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Potential in the Making

\Phi = \sum_{e \notin C} n^{2w_e}+n\exp\left(\frac{1}{2a}\left(\sum_{s \notin C^\prime} c_s - \sum_S 3x_sc_s \log(n)\right)\right)

This was the equation my professor put on the board. Magically. If you’ve seen algorithmic analysis before,  you may recognize the form as a potential function. Effectively, this is a very convenient and powerful way to performing an amortized analysis on an otherwise difficult-to-analyze algorithm. Said another way, it’s a way by which we can keep track of money coming in and going out of a metaphorical bank. Think of it this way: some weeks you need only $5 dollars, every once in a while you need $10 and one time you need $500 for a big expense. How much money do you usually need to go a week? A worst-case analysis will yield $500/wk, a gross over-estimate. A best-case analysis will yield $5, which doesn’t come close to covering the $500 you need that one time. Instead, we look at it through a type of averaging that allows us to retain the intricacies of the nature of the algorithm but more cleanly and tightly express bounds on its performance.

Here’s another example. Give a good estimate for a representative value of the “ruler function”. I put that in quotes because even I confused it with the more famous function which has differing continuities at rational and irrational numbers. The sequence I am here referring to is Sloane A001511. A graph of the function below reveals the reason behind its naming.

Now once more, a worst-case analysis or best-case analysis fail terribly (giving values of one and arbitrarily large as the sequence gets longer). But we can do better and that therefore is the purpose of amortized analysis.

There’s a great plenty that can be said about amortized analysis but I will focus on one particular aspect here: how the hell did we get that thing on the top of this page? I said it was magical but that was a lie. As it turns out, the rationale for this particular function derives from its formulation via a casting into a linear programming question. From this then were the characteristic properties drawn and put together. The problem for which this potential function applies is the Online Set Cover problem, which I will describe quickly below:

Online Set Cover

This is a pretty natural extension to a very common and simple problem called the Set Cover problem. The basic idea is this: you’d like to cover a finite set of elements using a subset from a set of subsets of the original elements. Formally, given E = \{e_1, \ldots, e_n\} and a family of subsets S =\{S_i\}_{i=1}^m with S_i \subseteq E, with possible overlaps between different sets. Each subset is assigned a non-negative weight c_i for each i=1, 2, \ldots, m and we wish to find the subset of family S that both covers every element in E and has the minimum weight. In the online variation of this problem, we are provided each S_i one at a time and we must immediately update our current best minimum weight set cover. The potential function above is used in the analysis of one algorithm to solve this problem.

The moral of the story, as I have learned through the course of this semester, is that potential functions are not as free-form as I had first thought. I was formally introduced to this type of analysis in both of my algorithms classes this semester and through seeing the utilization of this technique time and time again, I realize that reductions and re-castings of the problem in an entirely unrelated framework can eek out insights that otherwise fare well in avoiding the eye.

This is another part in my journey to learn about the structure of computation. As corny as it may sound, I like to think that numbers, in accordance to the laws of mathematics in this universe, hold deeper secrets than we like to usually give them credit for. My real aim here is to unravel the complexities of varied algorithms in different applications and find a common, basic, irreducible element of them all. It wonders me to see how much more we can see if we just look at a problem in the right way. While linear programming is something I’ve come to gain a great respect for this semester, it’s still only the tip of the iceberg. There is an even broader scope by which we may express a problem and perhaps other techniques will be the topic of future posts.

 
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Posted by on April 21, 2011 in Uncategorized

 

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Spring Semester Update

Today marks the beginning of Spring Break for us and the midway point of the semester as well. This is most definitely the hardest semester I’ve ever had so far and although it’s very stressful at times, I’m really enjoying it a lot. I guess in some ways this semester was a way for me to determine how much I actually loved computer science at the its core – to see how much I enjoyed it if I fully immersed myself in it. The results so far are better than what I had expected: Even though the great majority of the my homework and assignments are about computer science and even though I am studying or attending class on some CS subject nearly every single day, my interest in it has only increased. I feel very at home when learning about the theoretical aspects of computation and complexity, or knee-deep in the analysis of some algorithm. Even though the workload is just so very barely manageable, I’m glad I am taking what I decided to take and I’d like, to the extent possible, to complete all six courses I am enrolled in this semester. Six courses is a bit on the high side for an engineer here, but I guess it’s more about how interesting you find them. As I’ve said before, it only feels like work if you can actually distinguish it from play. In any case, most of the work that I have fits comfortably in the category of play, so it’s basically an excuse to do something I’d probably be doing – or wishing I could be doing – during that time anyway.

I had a few goals for the year. One to take at least one graduate class before the end of the year; to study lesser known data structures and algorithms; to learn at least one more popular language and one more obscure language; and to get an introduction on quantum computation. Of those, I am in the process of accomplishing the first few of them. Thanks to my class on approximation algorithms, which is both tough and very enjoyable, I am satisfying the first one. In my Theory of Algorithms course, we are covering quite a few different interesting data structures as well as algorithms. Although these are not exactly “lesser known” in the greater sense of computer science, they’re also not the ones you usually learn in an introductory course. Specifically, I was very happy to see that we covered Fibonacci heaps, rank-pairing heaps, AVL trees, and RAVL trees. These, especially the first, are among the more advanced data structures used for very large data sets in industry. If I recall correctly, I believe Google Maps uses some implementation of F-heaps in storing data. In general, self-adjusting data structures are extremely interesting and it’s so neat to get a pretty bound on their performance regardless of how convoluted the implementation and procedures may be. As far as the other goals, I’m making (slow) progress on picking up Python and we’re using C++ for our computer graphics course. And as far as lesser known languages, our compilers course is taught strictly in ML – a functional language. I am yet to fully wrap my head around those exotic things. More thought is necessary on that front.

Although I’m satisfied with the coverage of algorithms for trees and graphs so far, I’m now also interested in learning about probabilistic algorithms. I was turned in this direction after seeing the power of probabilistic rounding techniques in solving LP-relaxation problems for approximation algorithms for very common problems such as vertex cover. Although they may not always provide an optimal bound, they do sometimes make the code and analysis much simpler and offer better worst-case performance. And they are everywhere too. From the example I just mentioned to an implementation of MST in linear time (!) in the number of edges, randomized and probabilistic algorithms play a huge role. And looking at it from a more pure sense, it’s also particularly interesting to see how the nature of random numbers can reveal insight on the structure of computation. I’m starting to develop some ideas of my own; perhaps the better word for them would be “questions” or maybe just situations and combinations of known elements in a way I haven’t seen anywhere. Hopefully I will soon be able to gain the mathematical and analytical machinery to actually see if these ideas make any sense at all.

In other news, my copy of Knuth’s classic Art of Computer Programming Vol. 1-4A finally arrived this week! As you may know, the fourth volume was just released after something like 38 years. I’m very excited to check it out. It’s currently sitting, still shrink-wrapped, on the table beside me. During the next week I’ll see if I can make sense of it. On another note, I am very pleasantly surprised by the quality of CLRS, another very excellent algorithms book. The analysis is surprisingly clean and the writing is precise, which makes it a pleasure to read. And they tackle complicated concepts with ease, making it look like it takes hardly any effort, which is impressive. Over winter break, I had started on Michael Sipser’s Introduction to the Theory of Computation, another classic text. It turns out this gave me a very nice edge when we were covering DFA/NFAs, regular expressions, and context-free grammars in compilers. Although I had had an introduction to these concepts in a previous course, I accredit Sipser’s book for teaching them to me in a rigorous sense. Once again, I see the same ease with with Sipser explains these concepts and proofs and it’s quite impressive. It makes it as painless as possible to cover these proofs with sufficient depth. So I will have to continue on with that text, seeing as it coincides very closely with what I am interested in. And as for the last goal, it is yet unfulfilled. I need a whole lot more knowledge before I go near that subject. Perhaps next semester or next spring. There’s just so much learn.

 
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Posted by on March 12, 2011 in Uncategorized

 

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Spring Semester 2011

Today is Friday (well it was about an hour ago) and that means that this week is done. This week also happens to be my first week of my spring semester. And it was a hell of a ride. My current mood right now is a combination of extreme tiredness, sleepiness, and a great deal of excitement for weeks to come. [Pretty sure that last statement is grammatically incorrect but I can’t be bothered to fix it because it conveys my feelings well].  Going into the planning stages for this semester, I sought to take on a major challenge for myself: to out-do anything I’ve done before and push myself beyond any boundaries I have previously thought was the limitation of my abilities. Obviously, classes here are really hard, but I see them as springboards rather than roadblocks. And the springboard metaphor is quite fitting: you jump on-board and you sink and sink for a bit, very well aware of your weight and limitations and how long they’re dragging you down – but that’s only for the first half. Once you reach that critical state when things start clicking, it’s all uphill from there. Suddenly your own weight is actually helping you go higher. And guess what? You’ll end up higher than you could’ve ever jumped otherwise.

The moral of the story is obviously this: Avoid falling off on the way down and it’ll be worth it on your way up.

And that’s how it was for a few of my classes last semester too. I took an upper level math class on numerical analysis and found it pretty hard at first. It was intimidating to be in a room full of people a year or more older than me and a professor who walk into the room, give a half-hi gesture, and almost immediately begin lecturing without break for the entirety of the class. Although I do enjoy math a great deal, it does in no way imply that I’m particularly good at it. A math TA from last year put it aptly when he said mathematics was the process of banging your head on a table until 1) you passed out, in which case you called it a day, or 2) you eventually made a breakthrough. As convenient and elegant as it may to think that there’s some magical state beyond which the marginal difficulty of learning the next theorem/definition/proof/algorithm falls off to some small, constant value, I really am starting to doubt that’s the case. The more I make progress in my education on the fronts of both math and computer science, I’m thinking that what instead happens is that we, either by consciously looking for it or by our minds’ own doing, start seeing the very same patterns and paradigms again and again. Of course this isn’t a new idea, but it hits you hard when you make the realization for yourself. It’s interesting because it’s almost as if my brain is developing some form of auto-complete: give it some properties of some new entity and it “predicts” some of the others that will follow. There are obviously tons of exceptions to this and that’s where the fun comes in and keeps things interesting enough to continue pursuing (although the first time I heard matrix multiplication doesn’t commute was jarring to my pristine understanding of the world). And it’s this same notion of “auto-complete”, or intuition, that gives a better grip on the weird world out there and thus provides the illusion that the marginal difficulty is indeed decreasing.

Another metaphor which I particularly like derives from the use of the term “structure” when thinking about a problem: namely, in the context of a phrase like “now after X years of research, we have a better understanding of the inner structure of [complexity class/research problem/concept/etc]…”. In my mind, I see each of these concepts not quite as black boxes, but as dark rooms, the size of which is sometimes unknown from our current perspective. And so long as Erdös was just being metaphorical about his Book, there aren’t any lighting fixtures in the room. All we are given is a bunch of little candles. In fact we do have a few matches but it’s much harder to light a candle with a match than it is to use an already lit one. And so we go about setting up tiny little candles all about this room. They each illuminate brightly things in their immediate vicinity but the falloff of light is pretty drastic at times. And sometimes different candles illuminate the same portion of the room. Ideally, we’d like to use exactly one candle, so perfectly positioned so that it lights the entire room, but finding that position is almost definitely at least NP-hard or something… The idea is that there are rooms with candles from other people, in fact all of the other people in the world. And then there’s your own room, where you have to discover which candles are lit by yourself. You don’t have to necessarily light them yourself, but you have to discover that they do indeed exist. But of course, the room is too large to light up fully. So instead, we attack a certain direction. Perhaps we like what candles so far have illuminated or perhaps we think we’re on the edge of a major push forward. In either way, we are forced to narrow down our search. It’s pretty amazing how much brighter my room has gotten in just the past few months. (Baseless prediction: the room is a actually cylindrical. Interpret that as you understand it.).

And all of these thoughts are what are following me around these days. I love a good mystery and this is exactly that. I am consistently more amazed and in a state of awe than I ever expected. And I find the intricate complexity of the surface lit thus far extremely beautiful. Although theoretical computer science has a bit less of natural feel (that is to say, closeness to the ways of nature and the universe) than mathematics, it’s still astonishing to see how things fit together. Yes, computer science is a man-made field consisting of arguably arbitrary dichotomies depending on who you ask. And yes, this field is still so very much in its infancy. But nonetheless, they reveal something deeper than the definitions that we have given them. To put it shortly, there’s still some magic left which lays undiscovered waiting for us. As frustrating as it is that we do not understand some seemingly elementary relationships between things, it’s also exactly that which gives it its charm. I was sitting in class this week, with the professor writing theorem after theorem on the board, each of which had to do in some way with P vs. NP. And I thought how much more boring the class would have been if indeed we did know the answer. Or how even more boring it would be if P \neq NP. As much as I hope it’s resolved soon, it’s the idea of not knowing which is incredible in some strange way. It keeps the magic alive and I like it.

I considered what courses I wanted to take this semester. There are lots of things I want to learn about in computer science with only time being the limitation. I decided to go forward with a bold move by taking two very difficult theory classes together. They are both on algorithms: one on the theory of algorithms taught by the great R.E. Tarjan and the other a graduate course on advanced algorithm design – specifically approximation algorithms. They are fast-moving and the latter is extremely difficult (I don’t doubt the former will soon become so too!). But I’m not getting off the springboard, no matter how tempting it may be. I will continue to push forward, on until that pivotal moment hits where things start finally making sense. I’m learning an insane amount of things every single day and it’s amazing that a lot of things which I had read about casually in the past are all suddenly coming together with a much brighter luminance. It’s hard and I anticipate lots and lots of banging heads on tables ahead, but it’ll be worthwhile. This is one of those utterly invaluable experiences that I wouldn’t give up for anything.

I started the week inspired and now I am more inspired than I recall ever being. I live in an amazingly intricate and beautiful world and all I want to do is keep lighting candles.

 
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Posted by on February 5, 2011 in Uncategorized

 

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Summer Research!

Big news! I finally got an offer to do research this summer in Computer Science with one of my professors! =] I also accepted the offer this morning, so my summer plans are pretty much finalized! It honestly feels like a huge weight has been lifted from my chest because I have spent the past several months looking for something to do this summer. My professor was kind enough to spend over a half hour explaining everything to me and showing me what I will be working on. I can’t say too much (I still have a lot to read up on!) but it’s related to the resource allocation problem in a distributed computing environment. So that means I’ll be spending most of my summer here at Princeton. It’s pretty much the highlight of my week and I can’t stop smiling about it :). I know it’s gonna be really challenging and all, but that’s perfectly fine by me. I am eager to begin this challenge and welcome the next stage of my life. I know that I’ll learn a whole lot; much more than what I could pick up just by reading a textbook. This is pretty much my first stab at doing formal research, so I’m really quite nervous. I hope all goes well!

Oh and I totally messed up on physics 😦 But it’s alright, the points I lost were almost exclusively due to minor errors in computation or slight misinterpretations; most of what I wrote was right (although I should have perhaps completed a greater proportion of the exam..). I’m not too worried about grades at this point so I know it’s gonna be okay.

As I get more information about my summer research, I’ll keep this blog updated. My hope is that it could be helpful down the line for someone in my same place (I was completely clueless on how to go about doing it, and am only marginally less clueless, but hell it could help).

 
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Posted by on April 1, 2010 in Uncategorized

 

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