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<chapter>
  <header>
    <copyright>
      <year>2003</year><year>2016</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
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      Licensed under the Apache License, Version 2.0 (the "License");
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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>This section starts with a simple example, showing a generator and a filter:</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 is read as follows: The list of X such that X is taken from the list
        <c>[1,2,a,...]</c> and X is greater than 3.</p>
    <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, <c>integer(X)</c>, can be added 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> that 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> that are greater than or equal to <c>Pivot</c>.</p>
      <p>A list sorted as follows:</p>
      <list type="bulleted">
       <item>The first element in the list is isolated
       and the list is split into two sublists.</item>
       <item>The first sublist contains
      all elements that are smaller than the first element in
      the list.</item>
      <item>The second sublist contains all elements that are greater
      than, or equal to, the first element in the list.</item>
       <item>Then the sublists are sorted and the results are combined.</item>
     </list>
  </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>This takes <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 that occur in list
      comprehensions are as follows:</p>
    <list type="bulleted">
      <item>All variables that occur in a generator pattern are
       assumed to be "fresh" variables.</item>
      <item>Any variables that are defined before the list
       comprehension, and that are used in filters, have the values
       they had before the list comprehension.</item>
      <item>Variables cannot be exported from a list comprehension.</item>
    </list>
    <p>As an example of these rules, suppose you want to write
      the function <c>select</c>, which selects certain elements from
      a list of tuples. Suppose you 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 gives the following diagnostic:</p>
    <code type="none">
./FileName.erl:Line: Warning: variable 'X' shadowed in generate</code>
    <p>This diagnostic warns that the variable <c>X</c> in
      the pattern is not the same as the variable <c>X</c>
      that occurs in the function head.</p>
    <p>Evaluating <c>select</c> gives the following result:</p>
    <pre>
> <input>select(b,[{a,1},{b,2},{c,3},{b,7}]).</input>
[1,2,3,7]</pre>
    <p>This is not the wanted result. To achieve the desired
      effect, <c>select</c> must be written 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.</p>
      <p>This now works as expected:</p>
    <pre>
> <input>select(b,[{a,1},{b,2},{c,3},{b,7}]).</input>
[2,7]</pre>
    <p>A consequence of the rules for importing variables into a
      list comprehensions is that certain pattern matching operations
      must be moved into the filters and cannot be written directly
      in the generators.</p>
      <p>To illustrate this, do <em>not</em> 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>

