## Comparing Sequences Without Sorting

This is a post about a silly (and mostly pointless) optimization.  To motivate this, consider the following problem which you see all the time in mesh processing:

Given a collection of triangles represented as ordered tuples of indices, find all topological duplicates.

By a topological duplicate, we mean that the two faces have the same vertices.  Now you aren’t allowed to mutate the faces themselves (since orientation matters), but changing their order in the array is ok.  We could also consider a related problem for edges in a graph, or for tetrahedra in mesh.

There are of course many easy ways to solve this, but in the end it basically boils down to finding some function or other for comparing faces up to permutation, and then using either sorting or hashing to check for duplicates.  It is that comparison step that I want to talk about today, or in other words:

How do we check if two faces are the same?

In fact, what I really want to talk about is the following more general question where we let the lengths of our faces become arbitrarily large:

Given a pair of arrays, A and B, find a total order $\leq$ such $A \leq B \: \mathrm{and} B \leq A$ if and only if $A = \pi(B)$ up to some permutation $\pi$.

For example, we should require that:

[0, 1, 2] $\leq$ [2, 0, 1]

[2, 0, 1] $\leq$ [0, 1, 2]

And for two sequences which are not equal, like [0,1,2] and [3, 4, 5] we want the ordering to be invariant under permutation.

## Obvious Solution

An obvious solution to this problem is to sort the arrays element-wise and return the result:

function compareSimple(a, b) {
if(a.length !== b.length) {
return a.length - b.length;
}
var as = a.slice(0)
, bs = b.slice(0);
as.sort();
bs.sort();
for(var i=0; i<a.length; ++i) {
var d = as[i] - bs[i];
if(d) {
return d;
}
}
return 0;
}

This follows the standard C/JavaScript convention for comparison operators, where it returns -1 if a comes before b, +1 if a comes after b and 0 if they are equal.  For large enough sequences, this isn’t a bad solution.  It takes $O(n \log(n))$ time and is pretty straightforward to implement.  However, for smaller sequences it is somewhat suboptimal.  Most notably, it makes a copy of a and b, which introduces some overhead into the computation and in JavaScript may even trigger a garbage collection event (bad news).  If we need to do this on a large mesh, it could slow things down a lot.

An obvious way to fix this would be to try inlining the sorting function for small values of n (which is all we really care about), but doing this yourself is pure punishment.  Here is an optimized version for length 2 sets:

function compare2Brutal(a, b) {
if(a[0] < a[1]) {
if(b[0] < b[1]) {
if(a[0] === b[0]) {
return a[1] - b[1];
}
return a[0] - b[0];
} else {
if(a[0] === b[1]) {
return a[1] - b[0];
}
return a[0] - b[1];
}
} else {
if(b[0] < b[1]) {
if(a[1] === b[0]) {
return a[0] - b[1];
}
return a[1] - b[0];
} else {
if(a[1] === b[1]) {
return a[0] - b[0];
}
return a[1] - b[1];
}
}
}

If you have any aesthetic sensibility at all, then that code probably makes you want to vomit.   And that’s just for length two arrays!  You really don’t want to see what the version for length 3 arrays looks like.  But it is faster by a wide margin.  I ran a benchmark to try comparing how fast these two approaches were at sorting an array of 1000 randomly generated tuples, and here are the results:

• compareSimple:  5864 µs
• compareBrutal: 404 µs

That is a pretty good speed up, but it would be nice if there was a prettier way to do this.

Symmetry

The starting point for our next approach is the following screwy idea:  what if we could find a hash function for each face that was invariant under permutations?  Or even better, if the function was injective up to permutations, then we could just use the symmetrized hash compare two sets.  At first this may seem like a tall order, but if you know a little algebra then there is an obvious way to do this; namely you can use the (elementary) symmetric polynomials.  Here is how they are defined:

Given a collection of $n$ numbers, $a_0, a_1, ..., a_{n-1}$ the kth elementary symmetric polynomial, $S_{n,k}$ is the coefficient of $x^{n-k-1}$ in the polynomial:

$(x + a_0) (x + a_1) ... (x + a_{n-1}) = S_{n,n-1} + S_{n,n-2} x + ... + S_{n,0} x^{n-1} + x^n$

For example, if $n = 2$, then the symmetric polynomials are just:

$S_{2,0} = a_0 + a_1$

$S_{2,1} = a_0 a_1$

And for $n = 3$ we get:

$S_{3,0} = a_0 + a_1 + a_2$

$S_{3,1} = a_0 a_1 + a_0 a_2 + a_1 a_2$

$S_{3,2} = a_0 a_1 a_2$

The neat thing about these functions is that they contain enough data to uniquely determine $a_0, a_1, ..., a_{n-1}$ up to permutation.  This is a consequence of the fundamental theorem for elementary symmetric polynomials, which basically says that these $S_{n,k}$ polynomials form a complete independent basis for the ring of all symmetric polynomials.  Using this trick, we can formulate a simplified version of the sequence comparison for $n=2$:

function compare2Sym(a, b) {
var d = a[0] + a[1] - b[0] - b[1];
if(d) {
return d;
}
return a[0] * a[1] - b[0] * b[1];
}

Not only is this way shorter than the brutal method, but it also turns out to be a little bit faster.  On the same test, I got:

• compare2Sym: 336 µs

Which is about a 25% improvement over brute force inlining.  The same idea extends to higher n, for example here is the result for $n=3$:

function compare3Sym(a, b) {
var d = a[0] + a[1] + a[2] - b[0] - b[1] - b[2];
if(d) {
return d;
}
d = a[0] * a[1] + a[0] * a[2] + a[1] * a[2] - b[0] * b[1] - b[0] * b[2] - b[1] * b[2];
if(d) {
return d;
}
return a[0] * a[1] * a[2] - b[0] * b[1] * b[2];
}

Running the sort-1000-items test against the simple method gives the following results:

• compareSimple: 7637 µs
• compare3Sym:  352 µs

### Computing Symmetric Polynomials

This is all well and good, and it avoids making a copy of the arrays like we used in the basic sorting method.  However, it is also not very efficient.  If one were to compute the coefficients of a symmetric polynomial directly using the naive method we just wrote, then you would quickly end up with $O(2^n)$ terms!  That is because the number of terms in $\# S_{n,k} = { n \choose k+1 }$, and so the binomial theorem tells us that:

$\#S_{n,0} + \#S_{n,1} + \#S_{n,2} + ... + \#S_{n,n-1} = 2^n - 1$

A slightly better way to compute them is to use the polynomial formula and apply the FOIL method.  That is, we just expand the symmetric polynomials using multiplication.  This dynamic programming algorithm speeds up the time complexity to $O(n^2)$.  For example, here is an optimized version of the $n=3$ case:

function compare3SymOpt(a,b) {
var l0 = a[0] + a[1]
, m0 = b[0] + b[1]
, d  = l0 + a[2] - m0 - b[2];
if(d) {
return d;
}
var l1 = a[0] * a[1]
, m1 = b[0] * b[1];
d = l1 * a[2] - m1 * b[2];
if(d) {
return d;
}
return l0 * a[2] + l1 - m0 * b[2] - m1;
}

For comparison, the first version of compare3 used 11 adds and 10 multiplies, while this new version only uses 9 adds and 6 multiplies, and also has the option to early out more often.  This may seem like an improvement, but it turns out that in practice the difference isn’t so great.  Based on my experiments, the reordered version ends up taking about the same amount of time overall, more or less:

• compare3SymOpt: 356 µs

Which isn’t very good.  Part of the reason for the discrepancy most likely has something to do with the way compare3Sym gets optimized.  One possible explanation is that the expressions in compare3Sym might be easier to vectorize than those in compare3SymOpt, though I must admit this is pretty speculative.

But there is also a deeper question of can we do better than $O(n^2)$ asumptotically?  It turns out the answer is yes, and it requires the following observation:

Polynomial multiplication is convolution.

Using a fast convolution algorithm, we can multiply two polynomials together in $O(n \log(n))$ time.  Combined with a basic divide and conquer strategy, this gives an $O(n \log^2(n))$ algorithm for computing all the elementary symmetric functions. However, this is still worse than sorting the sequences!  It feels like we ought to be able to do better, but further progress escapes me.  I’ll pose the following question to you readers:

Question: Can we compute the n elementary symmetric polynomials in $O(n \log(n))$ time or better?

### Overflow

Now there is still a small problem with using symmetric polynomials for this purpose: namely integer overflow.  If you have any indices in your tuple that go over 1000 then you are going to run into this problem once you start multiplying them together.  Fortunately, we can fix this problem by just working in a ring large enough to contain all our elements.  In languages with unsigned 32-bit integers, the natural choice is $\mathbb{Z}_{2^{32}}$, and we get these operations for free just using ordinary arithmetic.

But life is not so simple in the weird world of JavaScript!  It turns out for reasons that can charitably be described as “arbitrary” JavaScript does not support a proper integer data type, and so every number type in the language gets promoted to a double when it overflows whether you like it or not (mostly not).  The net result:  the above approach messes up.  One way to fix this is to try applying a modulus operator at each step, but the results are pretty bad.  Here are the timings for a modified version of compare2Sym that enforces bounds:

That’s more than a 5-fold increase in running time, all because we added a few innocent bit masks!  How frustrating!

Semirings

The weirdness of JavaScript suggests that we need to find a better approach.  But in the world of commutative rings, it doesn’t seem like there are any immediately good alternatives.  And so we must cast the net wider.  One interesting possibility is to extend our approach to include semirings.  A semiring is basically a ring where we don’t require addition to be invertible, hence they are sometimes called “rigs” (which is short for a “ring without negatives”, get it? haha).

Just like a ring, a semiring is basically a set $S$ with a pair of operators $\oplus, \otimes$ that act as generalized addition and multiplication.  You also have a pair of elements, $0,1 \in S$ which act as the additive and multiplicative identity.  These things then have to satisfy a list of axioms which are directly analogous to those of the natural numbers (ie nonnegative integers).  Here are a few examples of semirings, which you may or may not be familiar with:

• The complex numbers are semiring (and more generally, so is every field)
• The integers are a semiring (and so is every other ring)
• The natural numbers are a semiring (but not a ring)
• The set of Boolean truth values, $\mathbb{B} = \{ \mathrm{true}, \mathrm{false} \}$ under the operations OR, AND is a semiring (but is definitely not a ring)
• The set of reals under the operations min,max is a semiring (and more generally so is any distributive lattice)
• The tropical semiring, which is the set of reals under (max, +) is a semiring.

In many ways, semirings are more fundamental than rings.  After all, we learn to count stuff using the natural numbers long before we learn about integers.  But they are also much more diverse, and some of our favorite definitions from ring theory don’t necessarily translate well.  For those who are familiar with algebra, probably the most shocking thing is that the concept of an ideal in a ring does not really have a good analogue in the language of semirings, much like how monoids don’t really have any workable generalization of a normal subgroup.

### Symmetric Polynomials in Semirings

However, for the modest purposes of this blog post, the most important thing is that we can define polynomials in a semiring (just like we do in a ring) and that we therefore have a good notion of an elementary symmetric polynomial.  The way this works is pretty much the same as before:

Let $R$ be a semiring under $\oplus, \otimes$; then we have the symmetric functions.  Then for two variables, we have the symmetric functions:

$S_{2,0} = a_0 \oplus a_1$

$S_{2,1} = a_0 \otimes a_1$

And for $n=3$ we get:

$S_{3,0} = a_0 \oplus a_1 \oplus a_2$

$S_{3,1} = (a_0 \otimes a_1) \oplus (a_0 \otimes a_2) \oplus (a_1 \otimes a_2)$

$S_{3,2} = a_0 \otimes a_1 \otimes a_2$

And so on on…

### Rank Functions and (min,max)

Let’s look at what happens if we fix a particular semiring, say the (min,max) lattice.  This is a semiring over the extended real line $\mathbb{R} \cup \{ - \infty, \infty \}$ where:

• $\oplus \mapsto \mathrm{min}$
• $\otimes \mapsto \mathrm{max}$
• $0 \mapsto \infty$
• $1 \mapsto -\infty$

Now, here is a neat puzzle:

Question: What are the elementary symmetric polynomials in this semiring?

Here is a hint:

$S_{2,0} = \min(a_0, a_1)$

$S_{2,1} = \max(a_0, a_1)$

And…

$S_{3,0} = \min(a_0, a_1, a_2)$

$S_{3,1} = \min( \max(a_0, a_1), \max(a_0, a_2), \max(a_1, a_2) )$

$S_{3,2} = \max(a_0, a_1, a_2)$

Give up?  These are the rank functions!

Theorem: Over the min,max semiring, $S_{n,k}$ is the kth element of the sorted sequence $a_0, a_1, ...$

In other words, evaluating the symmetric polynomials over the min/max semiring is the same thing as sorting.  It also suggests a more streamlined way to do the brutal inlining of a sort:

function compare2Rank(a, b) {
var d = Math.min(a[0],a[1]) - Math.min(b[0],b[1]);
if(d) {
return d;
}
return Math.max(a[0],a[1]) - Math.max(b[0],b[1]);
}

Slick!  We went from 25 lines down to just 5.  Unfortunately, this approach is a bit less efficient since it does the comparison between a and b twice, a fact which is reflected in the timing measurements:

• compare2Rank: 495 µs

And we can also easily extend this technique to triples as well:

function compare3Rank(a,b) {
var l0 = Math.min(a[0], a[1])
, m0 = Math.min(b[0], b[1])
, d  = Math.min(l0, a[2]) - Math.min(m0, b[2]);
if(d) {
return d;
}
var l1 = Math.max(a[0], a[1])
, m1 = Math.max(b[0], b[1]);
d = Math.max(l1, a[2]) - Math.max(m1, b[2]);
if(d) {
return d;
}
return Math.min(Math.max(l0, a[2]), l1) - Math.min(Math.max(m0, b[2]), m1);
}
• compare3Rank: 618.71 microseconds

### The Tropical Alternative

Using rank functions to sort the elements turns out to be much simpler than doing a selection sort, and it is way faster than calling the native JS sort on small arrays while avoiding the overflow problems of integer symmetric functions.  However, it is still quite a bit slower than the integer approach.

The final method which I will now describe (and the one which I believe to be best suited to the vagaries of JS) is to compute the symmetric functions over the (max,+), or tropical semiring.  It is basically a semiring over the half-extended real line, $\mathbb{R} \cup \{ - \infty \}$ with the following operators:

• $\oplus \mapsto \max$
• $\otimes \mapsto +$
• $0 \mapsto -\infty$
• $1 \mapsto 0$

There is a cute story for why the tropical semiring has such a colorful name, which is that it was popularized at an algebraic geometry conference in Brazil.  Of course people have known about (max,+) for quite some time before that and most of the pioneering research into it was done by the mathematician Victor P. Maslov in the 50s.  The (max,+) semiring is actually quite interesting and plays an important role in the algebra of distance transforms, numerical optimization and the transition from quantum systems to classical systems.

This is because the (max,+) semiring works in many ways like the large scale limit of the complex numbers under addition and multiplication.  For example, we all know that:

$\log(a b) = \log(a) + \log(b)$

But did you also know that:

$\log(a + b) = \max(\log(a), \log(b)) + O(1)$

This basically a formalization of that time honored engineering philosophy that once your numbers get big enough, you can start to think about them on a log scale.  If you do this systematically, then you eventually will end up doing arithmetic in the (max,+)-semiring.  Masolv asserts that this is essentially what happens in quantum mechanics when we go from small isolated systems to very big things.  A brief overview of this philosophy that has been translated to English can be found here:

Litvinov, G. “The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction” (2006) arXiv:math/0501038

The more detailed explanations of this are unfortunately all store in thick, inscrutable Russian text books (but you can find an English translation if you look hard enough):

Maslov, V. P.; Fedoriuk, M. V. “Semiclassical approximation in quantum mechanics”

But enough of that digression!  Let’s apply (max,+) to the problem at hand of comparing sequences.  If we expand the symmetric polynomials in the (max,+) world, here is what we get:

$S_{2,0} = \max(a_0, a_1)$

$S_{2,1} = a_0 + a_1$

And for $n = 3$:

$S_{3,0} = \max(a_0, a_1, a_2)$

$S_{3,1} = \max(a_0+a_1, a_0+a_2, a_1+a_2)$

$S_{3,2} = a_0+a_1+a_2$

If you stare at this enough, I am sure you can spot the pattern:

Theorem: The elementary symmetric polynomials on the (max,+) semiring are the partial sums of the sorted sequence.

This means that if we want to compute the (max,+) symmetric polynomials, we can do it in $O(n \log(n))$ by sorting and folding.

Working the (max,+) solves most of our woes about overflow, since adding numbers is much less likely to go beyond INT_MAX.  However, we will tweak just one thing: instead of using max, we’ll flip it around and use min so that the values stay small.  Both theoretically and practically, it doesn’t save much but it gives us a maybe a fraction of a bit of extra address space to use for indices.  Here is an implementation for pairs:

function compare2MinP(a, b) {
var d = a[0]+a[1]-b[0]-b[1];
if(d) {
return d;
}
return Math.min(a[0],a[1]) - Math.min(b[0],b[1]);
}

And it clocks in at:

• compare2MinP: 354 µs

Which is a bit slower than the symmetric functions, but still way faster than ranking.  We can also play the same game for $n=3$:

function compare3MinP(a, b) {
var l1 = a[0]+a[1]
, m1 = b[0]+b[1];
d = l1+a[2] - (m1+b[2]);
if(d) {
return d;
}
var l0 = Math.min(a[0], a[1])
, m0 = Math.min(b[0], b[1])
, d  = Math.min(l0, a[2]) - Math.min(m0, b[2]);
if(d) {
return d;
}
return Math.min(l0+a[2], l1) - Math.min(m0+b[2], m1);
}

Which hits:

• compare3MinP: 382 µs

Again, not quite as fast as integers, but pretty good for JavaScript.

Summary

You can get all the code to run these experiments on github:

https://github.com/mikolalysenko/set-compare

And here are the results that I got from running the experiment, all collated for easy reading:

### Dimension = 2

• compareSimple: 5982 µs
• compare2Brutal: 415 µs
• compare2Sym: 352 µs
• compare2Rank: 498 µs
• compare2MinP: 369 µs

### Dimension = 3

• compareSimple: 7737 µs
• compare3Sym: 361 µs
• compare3Sym_opt: 362 µs
• compare3Rank: 612 µs
• compare3MinP: 377 µs

As you can see, the (min,+) solution is nearly as fast as the symmetric version without having the same overflow problems.

I hope you enjoyed reading this as much as I enjoyed tinkering around!  Of course I still don’t know of an optimal way to compare two lists.  As a final puzzle, I leave you with the following:

Question: Is there any method which can test if two unordered sequences are equal in linear time and at most log space?

Frankly, I don’t know the answer to this question and it may very well be impossible.  If you have any thoughts or opinions on this, please leave a comment!

## CommonJS: Why and How

As I said last time, I want to start moving more of my stuff into npm, and so I think perhaps today would be a good time to explain why I think this is the right way to go, and how you can use it in your own JS projects.

The big picture reason that all of this stuff is important, is that there a huge trend these days towards single page browser based applications and JavaScript development in general.  Many factors have contributed to this situation, like the rise of fast JavaScript interpreters; widespread frustration and distrust of web plugins like SilverlightJava Applets and Flash; and the emergence of powerful new APIs like HTML 5 and WebGL.  Today, JavaScript runs on pretty much everything including desktops, cell phones, tablets and even game consoles.  Ignoring it is something you do at your own risk.

## Module Systems

However, while there are many good reasons to use JavaScript, it has a reputation for scaling poorly with project size.  In my opinion, the main cause of this is that JavaScript lacks anything even remotely resembling a coherent module system.  This omission makes it inordinately difficult to apply sensible practices like:

• Hiding implementation details behind interfaces
• Splitting large projects into multiple files
• Reusing functionality from libraries and other code bases

Ignoring these problems isn’t an option.  But despite its flaws, JavaScript deserves at least some credit for being a very flexible language, and with some ingenuity one can work around the lack of modules.  There are of course many ways you can do this, and for me at least as I’ve been learning JS I’ve found the bewildering array of solutions pretty confusing.  So in this post I’m going to summarize how I understand the situation right now, and give my assessment of the various options:

The obvious way to emulate a module in JavaScript would be to use a closure.  There is a fancy name for this in the JS community, and it is called the module design pattern:

var MyModule = (function() {
var exports = {};
//Export foo to the outside world
exports.foo = function() {
// ...
}

//Keep bar private
var bar = { /* ... */ };

//Expose interface to outside world
return exports;
})();

Here MyModule would become the interface for the module and the implementation of the module would be hidden away within the function() block.  This approach, when combined with features like “use strict”; can go a long way to mitigating the danger of the global-namespace free-for-all; and if applied with discipline it basically solves the first problem on our big list.

Unfortunately, it doesn’t really do much to fix our multiple files or importing problems.  Without bringing in third party tools or more complex module loaders, you are basically limited to two ways to do this within a browser:

• Or you can just concatenate all your files in a static build process.

The latter approach is generally more efficient since it requires fewer HTTP requests to load and there are tools like Google’s closure compiler which support this process.  The former approach is more dynamic since you can develop and test without having to do a full rebuild.  Unfortunately, neither method lets you to do selective imports, and they don’t handle external dependencies across libraries very well.

This style of coding is sometimes called monolithic, since the pressure to avoid code dependencies tends to push projects written in this fashion towards monumental proportions.  As an example, look at popular libraries like jQuery, YUI or THREE.js, each of which is packed with loads of features.  Now I don’t mean to especially pick on any of these projects, since if you aren’t going to accept external tooling their approach is actually quite reasonable.  Packaging extra functionality into a library (in the absence of a proper importing system) means you get a lot more features per <script>.  Sufficiently popular libraries (like jQuery) can leverage this using their own content distribution networks and can be cached across multiple sits.  This is great for load times, since users then don’t have to re-download all those libraries that their browser already has cached when they go from site-to-site.

The cost though is pretty obvious.  Because <script> tags have no mechanism to handle dependencies, the default way to include some other library in your project is copy-paste.  Heavy use of this style is a great way to proliferate bugs, since if you include a library with your project via this mechanism, you usually have no good way to automatically update it.

## CommonJS

The CommonJS module system solves all these problems.   The way it works is that instead of running your JavaScript code from a global scope, CommonJS starts out each of your JavaScript files in their own unique module context (just like wrapping it in a closure).  In this scope, it adds two new variables which you can use to import and export other modules: module.exports and require.  The first of these can be used to expose variables to other libraries.  For example, here is how to create a library that exports the variable “foo”:

//In library.js
exports.foo = function() {
//... do stuff
}

And you can import it in another module using the “require” function:

var lib = require("./library.js");
lib.foo();

The addition of this functionality effectively solves the module problem in JavaScript, in what I would consider the most minimally invasive way.  Today, CommonJS is probably the most widely adopted module system in JavaScript, in part due to the support from node.js.  The other major factor which has contributed to its success is the centralized npm package manager, which makes it easy to download, install and configure libraries written using CommonJS.

CommonJS packages should be lean and mean, with as little extraneous functionality as possible.  Compare for example popular modules like async or request to monolithic libraries like underscore.js or jQuery.  An interesting discussion of this philosophy can be found at James Halliday (substack)’s blog:

J. Halliday, “the node.js aesthetic” (2012)

Based on the success of node.js, this approach seems to be paying off for writing server side applications.  But what about browsers?

## CommonJS in the Browser

It turns out that doing this directly isn’t too hard if you are willing to bring in an extra build process.  The most tool to do this is browserify.  When you run browserify, it crawls all your source code starting from some fixed entry point and packages it up into a single .js file, which you can then minify afterwords (for example, using uglify.js).  Doing this reduces the number of http requests required to start up a page and reduces the overall size of your code, thus improving page load times.  The way it works is pretty automatic, just take your main script file and do:

browserify index.js > bundle.js

And then in your web page, you just add a single <script> tag:

<script src="bundle.js"></script>

browserify is compatible with npm and implements most of the node.js standard library (with the exception of some obviously impossible operating system specific stuff, like process.spawn).

### Rapid Development with browserify

The above solution is pretty good when you are looking to package up and distribute your program, but what if you just want to code and debug stuff yourself with minimal hassle? You can automate the bulk of this process using a Makefile, but one could argue reasonably that having to re-run make every time you edit something is an unnecessary distraction.  Fortunately, this is a trivial thing to automate, and so I threw together a tiny ~100-line script that runs browserify in a loop.  You can get it here:

serverify – Continuous development with browserify

To use it, you can just run the file from your project’s root directory and it will server up static HTML on port 8080 from ./www and bundle up your project starting at ./index.js. This behavior can of course be changed by command line options or configuration files.  As a result, you get both the advantages of a proper module system and the ease of continuous deployment for when you are testing stuff!

Other Formats

As it stands today, CommonJS/npm (especially when combined with tools like browserify) is an efficient and  comprehensive solution to the module problem in JavaScript.  In my opinion, it is definitely the best option if you are starting out from scratch.  But I’d be remiss if I didn’t at least mention some of the other stuff that’s out there, and how the situation may change over time:

• Looking ahead to the distant future, ECMA Script 6 has a proposal for some new module semantics that seems promising.  Unfortunately, we aren’t there yet, and no one really knows exactly what the final spec will look and how far out it will be until it gets there.  However, I hope that someday they will finally standardize all of this stuff and converge on a sane, language-level solution to the module problem.  For a balanced discussion of the relative merits of the current proposal,  I’d recommend reading the following post by Isaac Schlueter (current maintainer of node.js):

Schlueter, I. “On ES6 Modules” (2012)

1. Special syntax for specifying module imports (must be done at the top of each script)
2. No tooling required to use, works within browsers out of the box.

These choices are motivated by a preceived need to get modules to work within browsers with no additional tooling.  The standard reference implementation of AMD is the RequireJS library, and you can install it on your sever easily by just copy/pasting the script into your wwwroot.

The fact that AMD uses no tools is both its greatest strength and weakness.  On the one hand, you can use it in a browser right away without having to install anything. On the other hand, AMD is often much slower than CommonJS, since you have to do many async http requests to load all your scripts.  A pretty good critique of the AMD format can be found here:

Dale, T. “AMD is not the answer

I generally agree with his points, though I would also like to add that a bigger problem is that AMD does not have a robust package manager.  As a result, the code written for AMD is scattered across the web and relies on copy-paste code reuse.  There are some efforts under way to fix this, but they are nowhere near as advanced as npm.

## Conclusion

I’ve now given you my summary of the state of modules in JavaScript as I understand it, and hopefully explained my rationale for why I am picking CommonJS for the immediate future.  If you disagree with something I’ve said, have any better references or know of something important that I’ve missed, please leave a comment!

## New Year’s Resolution: Post more stuff on npm

First thing, I’d like to help announce/promote a project which I think is pretty cool (but am not directly involved in) which is voxeljs.  It is being developed by @maxogden and @substack, both of which are very active in the node.js community.  Internally, it uses some of the code from this blog, and it looks super fun to hack around with.  I highly recommend you check it out.

Second, I’d like to say that from here on out I’m going to try to put more of my code on npm so that it can be reused more easily.  I’m going to try to focus on making more frequent, smaller and focused libraries, starting with an npm version of my isosurface code:

https://github.com/mikolalysenko/isosurface

In the future, I hope to start moving more of my stuff on to npm, piece by piece.  Stay tuned for more details.

## Shapes and Coordinates

In a previous post, I talked a bit about solid modeling and discussed at a fairly high level what a solid object is.  This time I’m going to talk in generalities about how one actually represent shapes in practice.  My goal here is to say what these things are, and to introduce a bit of physical language which I think can be helpful when talking about these structures and their representations.

The parallel that I want to draw is that we should think of shape representations in much the same way that we think about coordinates.  A choice of coordinate system does not change the underlying reality of the thing we are talking about, but it can reveal to us different aspects of the nature of whatever it is we are studying.  Being conscious of the biases inherent in any particular point of view allows one to be more flexible when trying to solve some problem.

## Two Types of Coordinates

I believe that there are two general ways to represent shapes: Eulerian and LagrangianI’ll explain what this all means shortly, but as a brief preview here is a small table which contrasts these two approaches:

 Eulerian Lagrangian Coordinate System Global/fixed Local/moving Modeling Paradigm Implicit Parametric Transforms Passive Active

## Eulerian (Implicit)

The first of these methods, or the Eulerian representation, seeks to describe shapes in terms of the space they embedded in.  That is shapes are described by maps from an ambient space into a classifying set:

$f : \mathrm{Ambient}\:\mathrm{space} \to \mathrm{Classifying}\:\mathrm{space}$

More concretely, suppose we are working in n-dimensional Euclidean space $\mathbb{R}^n$.  As we have already seen, it is possible to describe solids using sublevel sets of functions.  At a high level, the way this works is that we have some map $f : \mathbb{R}^n \to S$ where $S$ is some set of labels, for example the real line $\mathbb{R}$.  Then we can define our shape as the preimage of some subset of these labels, $S_0 \subset S$:

$X = f^{-1}(S_0)$

Again, if $S = \mathbb{R}$ and $S_0 = (-\infty, 0]$, then we get the usual definition of a sublevel set:

$X = f^{-1}([-\infty, 0)) = \{ x \in \mathbb{R}^n : f(x) < 0 \} = V_0(f)$

But these are of course somewhat arbitrary choices.  We could for example replace $\mathbb{R}$ with the set $S = \{ 0,1 \}$ of truth values, and we might consider defining our shape as the preimage of 1:

$X = f^{-1}(1)$

Which would be the set of all points in $\mathbb{R}^n$ where f evaluates to 1.  In this situation f is what we could all point membership function.

### Motion in an Eulerian System

Say we have some Eulerian representation of our shape and we want to move it around.  One simple way to think about this is that we have the shape and we simply apply some map $w : \mathbb{R}^n \to \mathbb{R}^n$ to warp it into a new position:

I probably don’t need to convince you too much that motions are extremely useful in animation and physics, but they can also be a practical way to model shapes.  In fact, one simple way to construct more complicated shapes from simpler ones is to deform them along some kind of motion.  Now here is an important question:

Question: How do we apply a motion in an Eulerian coordinate system?

Or more formally, suppose that we have a set $S \subseteq \mathbb{R}^n$ and we have some warp $w : \mathbb{R}^n \to \mathbb{R}^n$.  If $S = f^{-1}( [-\infty, 0)$ is the 0-sublevel set of some potential function $f$, can we find another potential function $h$ such that,

$h^{-1}( [ -\infty, 0 ) ) = w(S)$

Done?  Ok, here is how we can arrive at the answer by pure algebraic manipulation.  Starting with the definition for $S$, we get:

$h^{-1}( . ) = w( f^{-1}( . ) )$

And taking inverses on both sides,

$h( x ) = f( w^{-1}( x ) )$

Which is of course only valid when $w$ is invertible.  So in summary,

In an Eulerian coordinate system, to move a shape $S = V_0(f)$ by $w$, we must precompose $f$ with $w^{-1}$:

$w(S) = V_0(f \circ w^{-1})$

Or said another way:

Eulerian coordinate systems transform passively.

### Pros and Cons

There are many advantages to an Eulerian coordinate system.  For example, point wise operations like Boolean set operations are particularly easy to compute.  (Just taking the min/max is sufficient to evaluate set intersection/union.)  On the other hand, because Eulerian coordinates are passive, operations like rendering or projection (which are secretly a kind of motion) can be much more expensive, since they require solving an inverse problem.  In general, this is as hard as root finding and so it is a non-trivial amount of work.  In 3D, there is also the bigger problem of storage, where representations like voxels can consume an enormous amount of memory.  However, in applications like image processing the advantages often outweigh the costs and so Eulerian representations are still widely used.

## Lagrangian (Parametric)

Lagrangian coordinates are in many ways precisely the dual of Eulerian coordinates.  Instead of representing a shape as a map from an ambient space to some set of labels, they instead map a parametric space into the ambient space:

$p : \mathrm{Parameter}\: \mathrm{space} \to \mathrm{Ambient}\: \mathrm{space}$

The simplest example of a Lagrangian representation is a parametric curve, which typically acts by mapping an interval (like $[0,1]$) into some ambient space, say $\mathbb{R}^n$:

$p(t) = ( x(t), y(t), z(t) )$

But there are of course more sophisticated examples.  Probably the most common place where people bump into Lagrangian coordinates is in the representation of triangulated meshes.  A mesh is typically composed of two pieces of data:

1. A cell complex (typically specified by the vertex-face incidence data)
2. And an embedding fo the cell complex into $\mathbb{R}^3$

The embedding is commonly taken to be piecewise linear over each face.  There are of course many examples of Lagrangian representations:

### Motion in Lagrangian Coordinates

Unlike in an Eulerian coordinate system, it is relatively easy to move an object in a Lagrangian system, since we can just apply a motion directly.  To see why, let $p : P \to \mathbb{R}^n$ be a parameterization of a shape $S$ and let $w : \mathbb{R}^n \to \mathbb{R}^n$ be a motion; then obviously:

$w(S) = w(p(P)) = (w \circ p)(P)$

And so we conclude,

Lagrangian coordinate systems transform actively.

### Pros and Cons

The main advantage to a Lagrangian formulation is that it is much more compact than the equivalent Eulerian form when the shape we are representing is lower dimensional than the ambient space.  This is almost always the case in graphics, where curves and surfaces are of course lower dimension than 3-space itself.  The other big advantage to Lagrangian coordinates is that they transform directly, which explains why they are much preferred in applications like 3D graphics.

On the other hand, Lagrangian coordinates are very bad at performing pointwise tests.  There is also a more subtle problem of validity.  While it is impossible to transform an Eulerian representation by a non-invertible transformation, this is quite easy to do in a Lagrangian coordinate system.  As a result, shapes can become non-manifold or self-intersecting which may break certain invariants and cause algorithms to fail.  This tends to make code that operates on Lagrangian systems much more complex than that which deals with Eulerian representations.