tag:blogger.com,1999:blog-37222332014-09-01T19:51:47.469-05:00Computational ComplexityComputational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill GasarchLance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.comBlogger2202125tag:blogger.com,1999:blog-3722233.post-6290405190522030302014-08-28T18:23:00.000-05:002014-08-28T18:24:33.030-05:00Sixteen Years in the MakingEvery paper has a story but Sunny Daniel's <a href="http://arxiv.org/abs/1408.6334">Arxiv paper</a> from yesterday deserves a blog post.<br />
<br />
We begin in 1982 when Ravi Kannan proved that Σ<sub>2</sub> (the problems computable in NP with an NP oracle) cannot have n<sup>2</sup> size circuits. Kannan's result hold for n<sup>k</sup>-size circuits but for this story we'll keep it simple.<br />
<br />
Kannan had an ingeniously simple proof. By diagonalization you can create a language L in Σ<sub>4 </sub>that does not have n<sup>2</sup>-size circuits. Now there are two cases:<br />
<ol>
<li>SAT doesn't have n<sup>2</sup>-size circuits. Since SAT is in Σ<sub>2</sub> we are done.</li>
<li>SAT has n<sup>2</sup>-size circuits. Then by <a href="http://dx.doi.org/10.1145/800141.804678">Karp-Lipton</a> Σ<sub>4</sub> = Σ<sub>2</sub> so L is in Σ<sub>2</sub> and we are done.</li>
</ol>
<div>
Kannan's proof is non-constructive and doesn't give an explicit Σ<sub>2</sub> language that we can show doesn't have n<sup>2</sup>-size circuits. Either SAT or L but one can't be sure which one.</div>
<div>
<br /></div>
<div>
In 1998, Sunny Daniels, a PhD student at Rutgers, took Eric Allender's complexity course. Eric offered up a constructive example of Kannan as an open problem. Sunny came up with a solution. He wrote up a draft in LaTeX but for personal reasons dropped out of academics and never published the paper.</div>
<div>
<br /></div>
<div>
In 2003, Jin-Yi Cai and Osamu Watanabe, not aware of Daniels' paper, came up with their own independent construction and presented <a href="http://dx.doi.org/10.1007/3-540-45071-8_22">their paper</a> at the COCOON conference in Montana. Word got back to Sunny but he thought he had lost the LaTeX file and didn't want to retypeset the whole proof.</div>
<div>
<br /></div>
<div>
Sunny had <a href="http://en.wikipedia.org/wiki/Zip_drive">Iomega Zip Drive</a> cartridges from his Rutgers days. Recently he found someone who had a Zip Drive reader and managed to recover the files. In there he discovered the original LaTeX, cleaned the paper up, and sixteen years after his proof put <a href="http://arxiv.org/abs/1408.6334">the paper</a> on ArXiv. Even if you don't care about the math, read the introduction for the complete version of this story.</div>
<div>
<br /></div>
<div>
Kannan's proof actually shows Σ<sub>2</sub>∩Π<sub>2</sub> does not have n<sup>2</sup>-size circuits and this was later <a href="http://dx.doi.org/10.1016/j.jcss.2003.07.015">improved</a> to S<sub>2</sub><sup>P</sup>. Whether we have any constructive language in Σ<sub>2</sub>∩Π<sub>2</sub> or S<sub>2</sub><sup>P</sup> without n<sup>2</sup>-size circuits still remains open. </div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com2tag:blogger.com,1999:blog-3722233.post-76180957040187341172014-08-25T18:50:00.000-05:002014-08-25T18:50:21.359-05:00A Deadly Side of ComplexityBetter algorithms can lead to better medicine and save lives. Just today Tim Gowers <a href="http://gowers.wordpress.com/2014/08/25/icm2014-emmanuel-cands-plenary-lecture/">discusses</a> Emmanuel Candès' ICM <a href="https://www.youtube.com/watch?v=W-b4aDGsbJk">Plenary Lecture</a>, which among other things describes how Candès' work on compressed sensing leads to shorter MRI scans for children, greatly reducing the risk of oxygen deprivation. Prove P = NP with a practical algorithm, and you'll conquer that worst of our diseases. Sounds great until you realize what we can't do.<div>
<br /></div>
<div>
I was talking to a cancer researcher recently and he points out that many of their challenges are indeed algorithmic. But he also brings up the contrapositive. Since we don't have great algorithms now, we don't know how to make sense of DNA sequences and in particular don't know how to map genetic markers to an appropriate cancer treatment. He works with cancer patients, knowing he can't give them the best possible treatment, not because of lack of data, but due to lack of ways to analyze that data. People die because we don't have the ability to break through the complexity of these algorithmic challenges.</div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com2tag:blogger.com,1999:blog-3722233.post-76056566662707215102014-08-21T08:38:00.001-05:002014-08-21T08:38:15.426-05:00Turing's Oracle<a href="http://www.newscientist.com/data/images/ns/covers/20140719.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://www.newscientist.com/data/images/ns/covers/20140719.jpg" height="200" width="152" /></a>My daughter had a summer project to read and summarize some popular science articles. Having heard me talk about Alan Turing more than a few times, she picked a <a href="http://www.newscientist.com/article/mg22329780.400-turings-oracle-the-computer-that-goes-beyond-logic.html">cover story</a> from a recent New Scientist. The cover copy says "Turing's Oracle: Will the universe let us build the ultimate thinking machine" sounds like an AI story but in fact more of an attack on the Church-Turing thesis. The story is behind a paywall but here is an excerpt:<br />
<blockquote class="tr_bq">
He called it the "oracle". But in his PhD thesis of 1938, Alan Turing specified no further what shape it might take...Turing has shown with his universal machine that any regular computer would have inescapable limitations. With the oracle, he showed how you might smash through them. </blockquote>
This is a fundamental misinterpretation of Turing's oracle model. Here is what Turing said in his paper <a href="http://www.turingarchive.org/viewer/?id=469&title=12">Systems of Logic Based on Ordinals</a>, Section 4.<br />
<blockquote class="tr_bq">
Let us suppose we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. We shall not go any further into the nature of the oracle <i>apart from saying it cannot be a machine.</i> (emphasis mine)</blockquote>
The rest of the section defines the oracle model and basically argues that for any oracle O, the halting problem relative to O is not computable relative to O. Turing is arguing here that there is no single hardest problem, there is always something harder.<br />
<br />
If you take O to be the usual halting problem then a Turing machine equipped with O can solve the halting problem, just by querying the oracle. But that doesn't mean that you have some machine that solves the halting problem for, as Turing has so eloquently argued in Section 9 of his <a href="http://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf">On Computable Numbers</a>, no machine can compute such an O. Turing created the oracle model, not because he thought it would lead to a process that would solve the halting problem, but because it allowed him to show there are problems even more difficult.<br />
<br />
Turing's oracle model, like so much of his work, has played a major role in both computability and computational complexity theory. But one shouldn't twist this model to think the oracle could lead to machines that solve non-computable problems and it is sacrilege to suggest that Turing himself would think that.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com3tag:blogger.com,1999:blog-3722233.post-9471532330682740272014-08-18T16:15:00.000-05:002014-08-18T16:15:19.406-05:00Complexity versus Algorithms: The FOCS ChallengeIn recent years, I've heard complaints from my complexity colleagues that FOCS and STOC are mostly algorithms and from the algorithm buddies that STOC and FOCS are mostly complexity. What exactly counts as a complexity or algorithms paper has become quite blurred in recent years. So let's try an experiment. Below is a poll I've created using titles from the upcoming <a href="http://www.boazbarak.org/focs14/">FOCS conference</a>. Which of these papers do you consider complexity? Does complexity in the title make them a complexity paper? <a class="OPP-powered-by" href="http://www.objectplanet.com/opinio/" style="text-decoration: none;"></a><br />
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If you are interested, you can find the manuscripts for most of these papers on the <a href="http://www.boazbarak.org/focs14/accepted.html">FOCS accepted papers list</a>.<br />
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<script type="text/javascript" src="http://www.easypolls.net/ext/scripts/emPoll.js?p=53f2696be4b0ecddd1ca4b67"></script><a class="OPP-powered-by" href="https://www.murvey.com" style="text-decoration:none;"><div style="font: 9px arial; color: gray;">survey tools</div></a>
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Disclaimer: This is a completely non-scientific poll solely for the interest of the readers of this blog. The results will have no effect on future conference papers.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com4tag:blogger.com,1999:blog-3722233.post-36203785674355445162014-08-14T16:02:00.000-05:002014-08-14T16:02:23.596-05:00Favorite Theorems: Limited IndependenceWhen can limited randomness act as well as true random bits? <br />
<blockquote class="tr_bq">
<a href="http://dx.doi.org/10.1145/1754399.1754401">Polylogarithmic independence fools AC<sup>0</sup> circuits</a> by Mark Braverman (JACM 2010)</blockquote>
To explain this result consider choosing uniformly from among the following four strings:<br />
<div style="text-align: center;">
000 110 101 011</div>
If we look at any two of the bits, say the first and third, all four possibilities 00 10 11 01 occur. The sequence is thus 2-wise independent. We can get 2-wise independence using only two random bits to choose one of the four strings. In general one can get k-wise independent in n-bit strings using O(k<sup>2</sup> log n) random bits.<br />
<br />
Braverman shows that any polylogarithmic independent distribution fools polynomial-size constant-depth circuits (AC<sup>0</sup>), or more precisely for a size s depth d circuit C, for k=(log (m/ε))<sup>O(d<sup>2</sup>)</sup>, the probability that C will output 1 with uniformly-random inputs will differ at most ε from the probability C outputs 1 from inputs chosen from a k-wise random distribution. Braverman's result followed after <a href="http://dx.doi.org/10.1137/070691954">Bazzi</a> and <a href="http://dx.doi.org/10.1145/1490270.1490273">Razborov</a> proved a similar result for depth-2 circuits (CNF formulas).
<br />
<br />
Another nice result along these lines: Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco Servedio and Emanuele Viola show that <a href="http://dx.doi.org/10.1137/100783030">Bounded Independence Fools Halfspaces</a>. A half-space is just a weighted threshold function (is the sum of w<sub>i</sub>x<sub>i</sub> at most some given θ). Diakonikolas et al. show that one can fool halfspaces with k=O(ε<sup>-2</sup>log<sup>2</sup>(1/ε)), in particular for constant ε, k is a constant independent of the number of variables.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com3tag:blogger.com,1999:blog-3722233.post-4944025742316149462014-08-12T13:36:00.002-05:002014-08-12T14:45:08.968-05:00Subhash Khot wins NevanlinnaAt the opening ceremonies of the <a href="http://www.icm2014.org/">International Congress of Mathematicians</a> in 2014, Subhash Khot was awarded the <a href="http://www.mathunion.org/general/prizes/nevanlinna/details/">Rolf Nevanlinna Prize</a>, given every four years to an under-40 researcher for outstanding contributions in Mathematical Aspects of Information Sciences. Subhash's citation reads<br />
<blockquote class="tr_bq">
<blockquote class="tr_bq">
Subhash Khot is awarded the Nevanlinna Prize for his prescient definition of the “Unique Games” problem, and leading the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems; his work has led to breakthroughs in algorithmic design and approximation hardness, and to new exciting interactions between computational complexity, analysis and geometry.</blockquote>
</blockquote>
Khot's work has indeed generated a large research agenda over the last decade. I highlighted his work in March's <a href="http://blog.computationalcomplexity.org/2014/03/favorite-theorems-unique-games.html">favorite theorems post</a>.<br />
<br />
In other big news, we have our first female Fields Medalist Maryam Mirzakhani for contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces. Still no female winners among the nine Nevanlinna winners. Artur Avila, Manjul Bhargava and Martin Hairer also received Fields medalists. Stanley Osher won the Gauss Prize, Phillip Griffiths the Chern Medal and Adrián Paenza the Leelavati Prize.<br />
<br />
<a href="http://www.mathunion.org/general/prizes/2014/">Pictures and press releases</a> and <a href="http://www.mathunion.org/general/prizes/2014/prize-citations/">citations</a> of all the prize winners.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com4tag:blogger.com,1999:blog-3722233.post-8629382914682215472014-08-11T06:39:00.000-05:002014-08-11T08:11:55.028-05:00Questions that arose teaching High School students cryptoI taught a 3-week, 3-hours-a-day course to High School student titled<br />
<br />
<i>Computer Science: A Hands Off Approach.</i><br />
<br />
Given that time constraints and the fact that some already know (say) Java and some don't know any language, this seemed like a good choice.<br />
<br />
I decided to teach mostly pre-RSA crypto with the following theme: Alice and Bob want to pass secret messages. How do they do it? I cover Shift, affine, general sub, Vigenere, Matrix, 1-time pad, Diffie-Helman (a highpoint of the course since Alice and Bob don't have to meet in a dark alley). In also did secret sharing with polynomials, error correcting codes (elementary), Huffman codes, and some applications of mod arithmetic.<br />
<br />
While teaching this course some points of interest came up. I suspect most are know and I appreciate polite comments telling me so.<br />
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<ol>
<li>A student suggested this cipher: code a,b,c,...,z into a 100-letter alapahbet and map each letter to a set of symbols that is the size of the freq. For example, if e occurs 9% of the time then map e to 9 letters. Then use those letters at random. This would seem to foil freq analysis? Does it? Has it been used? What are the downsides.</li>
<li>Many students suggested using Vigenere but instead of having every xth letter be done by a different shift, have it be affine or general. Of course this can be cracked the same way Vig is cracked. But it does raise an interesting point: Which ciphers are used and not used can be the based on when things were discovered. Martians may very well have used some kind of Vig where every xth letter is a different gen sub cipher.</li>
<li>Wikipedia and other sources say that the Vig cipher was unbroken for 300 years. A student pointed out that it might have been broken but the one who broke it didn't say. Jon Katz (crypto prof at UMCP) can't believe it wasn't broken immediately, but of course hindsight is 20-20.</li>
<li>(I have commented on this before) A matrix cipher with a 10x10 matrix seems uncrackable using plaintext only. I have made this question rigorous <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/matrix.pdf">here</a>.</li>
<li>I made the comment that 1-time pads are not used much (is this even true? Might depend on the definition of ``must'') because getting a perfect source of randomness is hard. During WW II they also would be hard to use because its hard to carry around a billion bits. But now that would not be a problem. Again--if history had been different we may use 1-time pads, or quasi-random ones today!</li>
<li>I told the students about arithmetic mod n. One of the students really really didn't like that (say) in mod 7 we have 1/3 = 5. He REALLY wants 1/3 to be between 0 and 1. I suspect he didn't care much for discrete logs either. This was a GOOD student- so his objection was not that it was hard.</li>
<li>For some of the students their first exposure to matrices was matrix codes over mod 26. I hope they can recover from that.</li>
<li>Most of the students know what logs were, but weren't that comfortable with them. And here I go and teach them discrete logs! I hope they can recover from that.</li>
<li>I showed the students that there were quadratic equations over mod 12 with more than 2 roots and challenged them to see how many roots they could come for other mods. One of my students ran massive computer tests and found stuff and in the end had a result that didn't need all of his computations: x^2 \equiv 0 mod n^2 has n roots. And I later had on a HW x^a \equiv 0 mod n^a. I am sure none of this is new, but its new to him when he discovered it and of course new to the class when I taught it.</li>
<li>I taught the class the Baby-Step/Giant-Step Discrete Log algorithm which has sqrt(p) prepocessing an sqrt(p) running time. Its not used because it also takes sqrt(p) space; however, it was good to show them that Discrete Log can be done in sqrt(p) time, much better than p time--- hence Alice and Bob need to pick their parameters larger than they may have thought when doing Diffie-Helman. That night I easily worked out that it can be modified to do p^{2/3} preprocessing (and space) but just p^{1/3} time. HW was p^a prep, p^{1-a} time. One of the students inquired if this algorithm has a name. I then looked over the web but couldn't find it anywhere so I told them to call it <i>The Gasarch Variant of Baby-Step, Giant-Step.</i> I also quickly told them to NOT be impressed--- and this helped me make a point I had made often, that CS is a NEW field, so NEW that one can present new or newish results to HS students. I also made the point that I am sure this variant is KNOWN to anyone who would care, but (1) they may not care since it takes even more space if x is larger than 0.5 and more time if x is less than 1/2 and (2) not everything is no the web. That last point freaked them out more than the quadratic equation mod 12 that had more than two roots.</li>
</ol>
UPSHOT- teaching HS students CS can bring up some issues that you had not thought of. Or at least that I had not thought of. <br />
<ol>
</ol>
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<br />GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com10tag:blogger.com,1999:blog-3722233.post-10023967642434045792014-08-07T23:00:00.003-05:002014-08-07T23:00:36.383-05:00The n^{1/3} barrier for 2-server PIR's broken! A lesson for us all!Recall: PIR stands for Private Information Retrieval. Here is the model: A database is an n-bit string (my wife tells me this is not true). The user wants to find the ith bit without the database knowing what bit the user wants. The user COULD just request ALL n bits. Can the user use less communication? See <a href="http://www.cs.umd.edu/~gasarch/TOPICS/pir/pir.html">here</a> for a website of many papers on PIR.<br />
<br />
<ol>
<li>If there is just one copy of the DB and there are no comp. constraints on the computational power of the DB then ANY protocol requires n bits comm.</li>
<li>If there is one copy of the DB then, with some comp constraints, you can do better. I state one result: if quadratic residue is hard then there is a protocol using n^epsilon bits of comm. (Kushilevitz and Ostrovsky). The rest of this post is about the info-theoretic case, so the DB has no comp. constraints.</li>
<li>If there are two copies of the DB then there is a protocol that uses n^{1/3} bits. (Chor, Kushilevitz, Goldreich, Sudan)</li>
<li>If there are three copies of the DB then there is a protocl that uses n^{1/32582658} bits. Really! (Yekhanin).</li>
<li>If a 2-server protocol is a bilinear-group protocol (which all prior constructions were) then it must take n^{1/3}.(Razborov and Yekhanin)</li>
<li>Most known constructions put into a geometric framework (Woodruff and Yekhanin). </li>
</ol>
Recently Dvir ahd Gopi posted a paper <a href="http://eccc.hpi-web.de/report/2014/094/">here</a> which gives a 2-server PIR protocol that uses n^P bits where P= sqrt( (log log n)/log n)). That breaks the barrier proven by RY! How is that possible?<br />
<br />
The barrier result was only for a certain type of PIR. It DID cover all the known PIR schemes at the time. But it does not cover this one.<br />
<br />
This is how Barrier SHOULD work--- they should not discourage, but they should point to difficulties so that someone can overcome them.<br />
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Also note, the new result used the some of the Framework of Woodruff and Yekhanin. So good to unify and obtain new proofs of old results. GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-50565893735091950552014-08-05T10:25:00.002-05:002014-08-05T10:25:31.741-05:00How I know Moneyball is a true story. Do we clean up theorems and life too much? A few years ago I saw the movie Moneyball about how the Oakland A's used intelligent statistics to... win?<br />
No, but to do better-than-expected. Even if I didn't know it was a true story I would have assumed it was because the following are very rare or odd in a fictional story:<br />
<br />
<ol>
<li>At the end of the team doesn't win- it just does better than expected. In the typical sports movie the underdog pulls it all together and wins. In some the underdogs loses but they are now better people or something. In an episode of Cheers where they were the underdog to Gary's Tavern in a bloody mary making contest the Cheers gang cheats and wins. But in Moneyball, and in NO other sports (or contest) movie that I know of, does the underdog do better-than-expected in an undramatic matter. This is NOT a complaint- just note that its real life.</li>
<li>In Moneyball the General Manager wants to try out mathematical methods and the Manager resists. In most movies its the suits that are wrong and the people on the ground that are right. This is even a theme of many articles about business that I read in magazines on airplanes. So this inversion is odd - but again, you can't argue that a true story is unrealistic or undramatic. </li>
<li>Bill Beane, the one who wants to use math techniques, thinks that what Baseball scounts look for is the wrong thing. In fact, they misjudged him when he was a young player. But in what direction?--- they thought he was BETTER than he was. If this was a fictional story surely the scouts would think he was WORSE than he was.</li>
</ol>
I am glad that (as far as I know) neither the book nor the movie tried to make the story more satisfying or dramatic.<br />
<br />
In academia we do clean things up for a better story line. If the true motivation for working on a problem doesn't really make sense when you see the final paper, we change our motivation. Our original proof is intuitive but ugly, so we change it to be polished but less clear where it came from. Historians often simplfiy to make sense of things. I am NOT complaining- merely asking, do we do it too much?<br />
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When I was in ninth grade and was told that you could solve a quadratic equation (I rederived the quadratic formula once a month to make sure I could), a cubic, a a quartic, but not quintic, I immediately said "I want to goto College to learn why you can't solve a quintic" That sparked my interest in math.<br />
<br />
Is the above story true? I am sure that in ninth grade I did learn that the quintic was unsolvable and that was of great interest to me, and I really did rederive the quadratic equation once a month. And I was interested to learn that the quintic was not solvable. But I really doubt the story is as clean as presented above. Even so, the story is true in spirit. However, I would not want to push the point.<br />
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How about you? Do you tell stories about yourself or about others that are just a little too polished? Not so much false, and not even to put yourself in a better light, but just a little to clean to have really happened.GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com8tag:blogger.com,1999:blog-3722233.post-28725548791844314762014-07-31T16:46:00.000-05:002014-07-31T16:46:12.379-05:00How NOT to bet(True Story, but names are changed for no good reason.)<br />
<br />
Alice, Bob, and Carol are betting on who will be the Republican Nominee in 2016.<br />
<br />
<ol>
<li>Alice bets on Jeb Bush. Alice's reasoning is that in the past the Republicans always end up with a moderate who they think can win. NOTE: From Alice's prediction you can tell NOTHING about her politics.</li>
<li>Bob bets on Paul Ryan. Bob's reasons that in the past the Republicans always end up with someone who they are familiar with, quite often someone who has run before. Normally this might be someone who ran in the primaries four years earlier; however, none of Herman Cain, Michelle Bachman, Rick Santorum, Newt Gingrich, seem likely. Rick Perry is possible but seems unlikely. That leaves Paul Ryan (Curiosity- people who lose, even if its close, do not run again. Not sure why. So it won't be Romney.) NOTE: From Bob's prediction you can tell NOTHING about his politics.</li>
<li>Carol bets on Rand Paul. Carol's reasoning is that the American Public is tired of Big Government and is ready for the Rand Paul Revolution! NOTE: From Carol's prediction you can tell EVERYTHING about her politics.</li>
</ol>
Carol is letting her opinion drive her betting. Can someone take advantage of this? YES- if someone is betting from sentiment or emotion or conviction rather than from facts, that gives you an advantage in a bet. This does not always work, but its a good bet. Now watch, I'll lose my bet to Carol (NOTE- I am Bob).<br />
One full dollar!<br />
<br />
I think I read that you are better off betting AGAINST a triple crown if a horse won the first two legs for the same reason- there will be emotion driving the bet FOR, and that increases the payoff for those betting AGAINST. But of course nothing is guaranteed. And there are some cases where its just ridiculous to vote against (Secretariat in 1973); however, to use my system you can't just skip one since you ``just know'' that there will be a triple crown. Its a long term strategy. And one could be wrong. That's why its called gambling!<br />
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Are there other cases where long term betting with facts and against sentiment can give you an edge? GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com5tag:blogger.com,1999:blog-3722233.post-84326468277519802242014-07-27T20:50:00.000-05:002014-07-29T18:31:51.939-05:00The Change Problem and the Gap between Recreational and Serious MathematicsIn a prior post (<a href="http://blog.computationalcomplexity.org/2014/06/how-many-ways-can-you-make-change-of-n.html">here</a>) I discussed the change problem: given a dollar, how many ways can you make change using pennies, nickels, dimes, and quarters (and generalize to n cents). Scouring the web I found either programs for it, the answer (242), and answers like `is the coefficient of x^100 in ... . What I didn't find is a formula for n cents. So I derived one using recurrences and wrote this up with some other things about the change problem, and I posted it on arXiv. (I have updated the paper many times after comments I got. It is still where it was, but updated, <a href="http://arxiv.org/abs/1406.5213">here</a>.)<br />
<br />
I then got some very helpful emails pointing me to the vast math literature on this problem. I was not surprised there was such a literature, though I was surprised I hadn't found any of it searching the web.<br />
The literature didn't have my formula, though it had several ways to derive it (some faster than mine).<br />
<br />
Why isn't the formula out there? Why couldn't I find the literature?<br />
<br />
<ol>
<li>The formula falls into a space right between recreational and serious math. I use a recurrence but not a generating function. Recreational math just wants to know the answer for 100, so a program suffices (or some clever by-hand recurrences, which are also in my paper). Serious math people are happy to show that a formula CAN be derived and to refine that in terms of how quickly the coefficients can be found.</li>
<li>There is a gap between recreational and serious math. In particular- the serious people aren't talking to (or posting on websites to) the rec people.</li>
<li>Different terminologies. I was looking for ``change problems'' and ``coin problems'' and things like that when I should have been looking for ``Sylvester Denumerant'' or ''Frobenius problem''</li>
</ol>
For some areas Google (and other tools) are still not as good as finding someone who knows about your area. My posting on the blog got some people who know stuff's attention, and my posting on arXiv got one more person (who knew A LOT!). I'm glad they emailed me comments and references so I improved the paper and cited the literature properly. But this seems so haphazard! Google didn't work, mathstack exchange and other similar sites didn't work (that is where I saw the Rec people post but not get a formula). Is it the case that no matter how good Google gets we'll still need to find people who know stuff? Thats fine if you can, but sometimes you can't.GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com11tag:blogger.com,1999:blog-3722233.post-39954068096934238272014-07-24T18:54:00.003-05:002014-07-25T14:57:51.881-05:00Need some book reviewed- faster than usualI have been the SIGACT NEWS book review editor since 1997. I have tried to get about 10 books reviewed per column. I have succeeded- perhaps too well! I have gotten so many reviews out that I only have six reviews left.<br />
<br />
I DO have many books that could be reviewed, and that is where YOU come in!<br />
<br />
List of books available for review: <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/booklist.pdf">Here</a><br />
<br />
Advice for reviewers: <a href="http://www.cs.umd.edu/~gasarch/bookrev/advice.txt">Here</a><br />
<br />
LaTeX Template for reviews: <a href="http://www.cs.umd.edu/~gasarch/bookrev/revtemplate.txt">Here</a><br />
<br />
ALL of my prior columns: <a href="http://www.cs.umd.edu/~gasarch/bookrev/bookrev.html">Here</a><br />
<br />
IF you want to review a book DO NOT leave a comment- just email me (ADDED LATER- EMAIL ME at gasarch@cs.umd.edu. Someone emailed Lance instead<br />
which you shouldn't do.)<br />
a set of books you want to review. I will pick out one for you--- it may be based<br />
on what I've already got requests for.<br />
<br />
Since it is summer you are not as busy as the regular hear (hmmm- I am running an REU program AND teaching an intense High School Class, AND going to a security conference, and mentoring three high school students hoping for some more free lunches, so I am actually MORE busy) so summer is a GOOD time to review a book.<br />
<br />
Deadline to ASK for a book: Request that you make your request BEFORE July 28 so that when I email in my column with the list-of-books-I-need-reviewed, it is accurate.<br />
<br />
Deadline for Review: About two months after you get it, though this can be flexible. GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-35260214701722907472014-07-22T10:42:00.002-05:002014-07-22T10:42:43.879-05:00The Burden of Large EnrollmentsThis week I'm at the <a href="http://cra.org/events/snowbird-2014/">CRA Snowbird conference</a>, the biennial meeting of CS chairs and other leaders in the field. In <a href="http://blog.computationalcomplexity.org/2012/08/moocs.html">2012</a> many of the discussion focused on MOOCS. This year the challenge facing most CS chairs are booming enrollments in CS courses. A nice problem to have, but a problem nevertheless.<br />
<br />
Last night we had a broad discussion about the burgeoning number of students. Ed Lazowska showed his <a href="http://lazowska.cs.washington.edu/NCWIT.pdf">NCWIT slides</a> giving anecdotal evidence. It's too early to get a complete survey of CS departments but hardly anyone in the audience felt that enrollments were not going up by double (or triple) digit percentages. Not only an increase in majors, but a number of non-majors, minors or others, who take upper level courses.<br />
At Georgia Tech it seems every engineering student wants to minor in CS.<br />
<br />
We all have to deal with the increase in the short term. Depending on the institution, we have more classes, larger classes, enrollment caps, increases in instructors, PhD lecturers, teaching postdocs, automated grading, undergrad and MS TAs and having other departments take up some of the load. All of this could hurt the undergrad experience but with careful planning we can mitigate those effects.<br />
<br />
What's driving the increase? Some suggested a change in the ubiquitous view of computing and the more positive opinion of geeks in society. More likely though, the driver is jobs, the great need for computer scientists and not just from computing companies, and the limited jobs for those graduating with non-technical majors.<br />
<br />
Is the big shifts in enrollment a permanent change or just another part of the boom and bust cycle in CS? More than a theoretical questions, a permanent change makes for a better argument with deans and provosts to increase faculty sizes in computer science. There are some signs of "this time is different," such as the increase in non-majors in upper level courses, but it's an argument that will have a hard sell unless the increase is sustained for several years. One good argument: If you draw a linear line through enrollments since the 70's you get a consistent 10% yearly increase averaged over booms and busts.Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-28362174389397926712014-07-17T08:24:00.001-05:002014-07-17T08:24:42.504-05:00Elfdrive<div>
New York Times, dateline June 11, 2019</div>
<div>
<blockquote>
With a near record-setting investment announced last week, the self-driving car service Elfdrive is the hottest, most valuable technology start-up on the planet. It is also one of the most controversial.<br />
The company, which has been the target of protests across Europe this week, has been accused of a reckless attitude toward safety, of price-gouging its customers, of putting existing cabbies out of work and of evading regulation. And it has been called trivial. In The New Yorker last year, George Packer huffed that Elfdrive typified Silicon Valley’s newfound focus on “solving all the problems of being 20 years old, with cash on hand.”<br />
It is impossible to say whether Elfdrive is worth the $117 billion its investors believe it to be; like any start-up, it could fail. But for all its flaws, Elfdrive is anything but trivial. It could well transform transportation the way Amazon has altered shopping — by using slick, user-friendly software and mountains of data to completely reshape an existing market, ultimately making many modes of urban transportation cheaper, more flexible and more widely accessible to people across the income spectrum.<br />
Elfdrive could pull this off by accomplishing something that has long been seen as a pipe dream among transportation scholars: It has the potential to decrease private car ownership.<br />
In its long-established markets, like San Francisco, using Elfdrive every day is already arguably cheaper than owning a private car. Elfdrive says that despite dust-ups about “elfpricing” at busy times, its cheapest service is usually 30 percent less expensive than taxis.<br />
The competition between Elfdrive and its rivals is likely to result in more areas of the country in which self-driving becomes both cheaper and more convenient than owning a car, a shift that could profoundly alter how people navigate American cities.<br />
Over the next few years, if Elfdrive do reduce the need for private vehicle ownership, they could help lower the cost of living in urban areas, reduce the environmental toll exacted by privately owned automobiles (like the emissions we spew while cruising for parking), and reallocate space now being wasted on parking lots to more valuable uses, like housing.<br />
Paradoxically, some experts say, the increased use of self-driving cars could also spawn renewed interest in and funding for public transportation, because people generally use taxis in conjunction with many other forms of transportation.<br />
In other words, if Elfdrive and its competitors succeed, it wouldn’t be a stretch to see many small and midsize cities become transportation nirvanas on the order of Manhattan — places where forgoing car ownership isn’t just an outré lifestyle choice, but the preferred way to live.</blockquote>
If you haven't guessed all I did was minor edits to a Farhod Manjoo <a href="http://www.nytimes.com/2014/06/12/technology/personaltech/with-ubers-cars-maybe-we-dont-need-our-own.html">piece</a> on Uber. My point is that it all fits, there really isn't that much difference between a car driven by a magical AI elf or that driving by a real person, if it's all controlled by an app. You can start to see the effects of self-driving cars, the good and the bad, from what's going on now with ride-sharing services. The big difference with self-driving cars will be the complaints won't come just from the taxi drivers, but the truckers and delivery people as well.</div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com5tag:blogger.com,1999:blog-3722233.post-80536880636651453172014-07-13T22:34:00.001-05:002014-07-14T12:38:08.663-05:00What to call the top and bottom part of (n choose k)In my last post I asked for candidates for names for the top and bottom part of (n choose k) . Here are the candidates and my comments on them and ... the winner!<br />
<br />
<ol>
<li>Top part: Degree, Bottom part: Index. </li>
<li>Top part: Bino, Bottom part: Mial</li>
<li>Top part: Numerator, Bottom part: Denominator</li>
<li>Top part: Outcomes, Bottom part: Possibilities</li>
<li>Top part: Binomerator, Bottom part: I've got nothing</li>
<li>Top part: *, Bottom part: *</li>
<li>Top part: Biponendo, Bottom part: Bividendo</li>
<li>Top part: Choosand, Bottom part: choosee</li>
<li>Top part: Set size, Bottom part: Subset size.</li>
</ol>
I leave out the explanations for these since one criteria is that they be self explanatory.<br />
While choices 4,8,9 are tempting along those lines, the winner is<br />
<br />
Numerator/Denominator<br />
<br />
Why? One of the people who suggested it gave a pointer. The pointer actually went to calling the top and bottom part of the Legendre symbol Numerator and Denominator. And thats just it- there are several<br />
other math things that have a top and bottom part. We could try to find a name for each one, OR<br />
just use Numerator and Denominator for all of them. That seems to work. SO- next time you write a paper and need to refer to the top part of a bin coeff or a leg symbol or something else, use the terms<br />
Numerator and Denominator.<br />
<br />
CODA: `The Denominator' could be an Arnold Schwarzenegger character. A Math teacher by day, a crimefighter by night. Catchphrase: I'm going to put you under!GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com6tag:blogger.com,1999:blog-3722233.post-47765689324423318682014-07-09T21:22:00.000-05:002014-07-09T21:22:03.646-05:00Is there a word for the top part of a binomial coefficient?Consider the sequence:<br />
<br />
x/y, (x+1)/y, (x+2)/y, ..., (x+z)/y<br />
<br />
one could say <i>in this sequence of fractions the numerators goes through z-1 consecutive numbers.</i><br />
<br />
Consider the sequence<br />
<br />
(x choose y), (x+1 choose y), (x+2 choose y),...,(x+z choose y)<br />
<br />
one could say <i>in this sequence of binomial coefficients the top-part goes through</i> <i>z-1 consecutive numbers.</i><br />
<br />
Is there a better way to say this? That is, is there a term for the top-part of a binomial coefficient? Or for that matter the bottom part? I have not been able to find one on the web. Hence I propose a contest:<br />
<i> </i><br />
<ol>
<li>Leave as a comment a proposed name for the top-part and for the bottom-part of a binomial coefficient.</li>
<li>If you find a website that has some official name, leave that as a comment.</li>
</ol>
I am not sure if there will be a winner of what he or she will get. But if we can agree upon a term then we are all winners!<br />
<br />GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com11tag:blogger.com,1999:blog-3722233.post-70455929922794544702014-07-07T07:48:00.000-05:002014-07-07T07:48:38.456-05:00Favorite Theorems: Compressing the Local LemmaNot only did Moser give a near optimal construction, but he did so in my favorite STOC talk of the decade.<br />
<br />
<div style="text-align: center;">
<a href="http://dx.doi.org/10.1145/1536414.1536462">A Constructive Proof of the Lovász Local Lemma</a> by Robin Moser</div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
The <a href="http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma">Lovász local lemma</a> informally states that if you have a large set of events with limited dependence and they individually have a reasonable chance of happening, then there is a positive probability that they all happen. Moser focused on the special case of Boolean formula satisfiability: If we have a k-CNF formula φ where each clause shares a variable with at most r other clauses with r<2<sup>k</sup>/8 then φ is satisfiable. Note the result does not depend on the number of variables or clauses.<br />
<br />
Moser gave this simple algorithm and a proof using entropy that I recognized as a Kolmogorov complexity argument and so excited me that I immediately wrote up the <a href="http://blog.computationalcomplexity.org/2009/06/kolmogorov-complexity-proof-of-lov.html">Kolmogorov proof</a> as a blog post.<br />
<br />
Moser's paper kickstarted a whole research agenda, with tight bounds using even simpler algorithms (though more complex proofs) and various generalizations. A whole <a href="http://www.columbia.edu/~cs2035/stoc/stoc2014/tutorials.shtml#lll">tutorial day</a> of STOC 2014 focused on the local lemma describing research that directly or indirectly came from Moser's most beautiful construction.</div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com3tag:blogger.com,1999:blog-3722233.post-19978707038615968712014-07-02T18:24:00.000-05:002014-07-02T18:25:05.316-05:00Four Centuries of Logarithms<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.nms.ac.uk/images/powerof10-book-348px.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://www.nms.ac.uk/images/powerof10-book-348px.jpg" height="320" width="236" /></a></div>
I just returned from visiting my former student Rahul Santhanam in Edinburgh. The National Museum of Scotland has an <a href="http://www.nms.ac.uk/our_museums/national_museum/exhibitions/power_of_ten.aspx">exhibit</a> on logarithms, first published in a book by John Napier in 1614.<br />
<br />
Napier invented logarithms to make calculations like multiplication, division and exponentiation easier, using identities like log(ab)=log(a)+log(b). Logarithmic tables and slide rules came soon thereafter. Slide rules became the standard way to do mathematical computations for your typical scientist/engineer until reasonably priced calculators appeared in the late 70's. My chemistry teacher in 1980 taught us how to use a slide rule, even though we all had calculators, and I even (foolishly) tried using my father's slide rule on a chemistry test.<br />
<br />
The exhibit struggled to find current uses for logarithms, mentioning only the Richter scale. In theoretical computer science we use logarithms all the time. Here's an incomplete list off the top of my head.<br />
<ul>
<li>As part of a running time usually as a result of divide and conquer. Sorting, for example, takes Θ(n log n) comparisons.</li>
<li>The size of a number, pointer or counter. To count up to n requires log n bits of storage.</li>
<li>The representation of a small amount of space as in the complexity classes L and NL.</li>
<li>To capture entropy, coupon collector and other topics in probability and information.</li>
<li>Roughly captures the sum of 1/i or exactly capturing the integral of 1/x.</li>
<li>The inverse of exponential growth.</li>
</ul>
<div>
Thanks John Napier for the logarithm and making our lives just a little less linear.</div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com4tag:blogger.com,1999:blog-3722233.post-24581321836395872582014-06-29T15:05:00.000-05:002014-06-29T15:05:03.583-05:003000 lunchesLast week fortune smiled on two of my friends:<br />
<br />
<ol>
<li>Ken Regan made the cover of Chess Life for his work on catching chess cheaters. <a href="http://rjlipton.wordpress.com/2014/06/18/the-problem-of-catching-chess-cheaters/">(See here.)</a></li>
<li>Jacob Lurie who I mentored when he was a high school student 1993-1995 won $3,000,000 for doing pure math. I am not kidding. See <a href="https://breakthroughprize.org/?controller=Page&action=news&news_id=18">here</a> and here. and <a href="http://www.scientificamerican.com/article/with-awards-like-the-breakthrough-prize-who-needs-a-nobel/">here. </a> This is a relatively new award called the <a href="http://en.wikipedia.org/wiki/Breakthrough_Prize">breakthrough prize.</a></li>
</ol>
The breakthrough prize started a few years ago in Life Sciences and Fundamental Physics (not sure what they mean-possibly Mathematical Physics or Theoretical Physics). This is the first year it is given for Mathematics. The other winners of the Breathrough prize in math are Simon Donaldson, Maxim Kontsevich, Terence Tao, and Richard Taylor.<br />
<br />
$3,000,000 is the largest amount of money for prizes of this sort (the Nobel is a paltry $1,200,000). Even so, I had not heard of the award until there was a math one. I wonder if it will become a household name like the Nobel has.<br />
<br />
A few weeks ago I told my student Tucker, who is going to Digipen- a 4-year college with a BIG emphasis on programming computer games- that he is more likely to become a millionaire then my student Sam who is going to CMU to study pure math. I could be wrong.<br />
<br />
I taught Jacob lots of recursion theory and he won the Westinghouse Award with a paper on Recursive Surreal Numbers; however, the work was all his own and didn't use much of what I taught him. After that I made a policy that for every $1000 a high school student of mine wins I get a free lunch. I emailed Jacob- and while he's not sure he'll give me 3000 lunches, he will treat me to lunch next time I am in Boston.<br />
<br />
To quote the article, he won it for <br />
<br />
Foundations of higher category theory and derived algebraic geometry for the classification of fully extended topological quantum field theories; and for providing a moduli-theoretic interpretation of elliptic co-homology.<br />
<br />
I know what some of those words mean. <br />
<br />
<br />
<br />GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com1tag:blogger.com,1999:blog-3722233.post-12024238515202264602014-06-26T03:36:00.000-05:002014-06-26T03:43:00.370-05:00Computer DatingMy brother went through his old stuff cleaning out his apartment in New York and stumbled upon the 1981 computer dating questionnaire I wrote in high school. Forgive the dated and adolescent nature of the questions.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-TgmZitxOQ4I/U6vUfusmCQI/AAAAAAAAvjw/JziTwdznkH4/s1600/IMG_1134.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-TgmZitxOQ4I/U6vUfusmCQI/AAAAAAAAvjw/JziTwdznkH4/s1600/IMG_1134.JPG" height="360" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
The computer science teacher showed us an example of a computer dating form created by some company, a new idea at the time. I decided to run computer dating at the high school, created a questionnaire and passed it around. I did this a few years, the 1981 form must have been the last as that's the year I graduated. In those ancient history days, my fellow students filled out these forms on paper and then I would type the answers into the big teletype machines in the computer lab. I wrote a program to create a matching, I have no idea what algorithm I used but I'm sure it was quite simplistic. I always got matched to the same person, but the pickup line "my computer dating program matched us up" didn't go very far. I did hear of one couple having one (and only one) date because of the program. There's a reason I became a theorist.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
My daughters tell me the whole scene has changed with traditional dating quite rare these days. "First you become boyfriend/girlfriend and then you date" which seems so backwards to me.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
Now we have websites that use sophisticated machine learning algorithms to connect people. I met my wife the old fashioned way--at a wine and cheese party sponsored by a Boston-area Jewish singles group.</div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com3tag:blogger.com,1999:blog-3722233.post-49385033008914761192014-06-23T08:04:00.002-05:002014-06-23T18:36:15.861-05:00How many ways can you make change of n cents?(For a related post see <a href="http://blog.computationalcomplexity.org/2013/06/quantum-tecniquesgen-functions-dont-be.html">my post on Schur's theorem</a>. The paper THIS post refers to is <a href="http://arxiv.org/abs/1406.5213">here</a>.) <br />
<br />
(ADDED LATER: A commenter pointed to Graham-Knuth-Patashnik for a closed form for pennies, nickels, dimes, quarters, half-dollars. I have wriitten up their account and it is available <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/knuthchange.pdf">here</a>. I also apply their technique to just get 1,5,10,25.)<br />
<br />
How many ways are there to make change of a dollar using pennies, nickels, dimes, and quarters? This is a well known question; however, the answers I found were disappointing. They were of two types<br />
<ol>
<li>There are 242 ways to make change. The author then point to a program he wrote or to the actual list of ways to do it.</li>
<li>The number of ways to make n cents change is the coeff of x^n in the power series for 1/(1-x)(1-x^5)(1-x^{10})(1-x^{25}). This can be worked out. I have never seen it worked out.</li>
</ol>
The first answer yields an actual number but is not interesting mathematically. The second answer is interesting mathematically but does not easily yield an actual number. <br />
<br />
I looked for a closed form on the web but could not find one. So I did it myself. The paper is pointed to above. I did not use generating functions, just recurrences. I do not know if it is new. Whether it is old or new I hope you enjoy it. The paper actually gives a closed form for the coin sets {1,x,kx} and {1,x,kx,rx}. Along the way we derive the answer for a dollar and the usual currency by hand.<br />
<br />
This raises some questions:<br />
<br />
<ol>
<li>Is my formula new? I'd be surprised either way. (Is it possible to be surprised at both statement A and statement NOT(A)?). If it's new I'll be surprised since the questions has been around for so long and surely someone would have done it. If it's known I'll be surprised since (a) I went to JSTOR and searched for all math journals they had for the words ``change'' and ``coin'', looked at the over 400 such articles (just the titles and abstracts) and didn't find it, (b) this came up in math.stackexchange <a href="http://math.stackexchange.com/questions/15521/making-change-for-a-dollar-and-other-number-partitioning-problems">here</a> and, true to form, every answer was either a program or a generating function and (c) other searches also did not turn up the result.(ADDED LATER- Since the 1,5,10,25,50 case was known, I guess I AM surprised.) </li>
<li>Even if its not new it is clearly not well known. Why is that? Its a natural problem that people seemed to be interested in. One conjecture: The COMP SCI people interested in it were happy to write a program. The MATH people interested in it were happy to say `its merely the nth coeff of...' So a closed form solution seems to not be of interest to either camp</li>
<li>Is it interesting? (a) Comp Sci: the closed form solution gives an answer in O(1) steps instead of the usual dynamic programming O(n) solution, and (b) Math: the closed form solution tells you the coefficients of the power series of 1/((1-x)(a-x^5)(1-x^{10})(1-x^{25}). </li>
</ol>
<br />
Side Bar: I asked the 10 students in my REU program to just GUESS how many ways there are to make change of a dollar with pennies, nickels, dimes, and quarters. Most made guesses under 100. The lowest guess was 30. One person guessed 100! and one guessed 100!/2. I then told them that it was between 30 and 100! and assigned them to derive it themselves by hand. Most were able to.GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com9tag:blogger.com,1999:blog-3722233.post-65522303321411062442014-06-18T15:12:00.000-05:002014-06-18T15:12:38.637-05:00Favorite Theorem: PCP SimplifiedThe famous <a href="http://dx.doi.org/10.1145/278298.278306">Probabilistically Checkable Proof Theorem</a> due to Arora, Lund, Motwani, Sudan and Szegedy states that every problem in NP has a proof that can be checked efficiently by a verifier using O(log n) random coins and a constant number of queries. Not only amazing in its own right, the PCP theorem has significant implications for hardness of approximation. In its <a href="http://dx.doi.org/10.1145/502090.502098">strongest form</a>, no efficient algorithm can maximize the number of satisfying clauses in a formula better than a 7/8 approximation unless P = NP. Both of these results have been on my previous <a href="http://people.cs.uchicago.edu/~fortnow/papers/topten.pdf">favorite</a> <a href="http://blog.computationalcomplexity.org/2004/05/favorite-theorems-probabilistically.html">theorems</a> lists so lets add one more.<br />
<br />
<div style="text-align: center;">
<a href="http://dx.doi.org/10.1145/1236457.1236459">The PCP Theorem by Gap Amplification</a> by Irit Dinur (<a href="http://www.wisdom.weizmann.ac.il/~dinuri/mypapers/combpcp.pdf">PDF</a>)</div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
Dinur's main result doesn't prove a new theorem per se but gives a much simplified and intuitively understandable proof than the original paper. She increases the approximation gap by using an expansion graph though this causes an increase in the alphabet size which she reduces using another transformation. Repeating these steps yields the theorem. Unless you need tight bounds, Dinur's proof is the one to teach or read. </div>
<div style="text-align: left;">
<br /></div>
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Dinur's paper followed shortly after <a href="http://blog.computationalcomplexity.org/2014/02/favorite-theorems-connecting-in-log.html">Omer Reingold's theorem on undirected connectivity</a>, and while these are very different proofs of different results, together they show the amazing power of expander graphs in computational complexity. </div>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com0tag:blogger.com,1999:blog-3722233.post-53754619193312701572014-06-16T08:01:00.004-05:002014-06-16T08:01:44.370-05:00Ref/info request for an obvious appraoch to GITalking about Graphi Isom with some students they came up with the following idea which is certainly not new; however, I couldn't quite find much about it on the web.<br />
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ALG(G,H): Let G(0,1) and H(0,1) be the usual adacency matrix using 0 for EDGE NOT THERE and 1 or EDGE IS THERE. Choose n random pairs of numbers (a1,b1), (a2,b2),...,(an,bn) all \le n^2 (happy to change the bound if it will make it work better). For all i see if DET(G(ai,bi))=DET(H(ai,bi)). If there is an i such that they are NOT equal than output NOT ISOM (with complete certainly). If they are ALWAYS equal then output GEE, THOSE GRAPHS ARE PROB ISOMORPHIC.<br />
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<li>I person who used to work on GI told me that he THINKS there are graphs G, H where the polynomials G(x,y) and H(x,y) are identical yet G and H are not isom. So their are some G,H where the above will definitely fail. Not surprising.</li>
<li>Are such graphs rare? Will this work on `almost all pairs of graphs' ? Not clear what that means.</li>
<li>If we also checked degree sequences and degrees-of-degrees, would that help? I doubt it since I think that graphs for which this fails are very similar in other ways, but I don't know that.</li>
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When I first heard the idea from my students it was new to me. Even so, I knew that it wasn't new and that it there has to be graphs it failes on (item 1 above). How did I know this? If there were no graphs that it failed on then GI would be in R. And if GI was in R, I would know that. And so would you. But this is not a good argument to give to students.<br />
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I think its a good idea and it might work for our purposes (which may be a topic of a later blog). GASARCHhttp://www.blogger.com/profile/06134382469361359081noreply@blogger.com7tag:blogger.com,1999:blog-3722233.post-52385143533936702442014-06-13T11:17:00.000-05:002014-06-13T11:17:01.945-05:00CCC 2014I'm returning from the <a href="http://www2.cs.sfu.ca/~kabanets/CCC14/CCC%202014%20%20Local%20Arrangements.htm">2014 IEEE Conference on Computational Complexity</a> in Vancouver, unfortunately missing the last day. Several very nice talks including a wonderful survey by Shubhangi Saraf on recent problems on lower bounds in algebraic circuits (addition and multiplication gates). In short, the interesting case is depth-4 algebraic circuits with bottom fan-in √n computing homogeneous polynomials. Kayal, Saha and Saptharishi <a href="http://eccc.hpi-web.de/report/2013/091/">showed</a> a 2<sup>Ω(√n log n)</sup> lower bound. Any improvement (changing the Ω to a ω) would separate VNP from VP (the algebraic version of the P v NP problem), or equivalently show the permanent cannot be computed by poly-size algebraic circuits.<br />
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At the business meeting, we discussed the <a href="http://computationalcomplexity.org/forum/">plans for independence</a> from IEEE. 60% of 189 voted for independence with open access as the main reason stated. There has been great support from the community both in volunteers to help with an independent conference and money ($42K) raised to get started. Future proceedings will likely be in the <a href="http://www.dagstuhl.de/en/publications/lipics">Dagstuhl LIPIcs</a> series. Timing and a new conference name are a few of the many issues still to be worked out. Whether or not you support independence, the conference organizers, particularly steering committee chair (and my former student) Dieter van Melkebeek, are doing a strong job coordinating the efforts.<br />
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Other highlights from the business meeting.<br />
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<li>29 accepted papers out of 94 submissions. A good number.</li>
<li>There was no best paper. The best student paper went to Alexander Belov for Quantum Algorithms for Learning Symmetric Juntas.</li>
<li>66 registered participants including 29 students. A bit low probably due to the proximity to STOC in time but not distance.</li>
<li>2015 CCC will be June 17-19 as part of the <a href="http://fcrc.acm.org/">FCRC conference</a> in Portland, Oregon.</li>
<li>There were three good bids for the 2016 CCC from Tokyo, Saarbrücken and Riga. Tokyo was the overwhelming winner.</li>
<li>CCC 2017 may be part of a federated theory conference including STOC, EC and SPAA.</li>
</ul>
Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com1tag:blogger.com,1999:blog-3722233.post-40286630378377637492014-06-12T10:08:00.000-05:002014-06-12T15:53:46.954-05:00Wassily Hoeffding 1914-1991In honor of the 100th anniversary of the birth of <a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hoeffding.html">Wassily Hoeffding</a> today, let's recall his incredibly useful <a href="http://en.wikipedia.org/wiki/Hoeffding%27s_inequality">inequality</a>.<br>
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Prob(|X-pn|>εn) < 2e<sup>-2ε<sup>2</sup>n</sup></center>
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where X is the random variable representing the number of heads from n independent coin flips each with a probability p of being heads. Informally the sum of many independent events will, with extremely high probability, be very close to the expected value. Notice that as long as ε = 1/o(√n) the probability goes down exponentially in n and this is tight as one expects X to be about O(√n) away from pn for constant p.</center>
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Back in the 80's, probabilistic computation started to play a major role in algorithms, cryptography, learning theory and interactive proofs. To analyze these computations one needed to deal with <a href="http://en.wikipedia.org/wiki/Chernoff_bound">Chernoff bounds</a> and it was difficult to find clean and easy statements of the bounds one could use in proofs. One of my officemates in grad school, Bob Sloan, took it upon himself to write up a short document giving Hoeffding's inequality and a few generalizations. Having that write-up was like holding Excalibur in my hands, ready to do battle with probabilistic complexity.</center>
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Later Alon and Spencer gave a nice treatment in their <a href="http://www.amazon.com/gp/product/0470170204?ie=UTF8&camp=213733&creative=393185&creativeASIN=0470170204&linkCode=shr&tag=computation09-20&qid=1402584592&sr=8-1&keywords=probabilistic+methods">Probabilistic Methods</a> text and these days you can get all the formulas you want on Wikipedia.</center>
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Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.com5