Object Mentor Blog: Imposing the Edges Later

Imposing the Edges Later

Michael Feathers — Sat, 08 Aug 2009 11:12:00 -0500

Here’s what wikipedia has to say about lazy evaluation:

The benefits of lazy evaluation include: performance increases due to avoiding unnecessary calculations, avoiding error conditions in the evaluation of compound expressions, the ability to construct infinite data structures, and the ability to define control structures as regular functions rather than built-in primitives.

It’s all true, but I think it misses a point. One of the nicest things about lazy evaluation is that it enables a different form of abstraction. Take a look at this code. It computes the Mandelbrot Set. There are many ways to code up this algorithm, but the code I just linked to is very typical – it’s a doubly nested loop which iterates over a specific rectangle in the complex plane. It’s hard to see anything that’s awkward about its iteration strategy in a procedural language – the algorithm does require starting and ending points in the plane.

Let’s look at a different way of approaching this. Here’s part of a Mandelbrot Set generator in Haskell:

mandelbrot :: Double -> [[Int]]
mandelbrot incrementSize =
  [[ escapeIterations $ translate offset . translate (x,y) . mandel
    | x <- increments]
    | y <- increments] 
   where increments = [0.0, incrementSize .. ]

window :: (Int, Int) -> (Int, Int) -> [[a]] -> [[a]]
window (x0, x1) (y0, y1) = range y0 y1 . map (range x0 x1)
  where range m n = take (n-m) . drop m

The interesting bit is the mandelbrot function. It contains a doubly-nested list comprehension which, essentially, mimics the behavior of a doubly-nested loop in the procedural algorithm. However, there is one interesting difference. This list comprehension actually represents the mandelbrot computation across a full quarter of the infinite plane – it goes from zero to infinity in x and y. All we have to do is pass it an increment size and it is configured to create that grid. If we want to zoom in to a particular place in the Mandelbrot Set, we can use the window function like this:

-- compute the mandelbrot from (10,10) inclusive to
-- (20,20) exclusive with an increment of 0.05 

window (10, 20) (10, 20) $ mandelbrot 0.05

The trick to this code is in window’s range computation. It takes a starting position, an ending position, and a list and it drops all of the elements of the list before the starting position, without evaluating them, and then it takes, from the front of the new list, end - start elements, giving us a sub-range from the original list. When that elements of that sub-list are evaluated, the computation for each point in the window occurs.

This code yields the computation that we want and the neat thing is, we are really only computing the piece we specified with window. We’ve essentially formed a general computation and imposed the edges later.

Looking back now at the imperative mandelbrot code, it’s easy to see that it was, in a way, mixing two concerns. The code which determined the range was mixed with the code which generated the set. Laziness allows us to separate the two concerns.

Note: There are much more direct ways of computing the Mandelbrot Set in Haskell. Here’s one which is particularly brief.

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