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<chapter>
  <header>
    <copyright>
      <year>2003</year><year>2009</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      The contents of this file are subject to the Erlang Public License,
      Version 1.1, (the "License"); you may not use this file except in
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    <title>List Comprehensions</title>
    <prepared></prepared>
    <docno></docno>
    <date></date>
    <rev></rev>
    <file>list_comprehensions.xml</file>
  </header>

  <section>
    <title>Simple Examples</title>
    <p>We start with a simple example:</p>
    <pre>
> <input>[X || X &lt;- [1,2,a,3,4,b,5,6], X > 3].</input>
[a,4,b,5,6]</pre>
    <p>This should be read as follows:</p>
    <quote>
      <p>The list of X such that X is taken from the list
        <c>[1,2,a,...]</c> and X is greater than 3.</p>
    </quote>
    <p>The notation <c><![CDATA[X <- [1,2,a,...]]]></c> is a generator and
      the expression <c>X > 3</c> is a filter.</p>
    <p>An additional filter can be added in order to restrict
      the result to integers:</p>
    <pre>
> <input>[X || X &lt;- [1,2,a,3,4,b,5,6], integer(X), X > 3].</input>
[4,5,6]</pre>
    <p>Generators can be combined. For example, the Cartesian product
      of two lists can be written as follows:</p>
    <pre>
> <input>[{X, Y} || X &lt;- [1,2,3], Y &lt;- [a,b]].</input>
[{1,a},{1,b},{2,a},{2,b},{3,a},{3,b}]</pre>
  </section>

  <section>
    <title>Quick Sort</title>
    <p>The well known quick sort routine can be written as follows:</p>
    <code type="none"><![CDATA[
sort([Pivot|T]) ->
    sort([ X || X <- T, X < Pivot]) ++
    [Pivot] ++
    sort([ X || X <- T, X >= Pivot]);
sort([]) -> [].]]></code>
    <p>The expression <c><![CDATA[[X || X <- T, X < Pivot]]]></c> is the list of
      all elements in <c>T</c>, which are less than <c>Pivot</c>.</p>
    <p><c><![CDATA[[X || X <- T, X >= Pivot]]]></c> is the list of all elements in
      <c>T</c>, which are greater or equal to <c>Pivot</c>.</p>
    <p>To sort a list, we isolate the first element in the list and
      split the list into two sub-lists. The first sub-list contains
      all elements which are smaller than the first element in
      the list, the second contains all elements which are greater
      than or equal to the first element in the list. We then sort
      the sub-lists and combine the results.</p>
  </section>

  <section>
    <title>Permutations</title>
    <p>The following example generates all permutations of
      the elements in a list:</p>
    <code type="none"><![CDATA[
perms([]) -> [[]];
perms(L)  -> [[H|T] || H <- L, T <- perms(L--[H])].]]></code>
    <p>We take take <c>H</c> from <c>L</c> in all possible ways.
      The result is the set of all lists <c>[H|T]</c>, where <c>T</c>
      is the set of all possible permutations of <c>L</c> with
      <c>H</c> removed.</p>
    <pre>
> <input>perms([b,u,g]).</input>
[[b,u,g],[b,g,u],[u,b,g],[u,g,b],[g,b,u],[g,u,b]]</pre>
  </section>

  <section>
    <title>Pythagorean Triplets</title>
    <p>Pythagorean triplets are sets of integers <c>{A,B,C}</c> such
      that <c>A**2 + B**2 = C**2</c>.</p>
    <p>The function <c>pyth(N)</c> generates a list of all integers
      <c>{A,B,C}</c> such that <c>A**2 + B**2 = C**2</c> and where
      the sum of the sides is equal to or less than <c>N</c>.</p>
    <code type="none"><![CDATA[
pyth(N) ->
    [ {A,B,C} ||
        A <- lists:seq(1,N),
        B <- lists:seq(1,N),
        C <- lists:seq(1,N),
        A+B+C =< N,
        A*A+B*B == C*C 
    ].]]></code>
    <pre>
> <input>pyth(3).</input>
[].
> <input>pyth(11).</input>
[].
> <input>pyth(12).</input>
[{3,4,5},{4,3,5}]
> <input>pyth(50).</input>
[{3,4,5},
 {4,3,5},
 {5,12,13},
 {6,8,10},
 {8,6,10},
 {8,15,17},
 {9,12,15},
 {12,5,13},
 {12,9,15},
 {12,16,20},
 {15,8,17},
 {16,12,20}]</pre>
    <p>The following code reduces the search space and is more
      efficient:</p>
    <code type="none"><![CDATA[
pyth1(N) ->
   [{A,B,C} ||
       A <- lists:seq(1,N-2),
       B <- lists:seq(A+1,N-1),
       C <- lists:seq(B+1,N),
       A+B+C =< N,
       A*A+B*B == C*C ].]]></code>
  </section>

  <section>
    <title>Simplifications with List Comprehensions</title>
    <p>As an example, list comprehensions can be used to simplify some
      of the functions in <c>lists.erl</c>:</p>
    <code type="none"><![CDATA[
append(L)   ->  [X || L1 <- L, X <- L1].
map(Fun, L) -> [Fun(X) || X <- L].
filter(Pred, L) -> [X || X <- L, Pred(X)].]]></code>
  </section>

  <section>
    <title>Variable Bindings in List Comprehensions</title>
    <p>The scope rules for variables which occur in list
      comprehensions are as follows:</p>
    <list type="bulleted">
      <item>all variables which occur in a generator pattern are
       assumed to be "fresh" variables</item>
      <item>any variables which are defined before the list
       comprehension and which are used in filters have the values
       they had before the list comprehension</item>
      <item>no variables may be exported from a list comprehension.</item>
    </list>
    <p>As an example of these rules, suppose we want to write
      the function <c>select</c>, which selects certain elements from
      a list of tuples. We might write
      <c><![CDATA[select(X, L) ->  [Y || {X, Y} <- L].]]></c> with the intention
      of extracting all tuples from <c>L</c> where the first item is
      <c>X</c>.</p>
    <p>Compiling this yields the following diagnostic:</p>
    <code type="none">
./FileName.erl:Line: Warning: variable 'X' shadowed in generate</code>
    <p>This diagnostic warns us that the variable <c>X</c> in
      the pattern is not the same variable as the variable <c>X</c>
      which occurs in the function head.</p>
    <p>Evaluating <c>select</c> yields the following result:</p>
    <pre>
> <input>select(b,[{a,1},{b,2},{c,3},{b,7}]).</input>
[1,2,3,7]</pre>
    <p>This result is not what we wanted. To achieve the desired
      effect we must write <c>select</c> as follows:</p>
    <code type="none"><![CDATA[
select(X, L) ->  [Y || {X1, Y} <- L, X == X1].]]></code>
    <p>The generator now contains unbound variables and the test has
      been moved into the filter. This now works as expected:</p>
    <pre>
> <input>select(b,[{a,1},{b,2},{c,3},{b,7}]).</input>
[2,7]</pre>
    <p>One consequence of the rules for importing variables into a
      list comprehensions is that certain pattern matching operations
      have to be moved into the filters and cannot be written directly
      in the generators. To illustrate this, do not write as follows:</p>
    <code type="none"><![CDATA[
f(...) ->
    Y = ...
    [ Expression || PatternInvolving Y  <- Expr, ...]
    ...]]></code>
    <p>Instead, write as follows:</p>
    <code type="none"><![CDATA[
f(...) ->
    Y = ...
    [ Expression || PatternInvolving Y1  <- Expr, Y == Y1, ...]
    ...]]></code>
  </section>
</chapter>

