Theory of Computing Blog Aggregator

Sundry and Various

2008-11-21T21:09:44Z

1. There’s now a popular article by Lisa Zyga at physorg.com, about my paper with John Watrous on quantum computing with closed timelike curves. On the whole, I think Zyga did an excellent job at getting the facts (such as they are) correct.

2. Challenged ballots in the Coleman vs. Franken race: you be the judge!

3. One of the unfortunate things about not updating your blog often, I find, is that people assume you’re still obsessed with the last thing you blogged about, weeks after you’ve all but forgotten about it. As it happens, I’ve now fully recovered from the joy of the election, and am back to my normal angst-ridden equilibrium. On the other hand, I’ve not yet recovered from the STOC deadline.

4. My quest to become more obamalike in temperament is now officially a miserable failure. I should try it again sometime.

Computer science in the news

2008-11-21T20:13:00Z

The New York Times magazine has a feature this week on the NetFlix challenge: http://www.nytimes.com/2008/11/23/magazine/23Netflix-t.html?pagewanted=1&hp

Its actually puts computer science in a pretty nice light -- it makes it seem very exciting. In particular, it seems to take the position that complex machine learning systems have something approaching real intelligence -- or at least that they are complex enough that they aren't fully understood by their own creators:

There’s a sort of unsettling, alien quality to their computers’ results. When the teams examine the ways that singular value decomposition is slotting movies into categories, sometimes it makes sense to them — as when the computer highlights what appears to be some essence of nerdiness in a bunch of sci-fi movies. But many categorizations are now so obscure that they cannot see the reasoning behind them. Possibly the algorithms are finding connections so deep and subconscious that customers themselves wouldn’t even recognize them. At one point, Chabbert showed me a list of movies that his algorithm had discovered share some ineffable similarity; it includes a historical movie, “Joan of Arc,” a wrestling video, “W.W.E.: SummerSlam 2004,” the comedy “It Had to Be You” and a version of Charles Dickens’s “Bleak House.” For the life of me, I can’t figure out what possible connection they have, but Chabbert assures me that this singular value decomposition scored 4 percent higher than Cinematch — so it must be doing something right.

The author is definitely giving the computer too much credit for these anomylous clusters (and holds up "singular value decomposition" throughout the article as some sort of magical technique), but the point made is the right one. The difference between real intelligence and "just an algorithm" does in large part seem to be whether or not you can anticipate what its going to do. When you create a program that can come up with output that surprises you, thats pretty cool.

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What is your best paper? Ambigous!

2008-11-21T16:55:00Z

I sometimes ask people

What is your best paper?

This question is actually ambigous. Here are criteria that you can use.

Most important to the public. E.g., a big breakthrough people care about.
Most important in your view E.g., a big breakthrough on a problem that you care about, but perhaps others do not.
Most citations. This might be the same as important to public but may take more time to be evident as such. This one is verifiable!
You are personally attatched to it ind. of what the public things. E.g., you are delighted that it used theorms in p-adic cohomology to solve a problem in number theory, but nobody else cares.
You had a big contribution to it.
You liked how it evolved.

Hence, if someone asks what your best paper is, ask them to clarify the question. Or give them 5 answers. ~

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Report on the Innovative Teaching Day at Reykjavik University held on 14 August 2008

2008-11-21T15:52:00Z

These notes have been lying in my folders since last August. I am posting them here just in case they may be of interest to some of my two readers, with apologies for the low-tech embedding of URLs in the running text.

On 14 August 2008, Reykjavik University held one of its Innovative Teaching Days for 2008. The programme featured two invited presentations by two academics from MIT: Janet Rankin (http://web.mit.edu/tll/about-tll/rankin.html), associate director for teaching initiatives at the Teaching and Learning Lab at MIT (http://web.mit.edu/tll/index.html), and Donald Sadoway (http://dmse.mit.edu/faculty/faculty/dsadoway/), who is John F. Elliott Professor of Materials Chemistry at MIT and is known as a star teacher within that institution. Overall, the establishment and the level of activity of the Teaching and Learning Lab at MIT indicate how important quality teaching is considered by that top-notch university.

The first presentation was delivered by Janet Rankin, who started by asking the question:

"What do we know about student learning and how can it inform our teaching?"

Janet Ranking stressed that every course/lecture must have a road map (a clear outline) and well defined objectives (clear results). She also said that the increasing impact of cognitive learning theories on teaching is producing a shift towards student-centred, active learning.

Message 1: When teaching try to raise the students' awareness of themselves as learners.

Message 2: When planning a course and each of its components, consider the learning objectives for your students.

Ask yourself: "What promotes learning by the students?" Typical answers are:

Engagement with questions.
Active learning opportunities.
Cognitive apprenticeship (http://en.wikipedia.org/wiki/Cognitive_apprenticeship).

It is also important "to teach for transfer", i.e., teach students in such a way that they can apply what they have learned in one class in another. This can be achieved by teaching in a variety of contexts and by providing students as many examples of applications as possible. One should try to use a varied collection of examples to clarify concepts, and it is advantageous to provide examples from different disciplines. Keep always in mind that learning is context dependent.

Involve the students in peer instruction. This involves making them solve problems, listen critically to solutions by their peers, evaluate the solutions, and argue about their appropriateness.

A useful tip: After each lecture/session make the student write down on a card the most confusing aspect of the meeting. Use the answers to reflect on what you can do to improve.

See this booklet for more information: http://web.mit.edu/tll/learning_guidelines_2007.pdf. Janet Rankin also recommended this book.

The second talk of the day was delivered by Donald Sadoway, who has been professor at MIT since 1992. (Personal comment: For what it's worth I have to say that this was one of the most entertaining presentations about any topic I have heard in my career. I have no doubt that he is indeed the star teacher the announcement claimed he was and that students flock to his classes in large numbers.)

Donald Sadoway's mission in teaching is to invigorate engineering education because it is typically boring! He stated right at the beginning that we need to make universities a better environment for teachers. We need to give our students the foundations that they will need to be successful in our future world that will be dominated by bio, nano and info sciences. (Ask yourself: How much of these foundational sciences do our students see in our degree courses right now?)

As a running example, Donald Sadoway mentioned his experience with the course 3.091 at MIT. This course

lays the foundation for more chemistry,
prepares students for their majors, and
provides scientific and technical literacy.

For many students, this is the only chemistry course they will take. Before he took over, this was a troublesome course. Now, there are 600 students taking it on average. His approach in planning the course can be summarized as follows.

When planning a course, begin with a clean slate. If you start by looking at what was there before or at a typical book, you will soon realize that there is too much material to be covered.
Less is actually better!
When selecting the material to be covered in the course, divide it into three categories (of decreasing order of importance):

What should the students recall from the course on their death bed?
What knowledge would be useful, but is not vital?
And what would be knowledge from the course they might recite at a cocktail party to show they have some advanced knowledge?

To evaluate student performance in the course and keep track of student progress, Donald Sadoway uses weekly 10-minute quizzes, monthly tests (with one A4 aid sheet) and a final exam, which he calls a celebration for final festival. The final celebration gives the student time to reflect on what they have learned and is an extra opportunity for improving their learning skills and mastery of the material. The final celebration should be a suitably challenging learning experience since, as Sadoway put it, "no pressure, no diamonds".

Each concept in the course is illustrated by suitable examples providing context. (See above.) Sadoway always offers references to history of science, music, arts and whatever else provides context and makes the material entertaining and catchy. He also uses parts of the lectures to touch upon the theme "chemistry and the world around us".

Note that the course has no laboratory component since there is no lab that can hold 600 students. To address this "shortcoming", Sadoway asked himself: "So, what is important?" The answer he came up with is:

Ethics,
Data analysis,
Communication, and
Teamwork.

The result was the development of a virtual lab. There is not need of physical contact during the "lab classes".

He encourages students to read the classics, and use that the university or departmental library to go back to the articles that shaped our understanding of a field. He also runs themes within a course. Examples of such themes are:

Women in science (studies of abuse),
History, society and solid state chemistry (he proposed a course on this topic, but his proposal died because the faculty of history did not want to give students credits for the course since it was taught by an engineer and was considered an engineering class).

His firm belief in making students go to the primary sources led him to develop the course "3.093 Information Exploration: Becoming a Savvy Scholar". See

http://ocw.mit.edu/OcwWeb/Materials-Science-and-Engineering/3-093Fall-2006/CourseHome/.

He said that he has reached the following conclusion:

"I'll do anything I want in that lecture room provided it is in good taste."

After all, if I may quote him again,

"Tenure means never say 'I am sorry'."

Sadoway said that a good university education should give our students a methodology for developing solutions to problems. On the other had, a great university education should provide them with a methodology for developing methodologies!

You can hear Sadoway present this course on YouTube at

http://www.youtube.com/watch?v=rzGmSWxhwM8&feature=related

and you can watch him deliver his first lecture in the course at

http://www.youtube.com/watch?v=R90sohp6h44&NR=1.

You can also read about Sadoway's involvement in the "Picturing to Learn" programme at MIT at

http://web.mit.edu/newsoffice/2006/picturing.html.

More on the programme is available at

http://www.picturingtolearn.org/.

Addendum

I encourage you to look at the short movie "Teaching Teaching & Understanding Understanding" conceived and directed by my Danish colleague Claus Brabrand. Info on the movie, which describes John Biggs' constructive alignment, is available at http://www.daimi.au.dk/~brabrand/short-film/. When I had a brief stint as head of the Computer Science Department at RU, I ordered 20 copies of the DVD and encouraged my colleagues to watch it and pay heed to its message. I encourage your to buy a copy of the DVD, which can also be viewed on YouTube in low quality format.

Aalborg University is a world leader in problem-based learning. You can read about problem-based learning at Aalborg University, with special emphasis on its implementation in engineering education, at http://www.ucpbl.org/Wismarpaper_finalversion%5B1%5D.pdf. Information on the European Consortium of Innovative Universities, of which Aalborg is a member, is at http://eciu.web.ua.pt/.

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Thomas Bayes and Probability

2008-11-21T00:03:52Z

How can we assign probabilities in cases of uncertainty? And what is the nature of probabilities, to start with? And what is the rational mechanism for making a choice under uncertainty?

Thomas Bayes lived in the eighteenth century. Bayes’ famous formula shows how to update probabilities given some new evidence. Following is an example for an application of Bayes’ rule:

Suppose that ninety percent of pedestrians cross a certain crosswalk when the light is green, and ten percent cross it when the light is red. Suppose also that the probability of being hit by a car is 0.1% for a pedestrian who crosses on a green light, but the probability of being hit by a car is 2% for a pedestrian who crosses on a red light. A pedestrian is hit by a car at this particular crossing and brought to the hospital. How likely is it that he crossed on a red light?

Well, to start with (or a priori), only ten percent of the people who cross the crosswalk cross it on a red light, but now that we are told that this person was hit by a car it makes the probability that he crossed illegally higher. But by how much? Bayes’ rule allows us to compute this (a posteriori) probability. I will not describe the mathematical formula, but I will tell you the outcome: the probability that this person crossed on a red light is 2/3.

The Bayesian approach can be described as follows. We start by assigning probabilities to certain events of interest and, as more evidence is gathered, we update these probabilities. This approach is applied to mundane decision-making and also to the evaluation of scientific claims and theories in philosophy of science.

Bayes’ rule tells us how to update probabilities but we are left with the question of how to assign probabilities in cases of uncertainty to begin with. What is the probability of success in a medical operation? What is the chance of your team winning the next baseball game? How likely is it that war will break out in the Middle East in the next decade? How risky are your stock-market investments?

One very early approach to probabilities, the principle of indifference (a.k.a. the principal of insufficient reason), asserts that given a certain number of mutually exclusive events, their probabilities are the same. The formulation of this principle goes back to Jakob Bernoulli and Pierre-Simon Laplace. This principle is an important very early appearance of the notion of symmetry. Of course, there are many cases where the principle of indifference fails miserably. Various other approaches to “subjective probabilities” and to the foundation of probability theory were developed in the twentieth century.

Decisions under uncertainty depend not only on the probabilities but also on the “stakes.” Crossing a crosswalk on a red light will get you to your destination more quickly ninety-eight percent of the time, and two percent of the time you will be hit by a car. To make a rational decision between crossing on a red light or not, you have to take into account how good it is for you to get to your destination earlier and how bad it is for you to get hit by a car. A theory of decisions under uncertainty, based on the notion of utility, was developed by John von Neumann and Oskar Morgenstern, the founders of “game theory.” In this theory, to each possible outcome we assign a numerical quantity called a “utility.” Rational decisions are based on combining the probabilities for various outcomes and the utility gained from each of these outcomes. The theory of von Neumann and Morgenstern has been the subject of intense debate in recent decades.

Thomas Bayes

Perhaps the major difficulty with the Bayesian point of view, whether relating to decisions under uncertainty or to the Bayesian philosophy of science, is that quite often, no one has a clue how to assign probabilities in cases of uncertainty.

The addition of probability thinking remarkably extends our understanding of reality. At the same time, we face the impossibility of understanding various phenomena, perhaps those about which we are most curious, even with the language and tools of probability at our disposal. Introducing the language of probability allowed us to ask many new questions that we cannot answer even using the tools of probability.

An experimental analysis of Lemke-Howson algorithm

2008-11-21T04:30:09Z

Authors: Bruno Codenotti , Stefano De Rossi, Marino Pagan
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Abstract: We present an experimental investigation of the performance of the Lemke-Howson algorithm, which is the most widely used algorithm for the computation of a Nash equilibrium for bimatrix games. Lemke-Howson algorithm is based upon a simple pivoting strategy, which corresponds to following a path whose endpoint is a Nash equilibrium. We analyze both the basic Lemke-Howson algorithm and a heuristic modification of it, which we designed to cope with the effects of a 'bad' initial choice of the pivot. Our experimental findings show that, on uniformly random games, the heuristics achieves a linear running time, while the basic Lemke-Howson algorithm runs in time roughly proportional to a polynomial of degree seven. To conduct the experiments, we have developed our own implementation of Lemke-Howson algorithm, which turns out to be significantly faster than state-of-the-art software. This allowed us to run the algorithm on a much larger set of data, and on instances of much larger size, compared with previous work.

Linear Time Approximation Schemes for the Gale-Berlekamp Game and Related Minimization Problems

2008-11-21T04:30:16Z

Authors: Marek Karpinski, Warren Schudy
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Abstract: We design a linear time approximation scheme for the Gale-Berlekamp Switching Game and generalize it to a wider class of dense fragile minimization problems including the Nearest Codeword Problem (NCP) and Unique Games Problem. Further applications include, among other things, finding a constrained form of matrix rigidity and maximum likelihood decoding of an error correcting code. As another application of our method we give the first linear time approximation schemes for correlation clustering with a fixed number of clusters and its hierarchical generalization. Our results depend on a new technique for dealing with small objective function values of optimization problems and could be of independent interest.

While in Paris...

2008-11-20T17:03:00Z

My latest reason for being off the air has to do with the amazingly bad internet capabilities of Paris hotels. Yes, I'm in Paris, city of lovers, but certainly not of lovers of wifi. There are at least 30 networks visibile wherever you go, but they're all secure, so no mooching. The hotel-provided wifi is actually a generic service that costs 22 E/day for connectivity, with all kinds of bandwidth caps and a very slow connection. If I were to splurge for the "business" level, I get the luxury of paying 27 E/day, with unclear benefits (presumably I can now download my bootleg bittorrents (just kidding)).

Other things I've noticed since I last came to Paris: (which is not to say that they are new, just that I just noticed them):

Every second store on the podunk street my hotel is on is a fancy clothing store. Clearly the world-wide economic collapse has not hit.
Speaking of world-wide economic collapses, it really hurts to have a weak dollar. $7 espressos, sigh...
....but it's always a pleasure to walk into a cafe and order a 'cafe' and just know that something good will appear. This is in contrast to the unbounded depth circuit needed to specify a proper cup of coffee at Starbucks.
Speaking of Starbucks, how on earth can they even survive in Paris ? I mean, you go to a Starbucks here, and you get the same experience as in the US, ending with a paper cup of coffee of questionable quality that you drink perched on a high bar stool. On the other hand, you go to a cafe, and they serve you with nice cups, and a little cookie, and let you sit there for hours nursing your coffee, and will even give you the WEP key for their secured WiFi. It's no contest !
You can change the world while nursing your coffee. I was staying in the 14th Arrondissement (the Montparnasse area) and had to have a coffee at the Dome cafe, a place apparently frequented by Lenin and Trotsky before the Revolution. I have to say that at the time I went, the clientele looked like they were plotting a revolution... in 1907.... I'd link to a verification of this, but I can't make any sense out of the search results on google.fr
Speaking of which, how does one tell google NOT to return results in french ? every time I edit the URL to go to google.com, it sends me back to google.fr. Suivant !!!

And why am I blogging from Paris, you might ask ? Well here's why. I'm an invited speaker, no less. Conference blogging was limited because there was no wifi at the conference site either. It's a long story involving military schools (and maybe even NASA). Details will appear shortly.

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Is the Medium the Message?

2008-11-20T14:18:00Z

Do you care about fonts?

In other words do you care about how your paper looks? Not the details of the proofs or the quality of the exposition but how the paper looks. The fonts, the margins, the spacing, the neatness of it all.

I used to care.

I was in graduate school when the great shift to LaTeX happened. All of a sudden our papers looked great, like finished journal versions right off the printer. I would spend hours on my papers and months on my thesis, making sure there were no overfull boxes, that equations lined up nicely, pagebreaks occurred at good places and hyphenations were done right. Didn't worry about fonts back in those days when we thought Computer Modern looked good.

Now I don't bother. I still fix the big ugly problems but who really cares if "nondeterministic" is hyphenated properly. As you can see from this beautiful green blog, I don't try that hard on the medium. Though the style of Bill's posts can sometimes make me look like a true artist.

Truth is you get little value in our field from looking good. So better to spend the time proving new theorems than putting the old ones in a pretty font.

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Three Days in San Francisco

2008-11-20T08:12:00Z

I started the week with a Misfits concert, those 31-year-old punk classics that have influenced all-time favorites like Slayer and Metallica.

Historical note: If the following are familiar to you...
"Die, die, die, my darling..."
"Here in this place lies the genie of death"
"I got something to say: I killed your baby today"
... chances are it is the Metallica inflections that are now ringing in your head. However, it is worth remembering that these are originally Misfits songs.

I continued with Alistair Sinclair's "Mixing times for the solid-on-solid model". As usual, I think this is great theory, but not theory of computer science. (Here, Mikkel tries to make a similar point, cutting even deeper to define theory of computer science.)

Personal note: Attending such talks is always very interesting to me. In a short previous life, I was a Physicist (I even won 1st and 2nd prizes in the Romanian Physics Olympiad in '96 - '97). This left me with great physics intuition, but not that much mathematical intuition on how to formalize "physics proofs" (at that age, my appreciation of Math was very limited). It is very rewarding to catch up on the mathematical intuition, whenever I get a chance.

Finally, I had a chance to listen to some great music from Nels Cline and his band. This led to the interesting exercise of trying to triangulate Cline's heavy, academic jazz relative to black jazz and the Béla Bartók classical music.

The U.S. is a big country, but there are not many places where you can hear the Misfits, Sinclair, and Cline in a span of three days.

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Trading GRH for algebra: algorithms for factoring polynomials and related structures

2008-11-20T04:30:05Z

Authors: Gábor Ivanyos, Marek Karpinski , Lajos Rónyai, Nitin Saxena
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Abstract: In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n^{\log n},\log |k|) time "either" a nontrivial factor of f(x) "or" a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various new GRH-free results, most striking of which are:

(1) Given a noncommutative algebra over a finite field, we can find a zero divisor in deterministic subexponential time.

(2) Given a positive integer r>4 such that either 4|r or r has two distinct prime factors. There is a deterministic polynomial time algorithm to find a nontrivial factor of the r-th cyclotomic polynomial over a finite field.

In this paper, following the seminal work of Lenstra (1991) on constructing isomorphisms between finite fields, we further generalize classical Galois theory constructs like cyclotomic extensions, Kummer extensions, Teichmuller subgroups, to the case of commutative semisimple algebras with automorphisms. These generalized constructs help eliminate the dependence on GRH.

An Almost Optimal Rank Bound for Depth-3 Identities

2008-11-20T04:30:04Z

Authors: Nitin Saxena, C. Seshadhri
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Abstract: We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most k^3\log d. The previous best rank bound known was 2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank \Omega(k\log d)).

Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic black-box identity tests for depth-3 circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-3 circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-3 circuit (over any field) is at most k^3\log d.

The novel feature of this work is a new notion of maps between sets of linear forms, called "ideal matchings", used to study depth-3 circuits. We prove interesting structural results about depth-3 identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.

Geometric properties of satisfying assignments of random $\epsilon$-1-in-k SAT

2008-11-20T04:30:03Z

Authors: Gabriel Istrate
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Abstract: We study the geometric structure of the set of solutions of random $\epsilon$-1-in-k SAT problem. For $l\geq 1$, two satisfying assignments $A$ and $B$ are $l$-connected if there exists a sequence of satisfying assignments connecting them by changing at most $l$ bits at a time.

We first prove that w.h.p. two assignments of a random $\epsilon$-1-in-$k$ SAT instance are $O(\log n)$-connected, conditional on being satisfying assignments. Also, there exists $\epsilon_{0}\in (0,\frac{1}{k-2})$ such that w.h.p. no two satisfying assignments at distance at least $\epsilon_{0}\cdot n$ form a "hole" in the set of assignments. We believe that this is true for all $\epsilon >0$, and thus satisfying assignments of a random 1-in-$k$ SAT instance form a single cluster.

Dynamic External Hashing: The Limit of Buffering

2008-11-20T04:30:21Z

Authors: Zhewei Wei, Ke Yi, Qin Zhang
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Abstract: Hash tables are one of the most fundamental data structures in computer science, in both theory and practice. They are especially useful in external memory, where their query performance approaches the ideal cost of just one disk access. Knuth gave an elegant analysis showing that with some simple collision resolution strategies such as linear probing or chaining, the expected average number of disk I/Os of a lookup is merely $1+1/2^{\Omega(b)}$, where each I/O can read a disk block containing $b$ items. Inserting a new item into the hash table also costs $1+1/2^{\Omega(b)}$ I/Os, which is again almost the best one can do if the hash table is entirely stored on disk. However, this assumption is unrealistic since any algorithm operating on an external hash table must have some internal memory (at least $\Omega(1)$ blocks) to work with. The availability of a small internal memory buffer can dramatically reduce the amortized insertion cost to $o(1)$ I/Os for many external memory data structures. In this paper we study the inherent query-insertion tradeoff of external hash tables in the presence of a memory buffer. In particular, we show that for any constant $c>1$, if the query cost is targeted at $1+O(1/b^{c})$ I/Os, then it is not possible to support insertions in less than $1-O(1/b^{\frac{c-1}{4}})$ I/Os amortized, which means that the memory buffer is essentially useless. While if the query cost is relaxed to $1+O(1/b^{c})$ I/Os for any constant $c<1$, there is a simple dynamic hash table with $o(1)$ insertion cost. These results also answer the open question recently posed by Jensen and Pagh.

Exact phase transition of backtrack-free search with implications on the power of greedy algorithms

2008-11-20T04:30:18Z

Authors: Liang Li, Tian Liu, Ke Xu
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Abstract: Backtracking is a basic strategy to solve constraint satisfaction problems (CSPs). A satisfiable CSP instance is backtrack-free if a solution can be found without encountering any dead-end during a backtracking search, implying that the instance is easy to solve. We prove an exact phase transition of backtrack-free search in some random CSPs, namely in Model RB and in Model RD. This is the first time an exact phase transition of backtrack-free search can be identified on some random CSPs. Our technical results also have interesting implications on the power of greedy algorithms, on the width of random hypergraphs and on the exact satisfiability threshold of random CSPs.

Evolutionary Construction of Geographical Networks with Nearly Optimal Robustness and Efficient Routing Properties

2008-11-19T00:00:00Z

Authors: Yuko Hayashi
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Abstract: Robust and efficient design of networks on a realistic geographical space is one of the important issues for the realization of dependable communication systems. In this paper, based on a percolation theory and a geometric graph property, we investigate such a design from the following viewpoints: 1) network evolution according to a spatially heterogeneous population, 2) trimodal low degrees for the tolerant connectivity against both failures and attacks, and 3) decentralized routing within short paths. Furthermore, we point out the weakened tolerance by geographical constraints on local cycles, and propose a practical strategy by adding a small fraction of shortcut links between randomly chosen nodes in order to improve the robustness to a similar level to that of the optimal bimodal networks with a larger degree $O(\sqrt{N})$ for the network size $N$. These properties will be useful for constructing future ad-hoc networks in wide-area communications.

Reflection on the old days- by Joseph Kruskal

2008-11-19T16:52:00Z

Guest Blog by Joe Kruskal. He is the Kruskal that did the MST algorithm, the Kruskal Tree Theorem and work on multidimensional scaling. However, this post is not on any of those topics. Its a response and reflection on my post about the monotone subequence theorem.

HIS POST:

In your post on the monotonic sequence theorem you said the following.

In those days it was harder to find out if someone else had done what you had done since they didn't have google, but it may have (in some cases) been easier since there were so many fewer researchers- you could just call them. OH- but long distance was expensive back then.

Yes, long-distance phone calls were expensive then. That's why mathematicians seldom used phone calls for that purpose. They used mail -- postal mail, of course. Now that email has become almost universal, and is seen as slow and stodgy compared with text messaging and other modes of communication that I haven't kept up with, people have no real idea what communication was like 50 years before.

The same thing was true 50 years ago. We didn't know then how communication was done 50 years before that. In England, at least, it was quite common for a well-to-do person to send a letter to a friend to propose having dinner out together, or going to a play together, or lots of other possibilities. They would expect to get a reply within say 4 hours, time enough to send another message confirming the arrangement for that evening.

In London at least, there were 4 deliveries/pickups per day, at least for the upper classes. When my wife and I visited England in the 1950s and stayed with my sister who had moved there with her British husband, we personally observed the following, which we had been told about. When a post office mail person come to the red "post box", which displayed the pick up times, he stood there waiting until the specified pick up time, to the minute (by his watch). Only then did he open the box and take out the letters and post cards. Everyone relied on the displayed pickup times, and would hurry to the box just in time, knowing that if they got there by the posted time the mail would go out right away. Watches were not so accurate then, so I imagine that the post office pick up people checked their watches against Big Ben or other large public clocks.

My own dissertation also indicates how things had changed:

Paul Erdos was telling lots of people about a conjecture due to a Hungarian mathematician, Vazsonyi, he was friendly with who he said "had died", meaning that he left mathematics for a well-paying job with some company -- I think it was an airplane manufacturer. I was one of many people who heard him describe this conjecture. Roughly a year later, I had put a lot of work into this problem, but was still not close to a solution. By chance I bumped into Erdos at the Princeton Junction station. We chatted. I don't know how the conversation turned to the Vazsonyi conjecture -- probably I told him I had been working on it. He said, oh, you must read a recent paper by Rado (a British mathematician, also from Hungary). I quickly went to the library and found his paper, which I read with fear and trembling. Had I been scooped? It turned out that he had made significant progress, but hadn't cracked the nut. His work combined with mine finally led to a solution.

Today, the equivalent of those two chance conversations can happen via Google and email. I feel certain that science of many kinds is developing much more rapidly than it used to for this reason (except, perhaps, in fields where progress is kept secret for reasons of financial gain).

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Secondary Indexing in One Dimension: Beyond B-trees and Bitmap Indexes

2008-11-19T04:30:14Z

Authors: Rasmus Pagh , S. Srinivasa Rao
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Abstract: Let S be a finite, ordered alphabet, and let x = x_1 x_2 ... x_n be a string over S. A "secondary index" for x answers alphabet range queries of the form: Given a range [a_l,a_r] over S, return the set I_{[a_l;a_r]} = {i |x_i \in [a_l; a_r]}. Secondary indexes are heavily used in relational databases and scientific data analysis. It is well-known that the obvious solution, storing a dictionary for the position set associated with each character, does not always give optimal query time. In this paper we give the first theoretically optimal data structure for the secondary indexing problem. In the I/O model, the amount of data read when answering a query is within a constant factor of the minimum space needed to represent I_{[a_l;a_r]}, assuming that the size of internal memory is (|S| log n)^{delta} blocks, for some constant delta > 0. The space usage of the data structure is O(n log |S|) bits in the worst case, and we further show how to bound the size of the data structure in terms of the 0-th order entropy of x. We show how to support updates achieving various time-space trade-offs.

We also consider an approximate version of the basic secondary indexing problem where a query reports a superset of I_{[a_l;a_r]} containing each element not in I_{[a_l;a_r]} with probability at most epsilon, where epsilon > 0 is the false positive probability. For this problem the amount of data that needs to be read by the query algorithm is reduced to O(|I_{[a_l;a_r]}| log(1/epsilon)) bits.

Generating Random Graphs with Large Girth

2008-11-19T04:30:11Z

Authors: Mohsen Bayati, Andrea Montanari, Amin Saberi
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Abstract: We present a simple and efficient algorithm for randomly generating simple graphs without small cycles. These graphs can be used to design high performance Low-Density Parity -Check (LDPC) codes. For any constant k, alpha<1/2k(k+3) and m=O(n^{1+alpha}), our algorithm generate s an asymptotically uniform random graph with n vertices, m edges, and girth larger tha n k in polynomial time. To the best of our knowledge this is the first polynomial-algorith m for the problem.

Our algorithm generates a graph by sequentially adding m edges to an empty graph with n vertices. Recently, these types of sequential methods for counting and random generation have been very successful.

About Conjectures: Shmuel Weinberger

2008-11-18T23:22:34Z

The following paragraph is taken from the original “too personal for publication draft” of an article entitled ” ‘Final values’ of functors” by Shmuel Weinberger for a volume in honor of Guido Mislin’s retirement from ETH. (L’enseignement Mathematique 54(2008), 180-182.) Shmuel’s remarks about making conjectures and the different types of conjectures appear here for the first time.

“Final Values” of Functors

Shmuel Weinberger

A personal letter for, and to, Guido Mislin, a mathematician who always epitomized for me elegance, taste, and precision.

Guido Mislin

I have, on very rare occasions, conjectured things I really believe with evidence by analogy and the religious belief in the essential simplicity of the mathematical universe (looked at correctly)[1]. Besides beauty, such a conjecture[2] pragmatically serves as a guide through unknown landscape. And even an unreliable guide helps to point in the right direction. In fact, an unreliable guide known to be unreliable can be useful indeed. Sometimes one makes conjectures knowing them to be false, but feeling that their falsity is a deep phenomenon and most of the predictions made with the conjecture as guide will be true[3]. On other times, I have conjectured to lay down the gauntlet: “See, you can’t even disprove this ridiculous idea.” [4] On yet other times, the conjectures come from daydreams: it would be so nice if this were true[5]. And, yet on others one makes a conjecture hoping to probe the landscape that other conjectures have already illuminated[6].

The conjecture (and speculations) that I would like to present to you, Guido, is motivated by ideas I learnt from Goodwillie and Weiss, but I think it is also much in the spirit of the way you sometimes approach mathematics and it is somewhere between the last two kinds…

[1] These include the package of conjectures about homology manifolds made in [BFMW], which incidentally flatly contradict other standard conjectures like the Bing-Borsuk conjecture.

[2] Here I have in mind things like the Riemann hypothesis, indeed huge swaths of number theory; in topology and geometry, Thurston’s geometricization conjecture, Baum-Connes conjecture and Novikov conjectures, and so on are examples.

[3] Here I am thinking of the equivariant version of the Borel conjecture, or the stratified version. See [W, Chapter 13].

[4] I think this is what Lott had in mind (although he was usually careful to call it a question) — but I have less concern for my reputation, and am willing to conjecture the opposite of what I believe -when he formulated the zero-in-the-spectrum problem for all complete manifolds. And it was disproved in [FW]. On the other hand, my dear friend and mentor Donald Newman made a conjecture of this same sort once in rational approximation, only to have it proven by V.A.Popov.

[5] This must be the case for, say, Gromov’s questions about large Riemannian manifolds [G] or the Weinberger conjecture discussed first in print in [D]

[6] The zero in the spectrum problem for universal covers of aspherical manifolds or even uniformly contractible manifolds, also discussed by Lott, is of this sort. The conjecture of rationality of the difference of twisted APS invariants for homotopy equivalent manifolds was made after realizing that it would follow from the Borel conjecture for torsion free groups. By now, it’s been proven three times. [FLW][K][GHW] and [HR].

As further afterthoughts Shmuel writes:I realize that I haven’t given enough thought to the affects of the different kinds of conjectures as they get lives of their own, over time. Also, I have treated all conjectures as if they were good things, when, in truth, I think some conjectures drain the surrounding areas of attention that they more importantly deserve.



References mentioned in the footnotes:

[BFMW] Bryant, Ferry, Mio, and Weinberger, Topology of homology manifolds. Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 324–328.

[D] Dranishnikov, Dimension theory and large Riemannian manifolds. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 423–432

[FLW] Farber, Levine, with an appendix by Weinberger, Jumps of the eta-invariant. Math. Z. 223 (1996), no. 2, 197-246.

[FW] Farber and Weinberger, On the zero-in-the-spectrum conjecture. Ann. of Math. (2) 154 (2001), no. 1, 139-154.

[GHW] Guentner, Higson, and Weinberger, The Novikov conjecture for linear groups, Publ Math d’IHES, 101 (2005) 243-268

[G] Gromov, Large Riemannian manifolds. Curvature and topology of Riemannian manifolds (Katata, 1985), 108–121, Lecture Notes in Math., 1201, Springer, Berlin, 1986.

[HR] Higson and Roe, K-homology, assembly, and rigidity theorems for relative eta invariants (preprint 2008)

[K] Keswani, Relative eta-invariants and C*-algebra K-theory. Topology 39 (2000), no. 5, 957-983.

[L] Lott, The zero-in-the-spectrum question. Enseign. Math. (2) 42 (1996), no. 3-4, 341-376.

[M] Mislin, Equivariant K-homology of the classifying space for proper actions. Proper group actions and the Baum-Connes conjecture, 1-78, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003.

[W] Weinberger, The topological classification of stratified spaces, University of Chicago Press 1994

William Browder, Sylvain Cappell, Shmuel Weinberger, and Stanley Chang

STOC notes

2008-11-18T17:31:00Z

It looks like we ended up with about 329 submissions. (Hopefully that won't change much at this point.) That's between 40-50 papers per PC member at 3 reviews per paper.

Thanks to everyone who withdrew the papers they weren't submitting. Otherwise, I had to go through and withdraw them manually myself.

Thanks to Shai Halevi for continuing to help me with the reviewing system.

Yes, I did get mail from about 10 people who hadn't known they'd need to file an abstract the week before, and I accommodated them. I think this should become a standard and everyone should get used to it. Again, keep in mind there's 40-50 papers per PC member; anything that makes their work easier is a good thing. (And Shai's interface for choosing preferences lets you see a paper's abstract pop up with a mouse rollover, so it's really nice to have abstracts early!)

Assuming things continue to go well, expect e-mail from PC members asking for you to handle a subreview before you head off for Thanksgiving....

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Secrets from the Future

2008-11-18T12:00:00Z

One of my crypto students pointed me to the "Nerdcore Hip-Hop" group MC Frontalot. Here's a trailer for their tribute film, Nerdcore Rising.

You can download their song Secrets from the Future that makes a good point about the life of crypto. From the (slightly explicit) lyrics
Best of all, your secret: nothing extant could extract it.
By 2025 a children's Speak & Spell could crack it.
You can't hide secrets from the future with math.
You can try, but I bet that in the future they laugh
at the half-a**ed schemes and algorithms amassed
to enforce cryptographs in the past.
I don't expect a general way to break RSA or factor numbers either on classical machines (for lack of algorithms) or quantum machines (for lack of controlled entanglement) in the next couple of decades. Nevertheless you can't count on say a 1024 or 2048 bit RSA key being safe in a decade or two. Better algorithms combined with faster highly parallelized machines may break those codes. Or maybe they won't—but you can't be sure.
Even the NSA gives expiration dates on encrypted data. If you used 100,000 bit keys your secrets should survive into the next century. But you have to wonder—how dark are your secrets that you need them to last?

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Fano

2008-11-18T07:28:32Z

On the off chance that anyone notices fano.ics.uci.edu being down: it had a little problem of not being able to boot up this afternoon and I resorted to booting from CD and trying a clean install. I hope I can bring it back up soon but I'm not sure I'll have any time to work on it tomorrow, so it may not be until Wednesday. If the reinstall works. If it doesn't, it could be longer.

I suppose I should do something about the bibliographic data served there being out of date, too, but at this point it will take actual programming effort on my part to make that happen (the server software wants the data in a format I've stopped using) so that will be even more slow.

ETA: The clean reinstall worked well. As did restoring my files from Time Machine, recovering everything but the half-dozen iTunes songs I'd added minutes before the crash. And even those are still recoverable from the USB key I used to put them on Fano, or if that fails by pulling them back off the iPod. Let this be a reminder to you all: do your backups!

Topological Dynamics of Cellular Automata: Dimension Matters

2008-11-18T04:30:03Z

Authors: Mathieu Sablik, Guillaume Theyssier
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Abstract: Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.

On models of a nondeterministic computation

2008-11-18T04:30:07Z

Authors: M. N. Vyalyi
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Abstract: In this paper we consider a nondeterministic computation by deterministic multi-head 2-way automata having a read-only access to an auxiliary memory. The memory contains additional data (a guess) and computation is successful iff it is successful for some memory content. Also we consider the case of restricted guesses in which a guess should satisfy some constraint. We show that the standard complexity classes such as L, NL, P, NP, PSPACE can be characterized in terms of these models of nondeterministic computation. These characterizations differ from the well-known ones by absence of alternation.

An Efficient Algorithm for Partial Order Production

2008-11-18T04:30:16Z

Authors: Jean Cardinal, Samuel Fiorini, Gwenaël Joret, Raphaël M. Jungers, J. Ian Munro
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Abstract: We consider the problem of partial order production: arrange the elements of an unknown totally ordered T into a target partially ordered set S, by comparing a minimum number of pairs in T. Special cases of this problem include sorting by comparisons, selection, multiple selection, and heap construction.

We give an algorithm performing ITLB + o(ITLB) + O(n) comparisons in the worst case. Here, n denotes the size of the ground sets, and ITLB denotes a natural information-theoretic lower bound on the number of comparisons needed to produce the target poset. The overall complexity of our algorithm is polynomial. This answers a question of Yao (SICOMP, 1989).

Our strategy is to extend the poset S to a weak order W whose corresponding information-theoretic lower bound is provably not much larger than that for S. Taking W instead of S as a target poset, we then solve the problem by applying a multiple selection algorithm that performs not much more than ITLB comparisons.

We base our analysis on the entropy of the target poset S, a quantity that can be efficiently computed and provides a good estimate of ITLB since the latter is, up to a linear error term, n times the entropy of S.

Phase transition for Local Search on planted SAT

2008-11-18T04:30:25Z

Authors: Andrei A. Bulatov , Evgeny S. Skvortsov
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Abstract: The Local Search algorithm (or Hill Climbing, or Iterative Improvement) is one of the simplest heuristics to solve the Satisfiability and Max-Satisfiability problems. It is a part of many satisfiability and max-satisfiability solvers, where it is used to find a good starting point for a more sophisticated heuristics, and to improve a candidate solution. In this paper we give an analysis of Local Search on random planted 3-CNF formulas. We show that if there is k<7/6 such that the clause-to-variable ratio is less than k ln(n) (n is the number of variables in a CNF) then Local Search whp does not find a satisfying assignment, and if there is k>7/6 such that the clause-to-variable ratio is greater than k ln(n)$ then the local search whp finds a satisfying assignment. As a byproduct we also show that for any constant r there is g such that Local Search applied to a random (not necessarily planted) 3-CNF with clause-to-variable ratio r produces an assignment that satisfies at least gn clauses less than the maximal number of satisfiable clauses.

Computing voting power in easy weighted voting games

2008-11-18T00:00:00Z

Authors: Haris Aziz, Mike Paterson
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Abstract: Weighted voting games are ubiquitous mathematical models which are used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. They model situations where agents with variable voting weight vote in favour of or against a decision. A coalition of agents is winning if and only if the sum of weights of the coalition exceeds or equals a specified quota. The Banzhaf index is a measure of voting power of an agent in a weighted voting game. It depends on the number of coalitions in which the agent is the difference in the coalition winning or losing. It is well known that computing Banzhaf indices in a weighted voting game is NP-hard. We give a comprehensive characterization of weighted voting games which can be solved in polynomial time. Among other results, we provide a polynomial ($O(k{(\frac{n}{k})}^k)$) algorithm to compute the Banzhaf indices in weighted voting games in which the number of weight values is bounded by $k$.

Perfect Matchings via Uniform Sampling in Regular Bipartite Graphs

2008-11-18T04:30:27Z

Authors: Ashish Goel , Michael Kapralov, Sanjeev Khanna
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Abstract: In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to K\"{o}nig's work in 1916 (here $m=nd$ is the number of edges in the graph, $2n$ is the number of vertices, and $d$ is the degree of each node). The currently most efficient algorithm takes time $O(m)$, and is due to Cole, Ost, and Schirra. We improve this running time to $O(\min\{m, \frac{n^{2.5}\ln n}{d}\})$; this minimum can never be larger than $O(n^{1.75}\sqrt{\ln n})$. We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a $d$-regular bipartite graph independently with a probability $p = O(\frac{n\ln n}{d^2})$ then the resulting graph has a perfect matching with high probability. The proof involves a decomposition of the graph into pieces which are guaranteed to have many perfect matchings but do not have any small cuts. We then establish a correspondence between potential witnesses to non-existence of a matching (after sampling) in any piece and cuts of comparable size in that same piece. Karger's sampling theorem for preserving cuts in a graph can now be adapted to prove our uniform sampling theorem for preserving perfect matchings. Using the $O(m\sqrt{n})$ algorithm (due to Hopcroft and Karp) for finding maximum matchings in bipartite graphs on the sampled graph then yields the stated running time. We also provide an infinite family of instances to show that our uniform sampling result is tight up to poly-logarithmic factors (in fact, up to $\ln^2 n$).

STOC

2008-11-17T23:03:00Z

I decided yesterday not to submit anything to STOC, so I spent my most relaxing deadline day yet. For one paper, I found a deadly bug, while for another I found an improvement (moving it back to the column of things I need to think more about).

The main lesson for me was that there are limits to everything, and that dealing with an endless stream of stuff (from my defense, to writing a thesis, to visa issues, to searching for an apartment, to IBM sign on stuff, to paperwork for buying a motorcycle...) can really kill research.

Now I look forward to a year without deadlines (I am on the FOCS program committee). I hope to spend the first month catching up with things, and the rest doing what I actually enjoy doing, i.e. thinking about my problems. If this plan fails, I might have to elope to Africa again for a few months :)

As for this blog, I will probably get back to writing interesting technical stuff soon. I expect the non-technical stuff to be kept to a minimum. (For one, we've got too much blogging in the community, and I'm bored. Plus, I've felt disappointed with the TCS community for quite a while, and there's not much I can say in between lying and insulting some folks.)

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A "well known" theorem

2008-11-17T18:16:00Z

In the excellent paper On the power of two, three, and four probes
It is well known that every graph with s vertices and at least 2s edges contains a cycle of length at most 2log s
My students puzzled over this one in two ways. (1) How to prove it? Two of them came up with correct proofs that were not that hard. (2) Is it really well known? Two of them searched the web for a proof. They could not find one.

The problem with the phrase It is well known that is that it may be well known to people who know it (duh) but not to others. People not in the know don't even know if its hard or not (its not). Perhaps they should have said It is easy to show that. Or give a hint to the proof.

I invite my readers to give proofs to see if they differ from my students, and also so that the next time someone needs to reference this, they can point to this blog and attibute it to some cute alias.

A student asked me if giving as a reference a blog site or an unpublished paper on line is legit. I would say its more legit then giving as a reference a paper that is not on-line. A paper that is refereed but not online had a few people look at it closely. A paper that is unrefereed but online might have a lot more people look at it (then again, it might not). But since the reader can access it, he or she might be able to tell for himself or herself whether the statement they need has been proven properly.

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Call for Nominations: Gödel Prize 2009

2008-11-17T16:30:00Z

The Call for Nominations for the 2009 Gödel Prize has been posted (see this pdf file). Nominations for the award should be submitted to the Award Committee Chair, Shafi Goldwasser. The deadline for nominations is January 31st, 2009.

Do nominate your favourite papers, and recall that any research paper, or series of papers, by a single author or by a team of authors is deemed eligible if the paper was published in a recognized refereed journal before the nomination, but the main results were not published (in either preliminary or final form) in a journal or conference proceedings before January 1st, 1995.

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Morning exercise - Turan’s theorem

2008-11-17T14:57:59Z

Here is a cute proof of Turan’s theorem. That is, a graph G with n vertices has an independent set of size n/(1+t), where t is the average degree in G.

Update: Following some readers comments, is a more elaborate write-up.