What algorithm blows your mind? (Reddit compsci) | Hacker News Reader

What algorithm blows your mind? (Reddit compsci)

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129 pointsbarrkel66 comments

I've always found error detecting/correcting codes to be lovely. A great example, that's easy to understand, is the Hamming(7,4) code.

http://en.wikipedia.org/wiki/Hamming(7,4)

I got interested in these codes because of the problems involved with deep space communication.

http://en.wikipedia.org/wiki/Error_detection_and_correction#...

Also, compression. Wikipedia got real good about error detection and correction and compression algorithms past few years http://en.wikipedia.org/wiki/Category:Error_detection_and_co...

I learned Hamming code in school this summer (digital systems) .It was probably the single most magical thing (even more than flipflops! Loops make memory!) I was exposed to that class.

I remember upon learning it someone went "wow, who came up with that". The prof answered dryly "some genius named Hamming".

All very nice suggestions below, but I found suffix trees missing: http://en.wikipedia.org/wiki/Suffix_tree

And the reasons why it blows my mind:

(1) Knuth called it "Algorithm of the year 1973" (possibly because it beat an intuitive lower bound that he had for a problem I can't remember).

(2) It's relatively new for a "core", first-principles algorithm. Ukkonen's algorithm is from 1995, although there are earlier versions.

(3) This BOOK is largely devoted to the many, many applications of suffix trees: http://www.amazon.com/Algorithms-Strings-Trees-Sequences-Com... It should be required reading for anyone interested in advanced algorithms.

(4) Longest common substring in linear time.

(5) Lempel-Ziv decomposition in linear time.

(6) Longest repeated substrings in linear time.

And too many more to list.

Bloom Filters: http://en.wikipedia.org/wiki/Bloom_filter

Sort of a probabilistic hash, where you trade space for accuracy. But it's also like a memory function - it can remember if it has seen a piece of data before.

What I love about bloom filters is that when checking to see if something is in a bloom filter, you can only get false positives, not false negatives. That property is just awesome to me.

Also the fact that you can use more bits per element and reduce the error rate for false positives. Using around 3 bytes per element, can bring the error rate down to 0.001%.

also the fact that you can use multiple hash functions to reduce the error rate even further is pretty nice. this is of course without incurring the overhead of extra storage space...

Using multiple hash functions requires that you use additional storage in order to maintain the same occupancy ratio.

I love that they're fast enough to use in hardware. Some of the newer schemes for hardware-accelerated transactional memory use Bloom filters to probabilistically detect when two transactions are using the same memory addresses. This is so much easier and more efficient than previous approaches, which had to use content associative memories and expensive broadcast operations.

Singular value decomposition. It continually amazes me how many problems -- from noise reduction to Netflix prediction -- SVD can be usefully applied to.

http://en.wikipedia.org/wiki/Singular_value_decomposition

The core ideas behind SVD and eigen values are very rich and profound. It has always fascinated me when I studying maths in undergrad.

Someone else already mentioned compressed sensing, which expands on some of those ideas. Terrence Tao had a pretty good presentation on the topic: http://terrytao.files.wordpress.com/2009/08/compressed-sensi...

I still do not completely understand it. I see the math, but I see it as symbols that I can memorize, rather than something intuitive (like when I see a moving cursor through a state tree during a binary search for instance).

Compressed Sensing is exciting - using a sequence of low res images to obtain a higher resolution sample. It was discussed in Wired (http://www.wired.com/magazine/2010/02/ff_algorithm/all/1) together with a compelling example (minimizing the time of a young patient in an MRI machine)

That sounds far too good to be true. I guess if you don't care about the fine details then its a decent technique, but to take the MRI example, what if the thing that was wrong was only visible in those small details?

There are other things which sound too good to be true and yet are true (the sampling theorem for exact reconstruction of periodic signals is pretty unintuitive IMO, and even more "magic")

Filling in missing information with a reasonable guess is vary different from having accurate information. Looking at a low res impressionist painting you are going to fill in based on reasonable estimates for the real world not what is actually there.

In compressed sensing, we don't do guesses. Guesses are reserved for inpainting.

The Wired articles gave the wrong impression. It gave an example of inpainting (the Obama picture) that is NOT compressed sensing. Compressed sensing on the other hand can be applied directly to MRI because the MRI machine actually picks up all the spatial information. It samples randomly in the Fourier domain thereby having access to all the spatial information needed to fully reconstruct an image. It used to be that the reconstruction algorithms were not that good before (they relied on SVD/least square). Candes, Tao, Romberg and Donoho then published papers showing that the reconstruction could be done in a totally different way AND it was exact. With these new reconstruction algorithms something like MRI data is acquired in a much more efficient manner than four years because of compressed sensing.

For mor eon the controversy with the Wired article: http://nuit-blanche.blogspot.com/2010/05/compressed-sensing-... http://nuit-blanche.blogspot.com/2010/03/why-compressed-sens...

The new reconstruction solvers: https://sites.google.com/site/igorcarron2/cs#reconstruction

Hardware that are implementing compressive sensing: https://sites.google.com/site/igorcarron2/compressedsensingh...

Compressed sensing was the subject of a recent talk which can be viewed at ee380.stanford.edu as well as via itunes and youtube.

I ahve compiled a long list of online talks here: https://sites.google.com/site/igorcarron2/csvideos

start from the bottom.

For me, it's boosting. Basically, if you make the assumption that your weak classifier can achieve X% correct on any random distribution over a data set, then you can create an ensemble of weak learners that together get Y% correct on the same set, where Y > X.

Question 3 here will walk you through the proof: http://bit.ly/cQ03na

Please don't post links through bit.ly and other URL indirection sites. It hurts the web by making it much more difficult to follow the link in the case that an intermediary goes out of business. Hacker News and your post will probably be around in a few years, but will bit.ly? (What's their business model, exactly...?)

For future reference, the unshortened version of his link is: http://www.stanford.edu/class/cs221/handouts/cs221-ps2.pdf

Not to mention the real URL shows that the link is going to Stanford whereas I have no clue whether the shortened link is going to Stanford, an XXX site, or rick rolling me.

Quake3's Fast InvSqrt function:

http://www.beyond3d.com/content/articles/8/

It's been discussed here before, and no doubt it'll be cited numerous times in the comments of that reddit post, but it was the first time I'd ever seen such trickery. When I got into low level DSP programming I saw many more examples of clever hacks, but this is the one which sticks in my mind above all of those.

My mind is blown by the algorithm for matching with mismatches which I present in the first chapter of my thesis. It shouldn't be, given that I discovered this algorithm -- but somehow "I take the Fourier Transfer of the FreeBSD kernel" sounds more like the punch line to a joke than the first step in an algorithm.

Speaking of incredibly cool algorithms based on Fourier transforms, the Schoenage-Strassen algorithm for multiplying integers is also amazing:

http://en.wikipedia.org/wiki/Sch%C3%B6nhage%E2%80%93Strassen...

At least to me, it seems hard to imagine that multiplying two N-bit integers could be done with less than O(N^2) operations. Schoenage-Strassen lets you do it using O(N log N log log N) operations!

This is the one that blew my mind. "If we treat the numbers as polynomials, then the multiplication can be reduced to a Fourier transform!" is also one of those things that sounds like a punchline. Each step of the algorithm just sounds stupider and stupider, and slower and slower... and then you get to the last step, where all the multiplications by exponentials of complex numbers get replaced by bitshifts. Suddenly the entire algorithm collapses in on itself to reveal something efficient.

Link?

http://www.daemonology.net/papers/thesis.pdf

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Lenstra-Lenstra-Lovasz:

http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%9...

High dimensional work is bloody hard, and this algorithm works amazingly well. I've spoken with Lenstra (one of them) and he's amazingly insightful on these things. He helped to crystalise my understanding of why high-dimensional spheres should be thought of as "spikey," rather than "round."

Why are high dimensional spheres spikey?

Ah. I really should write that up. Far too long to explain in a comment here, far to interesting (to me!) to forget or ignore.

I'll write it up and submit it. Anyone who cares to email me can get an early version to read, and your feedback would be useful.

Please.

Thanks.

Please do. My guess would be similar to Zaak's: the n-dimensional volume of an n-dimensional hypersphere approaches 0 as n approaches infinity. (http://www.mathreference.com/ca-int,hsp.html). A hypersphere with radius 1 centered at the origin has to get out to 1 along each axis (e.g. x=1, y=z=0 for a 3 dimensional sphere), and in order for that not to contribute much volume, it has to be a narrow spike rather than a gradual curve.

I believe it's because in high dimensions, most of the "volume" is near the center, whereas in low dimensions, most of the volume is near the surface.

Not at all. The expected distance between the center of an n-ball and a random point in the ball is n/(n+1) times the radius.

Now, if you look at an n-sphere with respect to orthogonal axes, you find that moving along an axis you get out as far as (1, 0, ... 0) and moving "away" from the axes you only get to (1/sqrt(n), 1/sqrt(n), ... 1/sqrt(n)); but this isn't due to the sphere being spiky -- rather, it's because orthogonal axes are spiky.

Now written up here:

http://news.ycombinator.com/item?id=1846682

In response to requests, the write-up is now available here:

http://news.ycombinator.com/item?id=1846682

How is generation of the Mandelbrot Set not on this list? That something so simple could produce something so indescribably beautiful...

Agreed, but while the notation's simple, the iterative effect is hard to follow -

it's like a spinning rod, angle doubled and length squared each iteration, with a rod of fixed length and angle added to it each iteration (if the spinner is facing away, it gets shorter, but if towards, it gets longer). Iterate til it goes over 2. If it doesn't then it's in the Set (or maybe you need to iterate more...). [that's my current understanding]

It's pretty clear to me that I can't visualize what it will do over a few iterations (apart from simple cases) - and I sure can't see how it would produce self-similar (ie fractal) shapes. Agreed that that's amazing.

The quantum search algorithm: http://en.wikipedia.org/wiki/Grovers_algorithm

It lets you search a completely unstructured N-item search space using the square root (!) of N queries, not the N queries you'd think were necessary.

Also, the algorithm is so simple that once you know it's possible, and provided you're very comfortable with basic quantum mechanics, it's almost trivial to find the algorithm.

Ok, not quite an algorithm, but the PCP theory (http://en.wikipedia.org/wiki/PCP_theorem) is easily the most mind-blowing result in comp sci in the past 20 years.

PCPs are true magic. I once learned how to construct a PCP-proof with verification using just 7 bits, but it still feels like magic to me. The 3-bit variants are just crazy.

The Burrows-Wheeler transform used in bzip2 is truly mind-blowing.

Many algorithms are ordinary genius (eventually you would have come up with it because the problem space dictates the solution), but this one is extraordinary genius (mind-blowingly original and non-obvious). Every time I see it, I'm amazed that it works.

Welzl's algorithm for minimum enclosing disk does it for me. Pick a point, recursively see if it is in the disk without the point - if not, it is part of the definition the minimum enclosing disk (forced to be on the boundary). Randomized, simple, clean and expected O(n).

Not enough people know about it! http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.46....

Kruskal, Prims and reverse-delete algorithms to find the Minimum Spanning Tree in a graph, are fun, not that hard, and very practical. http://en.wikipedia.org/wiki/Minimum_spanning_tree

Think about it, every time you get a google map direction/route, one of them is in play.

Also a bit different, but perhaps even more widespread application of spanning trees is STP family of L2 network bridging protocols.

http://en.wikipedia.org/wiki/Rapid_Spanning_Tree_Protocol

CORDIC (http://en.wikipedia.org/wiki/Cordic), used to efficiently calculate trig functions with only very basic hardware requirements (add, sub, shift, and table lookup).

. . . relatedly, from Jim Blinn, from Marvin Minsky -- drawing a circle (near enough), or doing incremental rotation, using just integer add, sub, and two shifts:

   N = 128 * 3.14159
   X = 1000
   Y = 0

   MOVE(X,Y)
   FOR I = 1 TO N
      X = X - (Y >> 6)
      Y = Y + (X >> 6)
      DRAW(X,Y)

Some of the sorting algorithms, as somebody learning them, it's almost entirely opaque how somebody could come up with those.

It seems like they must have sprung forth wholecloth to the inventor in the shower...it's almost impossible to have iteratively developed some of them because even small changes in the algorithms produce terrible results. I remember thinking over and over again, "how the hell could somebody come up with this?"

Searching in comparison looks very engineered, very studied, something that most people could come up with given need, motivation and time.

BSP trees blew my mind when I discovered they were under the hood of Doom. This was back in the era of 386 and 486 PCs.

The Genetic Algorithm. It totally blew my mind when I read about it's successful application to evolving satellite antennas for JPL, and the automated design of an analog->digital signal converter that relies on a component that isn't actually connected to anything else.

Poul-Henning Kamp (creator of Varnish) recently published the B-heap algorithm. It's a cache-friendly version of the binary heap storage we all learned as freshmen.

PHK points out that mapping node n to nodes 2n and 2n+1 will cause cache misses (and often page faults) as we vertically descend the tree. So instead, rearrange the mapping so that the child nodes are often very near their parent node. That way, heap comparisons are usually performed in the same virtual page, which cuts-down on the number of disk accesses the operating system must perform.

http://queue.acm.org/detail.cfm?id=1814327

More of a complexity result than an algorithm, but it is a really fun result anyway. In complexity class we were given the problem to simulate an n-tape non-deterministic Turing machine with a 2-tape non-deterministic Turing machine with just a constant slow-down (which turned out to be 3). It really gave an insight into the power of non-deterministic choice.

The binomial heap http://en.wikipedia.org/wiki/Binomial_heap

It's a heap supporting merging in log(n). While it's usage might not be very common, it is a beautiful data structure.

I might be cheating a little, but http://en.wikipedia.org/wiki/Grovers_algorithm blows my mind.

The heapsort. It sounds very strange.

http://www.youtube.com/watch?v=iXAjiDQbPSw

I'd go with Tarjan and Hopcroft's planarity testing algorithm, whose runtime is linear in the number of vertices.

FWIW, some of these algorithms: Bloom filters, Error Correcting Codes are connected to the Compressive Sensing.

while looking thru some 360 assembler code one day, I came across this ...

XC loc1,loc2

XC loc2,loc1

XC loc1,loc2

It took a day of head scratching, and various notes, before I understood that it was a clever way of exchanging the contents of two storage locations, without a third intermediate location. Three consecutive exclusive-Ors.

for me simplex (george-dantzig) is very very cool.

It surprised me that matching is not NP-complete.

But maximal matroid matching is NP-hard. ;)

Approximating the minimum spaning tree weight in subliner time is very unexpected algorithm.

Arithmetic coding: http://en.wikipedia.org/wiki/Arithmetic_coding#Encoding_and_...

The entire message is coded into a single number and it approaches the theoretical maximum for compression.

Too bad about software patents!

In heuristics, I like Asynchronous teams [ http://www.cs.cmu.edu/afs/cs/project/edrc-22/project/ateams/... ]

In multi-robot coordination I like the free market system [ http://www.frc.ri.cmu.edu/projects/colony/architecture.shtml ]

And regarding machine learning I like neural nets (MLP), however the algorithms that currently blow my mind are convoluted neural networks (CNNs) and deep belief networks trained with auto-encoders. http://www.youtube.com/watch?v=AyzOUbkUf3M on the 21-minute mark to see it in action.

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