This is the first time in my career that I’ve used JavaScript extensively, so I’m trying to learn some best practices. A few people have recommended Crockford’s [book: JavaScript: The Good Parts], so I picked it up. While skimming through it looking for something else, I ran into his description (on page 44) of using memoization (essentially, caching) to speed up the computation of certain functions.

The example people always use here — and Crockford is no exception — is the Fibonacci sequence. The shortest way to implement it is recursively. As Crockford points out, the recursive solution doesn’t scale: the number of recursive calls necessary to compute the nth Fibonacci number is proportional to the nth power of the golden ratio. (From what I can see, the constant of proportionality converges very rapidly to about 1.11803. That number must have some special name.) I’ve coded both versions up in Python; the recursive version keeps track of how many function calls it had to make.

So then Crockford’s solution, and the solution written in lots of places ([book: Higher-Order Perl], for instance) is to use memoization: cache the results of fib(n) for smaller n, then use the cached results rather than computing those values anew when you need them later on.

This isn’t really a solution, though, as my friend Seth pointed out to me some months ago. It’s certainly clever, and languages like Perl or JavaScript make it very easy. In Perl, you can automatically memoize any of your functions with the Memoize module: just do

use Memoize;

memoize(‘fib’);

and voilà: all of your calls to `fib()` will be memoized from then on. Pretty cool, really. Manipulating the symbol table is kind of neat.

But this has only temporarily disguised the problem. Let’s look at the time and space requirements for all three of our `fib(n)` implementations:

__Recursive, non-memoized__: exponentially many function calls, hence exponentially many stack frames, hence exponential memory usage. Linear running time.

__Recursive, memoized__: linear memory usage (one cache entry for every [math: i], for [math: i] less than or equal to [math: n]). Linear running time.

__Iterative, non-memoized__: constant memory usage (must keep the [math: n]th, [math: (n-1)]st, and [math: (n-2)]th values of the sequence in memory at all times, but that’s it), linear running time.

By using some language cleverness, you’ve made the problem harder than you need to: memoization increases your memory usage from constant to linear, and does nothing to your running time. By thinking harder about the problem, you can improve both performance aspects and not need any clever language business.

Seth has told me for quite a long time that recursion and higher-order programming (closures and so forth) are interesting but make debugging a lot harder. His contention would probably be that you can often replace a clever recursive solution with a non-clever, easier-to-debug, better-performing one.

That said, at least some higher-order-programming tools make my life easier. In Python, I love list comprehensions:

print [x**2 for x in xrange(1,11) if (x**2 % 2 == 0)]

or the older-school but perhaps more familiar `map()` and `filter()`:

print filter(lambda x : x % 2 == 0, map(lambda x : x**2, xrange(1,11)))

(because I favor readability over concision, in practice I would expand this into several lines: a `map()` line, and a `filter()` line, and so forth)

As I’ve mentioned, I lament the absence of first-class functions in Java. Then again, I’ve seen enough people shoot themselves (and me) in the foot with unreadable Perl or Python, using all the cleverness that the languages provide to them, that I think I’d be okay with a less expressive language that forces programmers to be really boring.

Of course, all of these are sub-optimal. Using matrix exponentiation you can compute fibonacci in constant space and logarithmic running time.

My experience with functional programming is that it may make debugging harder, but it also makes it far less necessary. Surely it’s better to write a correct program the first time than to spend hours chasing bugs… It was several years after I started functional programming that I realised that I hadn’t been using a debugger, and didn’t miss it…

(Debuggers do exist, and functional programming makes some techniques easier, like stepping backwards in time, but I’ve never felt the need to learn them.)

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It seems that you’re only considering the performance of one invocation of fib. Someone using the recursive memoized version is probably considering the average performance of fib after several invocations.

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