu.edu/?p=237#comment-184 Haha, gotcha! In a way this post is arguing that the constant in front of polylog matters. So while I don't exactly consider big-O to be the enemy all the time, for real-life applications, I can imagine we all have our doubts when choosing between, say, $2 \log^2 n$ and $20 \log n \log \log n$. Haha, gotcha!

In a way this post is arguing that the constant in front of polylog matters. So while I don’t exactly consider big-O to be the enemy all the time, for real-life applications, I can imagine we all have our doubts when choosing between, say, $2 \log^2 n$ and $20 \log n \log \log n$.

]]> By: Ryan Williams http://magic.aladdin.cs.cmu.edu/2006/02/11/a-small-constant/comment-page-1/#comment-183 Ryan Williams Sun, 12 Feb 2006 19:58:07 +0000 http://magic.aladdin.cs.cmu.edu/?p=237#comment-183 Err, I meant that the proof length is bounded by 2^r... big-O notation is the enemy, eh? Err, I meant that the proof length is bounded by 2^r… big-O notation is the enemy, eh?

]]> By: Ryan Williams http://magic.aladdin.cs.cmu.edu/2006/02/11/a-small-constant/comment-page-1/#comment-181 Ryan Williams Sun, 12 Feb 2006 19:56:51 +0000 http://magic.aladdin.cs.cmu.edu/?p=237#comment-181 Your point comes up in measuring the parameters of PCPs, in particular in the amount of random bits r used, since the length of the proof is bound by 2^{O(r)}. E.g., a corollary of Irit Dinur's proof of the PCP theorem is that there are PCPs for SAT that use log_2(n * poly log n) random bits and only O(1) queries suffice. There is <i>no</i> constant in front of the log_2, and the logarithm really is base-two. So, the resulting proof length is n*polylog n, as that's the maximum length that it could possibly be. Your point comes up in measuring the parameters of PCPs, in particular in the amount of random bits r used, since the length of the proof is bound by 2^{O(r)}.

E.g., a corollary of Irit Dinur’s proof of the PCP theorem is that there are PCPs for SAT that use log_2(n * poly log n) random bits and only O(1) queries suffice. There is no constant in front of the log_2, and the logarithm really is base-two. So, the resulting proof length is n*polylog n, as that’s the maximum length that it could possibly be.

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