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    <p>I have finally figured this out after much messing about:</p> <pre><code>scala&gt; def findMatches(divs: List[Int], nums: List[Int]): List[(Int, Int)] = { | (divs.traverse[({type l[a] = State[List[Int], a]})#l, Option[(Int, Int)]]( div =&gt; | state { (rem: List[Int]) =&gt; | rem.find(_ % div == 0).map(n =&gt; rem.filterNot(_ == n) -&gt; Some(div -&gt; n)).getOrElse(rem -&gt; none[(Int, Int)]) | } | ) ! nums).flatten | } findMatches: (divs: List[Int], nums: List[Int])List[(Int, Int)] </code></pre> <p>I think I'll be looking at Eric's answer for more insight into what is actually going on, though.</p> <hr/> <h3>Iteration #2</h3> <p>Exploring Eric's answer using scalaz6</p> <pre><code>scala&gt; def findMatches2(divs: List[Int], nums: List[Int]): List[(Int, Int)] = { | case class S(matches: List[(Int, Int)], remaining: List[Int]) | val initialState = S(nil[(Int, Int)], nums) | def find(div: Int, s: S) = { | val newState = s.remaining.find(_ % div == 0).map { (n: Int) =&gt; | S(s.matches :+ div -&gt; n, s.remaining filterNot (_ == n)) | }.getOrElse(s) | newState -&gt; newState.matches | } | val findDivs = (div: Int) =&gt; state((s: S) =&gt; find(div, s)) | (divs.traverse[({type l[a]=State[S, a]})#l, List[(Int, Int)]](findDivs) ! initialState).join | } findMatches2: (divs: List[Int], nums: List[Int])List[(Int, Int)] scala&gt; findMatches2(List(2, 3, 4), List(1, 6, 7, 8, 9)) res11: List[(Int, Int)] = List((2,6), (2,6), (3,9), (2,6), (3,9), (4,8)) </code></pre> <p>The <code>join</code> on the <code>List[List[(Int, Int)]]</code> at the end is causing grief. Instead we can replace the last line with:</p> <pre><code>(divs.traverse[({type l[a]=State[S, a]})#l, List[(Int, Int)]](findDivs) ~&gt; initialState).matches </code></pre> <hr/> <h3>Iteration #3</h3> <p>In fact, you can do away with the extra output of a state computation altogether and simplify even further:</p> <pre><code>scala&gt; def findMatches2(divs: List[Int], nums: List[Int]): List[(Int, Int)] = { | case class S(matches: List[(Int, Int)], remaining: List[Int]) | def find(div: Int, s: S) = | s.remaining.find(_ % div == 0).map( n =&gt; S(s.matches :+ div -&gt; n, s.remaining filterNot (_ == n)) ).getOrElse(s) -&gt; () | (divs.traverse[({type l[a]=State[S, a]})#l, Unit](div =&gt; state((s: S) =&gt; find(div, s))) ~&gt; S(nil[(Int, Int)], nums)).matches | } findMatches2: (divs: List[Int], nums: List[Int])List[(Int, Int)] </code></pre> <h3>Iteration #4</h3> <p><code>modify</code> described above by <em>Apocalisp</em> is also available in scalaz6 and removes the need to explicitly supply the <code>(S, ())</code> pair (although you do need <code>Unit</code> in the type lambda):</p> <pre><code>scala&gt; def findMatches2(divs: List[Int], nums: List[Int]): List[(Int, Int)] = { | case class S(matches: List[(Int, Int)], remaining: List[Int]) | def find(div: Int) = modify( (s: S) =&gt; | s.remaining.find(_ % div == 0).map( n =&gt; S(s.matches :+ div -&gt; n, s.remaining filterNot (_ == n)) ).getOrElse(s)) | (divs.traverse[({type l[a]=State[S, a]})#l, Unit](div =&gt; state(s =&gt; find(div)(s))) ~&gt; S(nil, nums)).matches | } findMatches2: (divs: List[Int], nums: List[Int])List[(Int, Int)] scala&gt; findMatches2(List(2, 3, 4), List(1, 6, 7, 8, 9)) res0: List[(Int, Int)] = List((2,6), (3,9), (4,8)) </code></pre>
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