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    <p>It is a common myth that Haskell doesn't offer mutation at all. In reality, it offers a very special kind of mutation: a value can mutate exactly once, from un-evaluated to evaluated. The art of taking advantage of this special kind of mutation is called <a href="http://www.haskell.org/haskellwiki/Tying_the_knot" rel="noreferrer">tying the knot</a>. We will start with a data structure just like your one from C++:</p> <pre><code>data Vector -- held abstract data Point = Point { position :: Vector , v, w :: Double , neighbors :: [Point] } </code></pre> <p>Now, what we're going to do is build an <code>Array Point</code> whose <code>neighbors</code> contain pointers to other elements within the same array. The key features of <code>Array</code> in the following code are that it's spine-lazy (it doesn't force its elements too soon) and has fast random-access; you can substitute your favorite alternate data structure with these properties if you prefer.</p> <p>There's lots of choices for the interface of the neighbor-finding function. For concreteness and to make my own job simple, I will assume you have a function that takes a <code>Vector</code> and a list of <code>Vectors</code> and gives the indices of neighbors.</p> <pre><code>findNeighbors :: Vector -&gt; [Vector] -&gt; [Int] findNeighbors = undefined </code></pre> <p>Let's also put in place some types for <code>computeV</code> and <code>computeW</code>. For the nonce, we will ask that <code>computeV</code> live up to the informal contract you stated, namely, that it can look at the <code>position</code> and <code>neighbors</code> fields of any <code>Point</code>, but not the <code>v</code> or <code>w</code> fields. (Similarly, <code>computeW</code> may look at anything but the <code>w</code> fields of any <code>Point</code> it can get its hands on.) It is actually possible to enforce this at the type level without too many gymnastics, but for now let's skip that.</p> <pre><code>computeV, computeW :: Point -&gt; Double (computeV, computeW) = undefined </code></pre> <p>Now we are ready to build our (labeled) in-memory graph.</p> <pre><code>buildGraph :: [Vector] -&gt; Array Int Point buildGraph vs = answer where answer = listArray (0, length vs-1) [point pos | pos &lt;- vs] point pos = this where this = Point { position = pos , v = computeV this , w = computeW this , neighbors = map (answer!) (findNeighbors pos vs) } </code></pre> <p>And that's it, really. Now you can write your</p> <pre><code>newPositions :: Point -&gt; [Vector] newPositions = undefined </code></pre> <p>where <code>newPositions</code> is perfectly free to inspect any of the fields of the <code>Point</code> it's handed, and put all the functions together:</p> <pre><code>update :: [Vector] -&gt; [Vector] update = newPositions &lt;=&lt; elems . buildGraph </code></pre> <p>edit: ...to explain the "special kind of mutation" comment at the beginning: during evaluation, you can expect when you demand the <code>w</code> field of a <code>Point</code> that things will happen in this order: <code>computeW</code> will force the <code>v</code> field; then <code>computeV</code> will force the <code>neighbors</code> field; then the <code>neighbors</code> field will mutate from unevaluated to evaluated; then the <code>v</code> field will mutate from unevaluated to evaluated; then the <code>w</code> field will mutate from unevaluated to evaluated. These last three steps look very similar to the three mutation steps of your C++ algorithm!</p> <p>double edit: I decided I wanted to see this thing run, so I instantiated all the things held abstract above with dummy implementations. I also wanted to see it evaluate things only once, since I wasn't even sure I'd done it right! So I threw in some <code>trace</code> calls. Here's a complete file:</p> <pre><code>import Control.Monad import Data.Array import Debug.Trace announce s (Vector pos) = trace $ "computing " ++ s ++ " for position " ++ show pos data Vector = Vector Double deriving Show data Point = Point { position :: Vector , v, w :: Double , neighbors :: [Point] } findNeighbors :: Vector -&gt; [Vector] -&gt; [Int] findNeighbors (Vector n) vs = [i | (i, Vector n') &lt;- zip [0..] vs, abs (n - n') &lt; 1] computeV, computeW :: Point -&gt; Double computeV (Point pos _ _ neighbors) = sum [n | Point { position = Vector n } &lt;- neighbors] computeW (Point pos v _ neighbors) = sum [v | Point { v = v } &lt;- neighbors] buildGraph :: [Vector] -&gt; Array Int Point buildGraph vs = answer where answer = listArray (0, length vs-1) [point pos | pos &lt;- vs] point pos = this where { this = Point { position = announce "position" pos $ pos , v = announce "v" pos $ computeV this , w = announce "w" pos $ computeW this , neighbors = announce "neighbors" pos $ map (answer!) (findNeighbors pos vs) } } newPositions :: Point -&gt; [Vector] newPositions (Point { position = Vector n, v = v, w = w }) = [Vector (n*v), Vector w] update :: [Vector] -&gt; [Vector] update = newPositions &lt;=&lt; elems . buildGraph </code></pre> <p>and a run in ghci:</p> <pre><code>*Main&gt; length . show . update . map Vector $ [0, 0.25, 0.75, 1.25, 35] computing position for position 0.0 computing v for position 0.0 computing neighbors for position 0.0 computing position for position 0.25 computing position for position 0.75 computing w for position 0.0 computing v for position 0.25 computing neighbors for position 0.25 computing v for position 0.75 computing neighbors for position 0.75 computing position for position 1.25 computing w for position 0.25 computing w for position 0.75 computing v for position 1.25 computing neighbors for position 1.25 computing w for position 1.25 computing position for position 35.0 computing v for position 35.0 computing neighbors for position 35.0 computing w for position 35.0 123 </code></pre> <p>As you can see, each field is computed at most once for each position.</p>
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    1. USDaniel Wagner
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    1. COVery interesting and simple solution. The kind of thing I was hoping for. Thank you very much.
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      1. USCharles Welton
 

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