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<chapter>
  <header>
    <copyright>
      <year>2021</year><year>2021</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      Licensed under the Apache License, Version 2.0 (the "License");
      you may not use this file except in compliance with the License.
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          http://www.apache.org/licenses/LICENSE-2.0

      Unless required by applicable law or agreed to in writing, software
      distributed under the License is distributed on an "AS IS" BASIS,
      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
      See the License for the specific language governing permissions and
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    <title>Maps</title>
    <prepared>Bjorn Gustavsson</prepared>
    <docno></docno>
    <date>2021-07-01</date>
    <rev></rev>
    <file>maps.xml</file>
  </header>

  <p>This guide to using maps efficiently starts with a brief section
  on the choice between records or maps, followed by three sections
  giving concrete (but brief) advice on using maps as an alternative to
  records, as dictionaries, and as sets. The remaining sections dig
  deeper, looking at how maps are implemented, the map syntax, and
  finally the functions in the <seeerl
  marker="stdlib:maps">maps</seeerl> module.</p>

  <p>Terminology used in this chapter:</p>
  <list type="bulleted">
    <item>A map with at most 32 elements will informally be called a
    <em>small map</em>.</item>
    <item>A map with more than 32 elements will informally be called a
    <em>large map</em>.</item>
  </list>

  <section>
    <title>Maps or Records?</title>
    <p>If the advice in this chapter is followed, the performance of
    records compared to using small maps instead of records is
    expected to be similar. Therefore, the choice between records and
    maps should be based on the desired properties of the data
    structure and not performance.</p>

    <p>The advantages of records compared to maps are:</p>

    <list type="bulleted">
      <item>If the name of a record field is misspelled, there will be
      a compilation error. If a map key is misspelled, the compiler
      will give no warning and program will fail in some way when it
      is run.</item>

      <item>Records will use slightly less memory than maps, and
      performance is expected to be <em>slightly</em> better than maps
      in most circumstances.</item>
    </list>

    <p>The disadvantage of records compared to maps is that if a new
    field is added to a record, all code that uses that record must be
    recompiled. Because of that, it is recommended to only use records
    within a unit of code that can easily be recompiled all at once, for
    example within a single application or single module.</p>
  </section>

  <section>
    <title>Using Maps as an Alternative to Records</title>

    <list type="bulleted">
      <item><p>Use the map syntax instead of the functions in
      the <seeerl marker="stdlib:maps">maps</seeerl> module.</p></item>

      <item><p>Avoid having more than 32 elements in the
      map. As soon as there are more than 32 elements in the map,
      it will require more memory and keys can no longer be shared
      with other instances of the map.</p></item>

      <item><p>When creating a new map, always create it with all keys
      that will ever be used. To maximize sharing of keys (thus
      minimizing memory use), create a single function that constructs
      the map using the map syntax and always use it.</p></item>

      <item><p>Always update the map using the
      <c>:=</c> operator (that is, requiring that an element with
      that key already exists). The <c>:=</c> operator is slightly
      more efficient, and it helps catching mispellings of
      keys.</p></item>

      <item><p>Whenever possible, match multiple map elements at
      once.</p></item>

      <item><p>Whenever possible, update multiple map elements at
      once.</p></item>

      <item><p>Avoid default values and the <seemfa
      marker="stdlib:maps#get/3">maps:get/3</seemfa> function. If
      there are default values, sharing of keys between different
      instances of the map will be less effective, and it is not
      possible to match multiple elements having default values in one
      go. The <c>maps:get/3</c> function is <seeguide
      marker="#maps_get_3">implemented in Erlang</seeguide>, making it
      less efficient than <c>maps:get/2</c> or the map matching
      syntax.</p></item>

      <item><p>To avoid having to deal with a map that may lack some keys,
      <seemfa marker="stdlib:maps#merge/2">maps:merge/2</seemfa>
      can efficiently add multiple default values. For example:</p>
      <code type="erl"><![CDATA[
      DefaultMap = #{shoe_size => 42, editor => emacs},
      MapWithDefaultsApplied = maps:merge(DefaultMap, OtherMap)]]></code>
      </item>
    </list>
  </section>

  <section>
    <title>Using Maps as Dictionaries</title>

    <p>Using a map as a dictionary implies the following usage pattern:</p>
    <list type="bulleted">
      <item><p>Keys are usually variables not known at compile-time.</p></item>
      <item><p>There can be any number of elements in the map.</p></item>
      <item><p>Usually, no more than one element is looked up or
      updated at once.</p></item>
    </list>

    <p>Given that usage pattern, the difference in performance between
    using the map syntax and the maps module is usually
    small. Therefore, which one to use is mostly a matter of
    taste.</p>

    <p>Maps are usually the most efficient dictionary data structure,
    with a few exceptions:</p>

    <list type="bulleted">
      <item><p>If it is necessary to frequently convert a
      dictionary to a sorted list, or from a sorted list to a
      dictionary, using <seeerl marker="stdlib:gb_trees">gb_trees</seeerl>
      can be a better choice.</p></item>

      <item><p>If all keys are non-negative integers, the <seeerl
      marker="stdlib:array">array</seeerl> module can be a better
      choice.</p></item>
    </list>
  </section>

  <section>
    <title>Using Maps as Sets</title>

    <p>Starting in OTP 24, the <seeerl marker="stdlib:sets">sets</seeerl> module
    has an option to represent sets as maps. Examples:</p>

    <code type="erl"><![CDATA[
1> sets:new([{version,2}]).
#{}
2> sets:from_list([x,y,z], [{version,2}]).
#{x => [],y => [],z => []}]]></code>

    <p><c>sets</c> backed by maps is generally the most efficient set
    representation, with a few possible exceptions:</p>

    <list type="bulleted">
      <item><p><seemfa
      marker="stdlib:ordsets#intersection/2">ordsets:intersection/2</seemfa>
      can be more efficient than <seemfa
      marker="stdlib:sets#intersection/2">sets:intersection/2</seemfa>.
      If the intersection operation is frequently used and operations
      that operate on a single element in a set (such as
      <c>is_element/2</c>) are avoided, <seeerl
      marker="stdlib:ordsets">ordsets</seeerl> can be a better
      choice than <seeerl marker="stdlib:sets">sets</seeerl>.</p></item>

      <item><p>If the intersection operation is frequently used and operations
      that operate on a single element in a set (such as
      <c>is_element/2</c>) must also be efficient, <seeerl
      marker="stdlib:gb_sets">gb_sets</seeerl> can potentially be a
      better choice than <seeerl marker="stdlib:sets">sets</seeerl>.</p></item>

      <item><p>If the elements of the set are integers in a fairly
      compact range, the set can be represented as an integer where
      each bit represents an element in the set. The union operation
      is performed by <c>bor</c> and the intersection operation by
      <c>band</c>.</p></item>
    </list>
  </section>

  <section>
    <title>How Maps are Implemented</title>
    <p>Internally, maps have two distinct representations depending on
    the number of elements in the map. The representation changes
    when a map grows beyond 32 elements, or when it shrinks to 32
    elements or less.</p>

    <list type="bulleted">
      <item>A map with at most 32 elements has a compact
      representation, making it suitable as an alternative to
      records.</item>
      <item>A map with more than 32 elements is represented as a tree
      that can be efficiently searched and updated regardless of how
      many elements there are.</item>
    </list>

    <section>
      <title>How Small Maps are Implemented</title>
      <p>A small map looks like this inside the runtime system:</p>
      <table align="left">
        <row>
          <cell align="center"><c>FLATMAP</c></cell>
          <cell align="center"><em>N</em></cell>
          <cell align="center"><em>Keys</em></cell>
          <cell align="center"><em>Value1</em></cell>
          <cell align="center"><em>. . .</em></cell>
          <cell align="center"><em>ValueN</em></cell>
        </row>
        <tcaption>The representation of a small map</tcaption>
      </table>

      <taglist>
        <tag><c>FLATMAP</c></tag>
        <item>The tag for a small map (called <em>flat map</em> in
        the source code for the runtime system).</item>

        <tag>N</tag>
        <item>The number of elements in the map.</item>

        <tag>Keys</tag>
        <item>A tuple with keys of the map: <c>{Key1,...,KeyN}</c>. The
        keys are sorted.</item>

        <tag>Value1</tag>
        <item>The value corresponding to the first key in the key tuple.</item>

        <tag>ValueN</tag>
        <item>The value corresponding to the last key in the key tuple.</item>
      </taglist>

      <p>As an example, let us look at how the map <c>#{a => foo, z => bar}</c>
      is represented:</p>

      <table align="left">
        <row>
          <cell align="center"><c>FLATMAP</c></cell>
          <cell align="center"><em>2</em></cell>
          <cell align="center"><em>{a,z}</em></cell>
          <cell align="center"><em>foo</em></cell>
          <cell align="center"><em>bar</em></cell>
        </row>
        <tcaption>#{a => foo, z => bar}</tcaption>
      </table>

      <p>Let us update the map: <c>M#{q => baz}</c>. The map now looks
      like this:</p>

      <table align="left">
        <row>
          <cell align="center"><c>FLATMAP</c></cell>
          <cell align="center"><em>3</em></cell>
          <cell align="center"><em>{a,q,z}</em></cell>
          <cell align="center"><em>foo</em></cell>
          <cell align="center"><em>baz</em></cell>
          <cell align="center"><em>bar</em></cell>
        </row>
        <tcaption>#{a => foo, q => baz, z => bar}</tcaption>
      </table>

      <p>Finally, change the value of one element: <c>M#{z :=
      bird}</c>. The map now looks like this:</p>

      <table align="left">
        <row>
          <cell align="center"><c>FLATMAP</c></cell>
          <cell align="center"><em>3</em></cell>
          <cell align="center"><em>{a,q,z}</em></cell>
          <cell align="center"><em>foo</em></cell>
          <cell align="center"><em>baz</em></cell>
          <cell align="center"><em>bird</em></cell>
        </row>
        <tcaption>#{a => foo, q => baz, z => bird}</tcaption>
      </table>

      <p>When the value for an existing key is updated, the key tuple is not
      updated, allowing the key tuple to be shared with other instances of the
      map that have the same keys. In fact, the key tuple can be shared between all
      maps with the same keys with some care. To arrange that, define a function that
      returns a map. For example:</p>

      <code type="erl"><![CDATA[
new() ->
    #{a => default, b => default, c => default}.]]></code>

      <p>Defined like this, the key tuple <c>{a,b,c}</c> will be a
      global literal. To ensure that the key tuple is shared
      when creating an instance of the map, always call
      <c>new()</c> and modify the returned map:</p>

      <code type="erl"><![CDATA[
    (SOME_MODULE:new())#{a := 42}.]]></code>

      <p>Using the map syntax with small maps is particularly
      efficient. As long as the keys are known at compile-time, the
      map is updated in one go, making the time to update a map
      essentially constant regardless of the number of keys
      updated. The same goes for matching. (When the keys are
      variables, one or more of the keys could be identical, so the
      operations need to be performed sequentially from left to
      right.)</p>

      <p>The memory size for a small map is the size of all keys and values
      plus 5 words. See <seeguide marker="advanced#memory">Advanced</seeguide>
      for more information about memory sizes.</p>
    </section>

    <section>
      <title>How Large Maps are Implemented</title>

      <p>A map with more than 32 elements is implemented as a <url
      href="https://en.wikipedia.org/wiki/Hash_array_mapped_trie">Hash
      array mapped trie (HAMT)</url>. A large map can be efficiently
      searched and updated regardless of the number of elements in the
      map.</p>

      <p>There is less performance to be gained by matching or
      updating multiple elements using the map syntax on a large map
      compared to a small map. The execution time is roughly proportional
      to the number of elements matched or updated.</p>

      <p>The storage overhead for a large map is higher than for a
      small map. For a large map, the extra number of words besides
      the keys and values is roughly proportional to the number of
      elements. For a map with 33 elements the overhead is at
      least 53 heap words according to the formula in <seeguide
      marker="advanced#memory">Advanced</seeguide> (compared to 5
      extra words for a small map regardless of the number of
      elements).</p>

      <p>When a large map is updated, the updated map and the original
      map will share common parts of the HAMT, but sharing will never
      be as effective as the best possible sharing of the key tuple
      for small maps.</p>

      <p>Therefore, if maps are used instead of records and it is
      expected that many instances of the map will be created, it is
      more efficient from a memory standpoint to avoid using large
      maps (for example, by grouping related map elements into sub
      maps to reduce the number of elements).</p>
    </section>
  </section>

  <section>
    <title>Using the Map Syntax</title>

    <p>Using the map syntax is usually slightly more efficient than
    using the corresponding function in the <seeerl
    marker="stdlib:maps">maps</seeerl> module.</p>

    <p>The gain in efficiency for the map syntax is more noticeable
    for the following operations that can only be achieved using the
    map syntax:</p>

    <list type="bulleted">
      <item><p>Matching multiple literal keys</p></item>
      <item><p>Updating multiple literal keys</p></item>
      <item><p>Adding multiple literal keys to a map</p></item>
    </list>

    <p>For example:</p>

    <p><em>DO</em></p>
    <code type="erl"><![CDATA[
    Map = Map1#{x := X, y := Y, z := Z}]]></code>

    <p><em>DO NOT</em></p>
    <code type="erl"><![CDATA[
    Map2 = maps:update(x, X, Map1),
    Map3 = maps:update(y, Y, Map2),
    Map = maps:update(z, Z, Map3)]]></code>

    <p>If the map is a small map, the first example runs roughly
    three times as fast.</p>

    <p>Note that for variable keys, the elements are updated
    sequentially from left to right. For example, given the following
    update with variable keys:</p>

    <code type="erl"><![CDATA[
    Map = Map1#{Key1 := X, Key2 := Y, Key3 := Z}]]></code>

    <p>the compiler rewrites it like this to ensure that the updates
    are applied from left to right:</p>

    <code type="erl"><![CDATA[
    Map2 = Map1#{Key1 := X},
    Map3 = Map2#{Key2 := Y},
    Map = Map3#{Key3 := Z}]]></code>

    <p>If a key is known to exist in a map, using the <c>:=</c>
    operator is slightly more efficient than using the <c>=></c>
    operator for a small map.</p>
  </section>

  <section>
    <title>Using the Functions in the maps Module</title>

    <p>Here follows some notes about most of the functions in the
    <c>maps</c> module. For each function, the implementation language
    (C or Erlang) is stated. The reason we mention the language is
    that it gives an hint about how efficient the function is:</p>

    <list type="bulleted">
      <item><p>If a function is implemented in C, it is pretty much
      impossible to implement the same functionality more efficiently
      in Erlang.</p></item>

      <item><p>However, it might be possible to beat the <c>maps</c>
      modules functions implemented in Erlang, because they are
      generally implemented in a way that attempts to make the performance
      reasonable for all possible inputs.</p>

      <p>For example, <seemfa
      marker="stdlib:maps#map/2">maps:map/2</seemfa> iterates over
      all elements of the map, calling the mapping fun, collects the
      updated map elements in a list, and finally converts the list
      back to a map using <seemfa
      marker="stdlib:maps#from_list/1">maps:from_list/1</seemfa>. If
      it is known that at most one percent of the values in the map
      will change, it can be more efficient to update only the
      changed values.</p></item>
    </list>

    <note><p>The implementation details given in this section can
    change in the future.</p></note>

    <section>
      <title>maps:filter/2</title>
      <p><seemfa marker="stdlib:maps#filter/2">maps:filter/2</seemfa>
      is implemented in Erlang. It creates a new map using <seemfa
      marker="stdlib:maps#from_list/1">maps:from_list/1</seemfa>. If
      it is known that only a minority of the values will be removed,
      it can be more efficient to avoid <c>maps:filter/2</c> and write
      a function that will use <seemfa
      marker="stdlib:maps#remove/2">maps:remove/3</seemfa> to remove
      the unwanted values.</p>
    </section>

    <section>
      <title>maps:filtermap/2</title>
      <p><seemfa
      marker="stdlib:maps#filtermap/2">maps:filtermap/2</seemfa> is
      implemented in Erlang. It creates a new map using <seemfa
      marker="stdlib:maps#from_list/1">maps:from_list/1</seemfa>. See
      the notes for <c>maps:map/2</c> and <c>maps:filter/2</c> for
      hints on how to implement a more efficient version.</p>
    </section>

    <section>
      <title>maps:find/2</title>
      <p><seemfa marker="stdlib:maps#find/2">maps:find/2</seemfa> is
      implemented in C.</p>
      <p>Using the map matching syntax instead of <c>maps:find/2</c>
      will be slightly more efficient since building an
      <c>{ok,Value}</c> tuple will be avoided.</p>
    </section>

    <section>
      <title>maps:get/2</title>
      <p>As an optimization, the compiler will rewrite a call to <seemfa
      marker="stdlib:maps#get/2">maps:get/2</seemfa> to a call to the
      guard BIF <seemfa
      marker="erts:erlang#map_get/2">map_get/2</seemfa>. A call to a
      guard BIF is more efficient than calls to other BIFs, making the
      performance similar to using the map matching syntax.</p>

      <p>If the map is small and the keys are constants known at
      compile-time, using the map matching syntax will be more
      efficient than multiple calls to <c>maps:get/2</c>.</p>
    </section>

    <section>
      <marker id="maps_get_3"/>
      <title>maps:get/3</title>
      <p><seemfa marker="stdlib:maps#get/3">maps:get/3</seemfa>
      is implemented in Erlang essentially like this:</p>

      <code type="erl"><![CDATA[
get(Key, Map, Default) ->
    case Map of
        #{Key := Value} -> Value;
        #{} -> Default
    end.]]></code>

      <p>Therefore, a call <c>maps:get/3</c> is more expensive than a
      call to <c>maps:get/2</c>.</p>

      <p>If a small map is used as alternative to using a record,
      instead of calling <c>maps:get/3</c> multiple times to handle
      default values, consider putting the default values in a map and
      merging that map with the other map:</p>

      <code type="erl"><![CDATA[
    DefaultMap = #{Key1 => Value2, Key2 => Value2, ..., KeyN => ValueN},
    MapWithDefaultsApplied = maps:merge(DefaultMap, OtherMap)]]></code>

      <p>Whether that is faster than calling <c>maps:get/3</c>
      multiple times depends on the size of the map and the number of
      default values.</p>
    </section>

    <section>
      <title>maps:intersect/2, maps:intersect_with/3</title>
      <p><seemfa marker="stdlib:maps#intersect/2">maps:intersect/2</seemfa> and
      <seemfa marker="stdlib:maps#intersect_with/3">maps:intersect_with/3</seemfa>
      are implemented in Erlang. They both create new maps using <seemfa
      marker="stdlib:maps#from_list/1">maps:from_list/1</seemfa>.</p>

      <note><p>A map is usually the most efficient way to implement a
      set, but an exception is the intersection operation, where
      <seemfa
          marker="stdlib:ordsets#intersection/2">ordsets:intersection/2</seemfa>
      used on <seeerl marker="stdlib:ordsets">ordsets</seeerl> can be more
      efficient than <c>maps:intersect/2</c> on sets implemented as maps.</p></note>
    </section>

    <section>
      <title>maps:from_list/1</title>
      <p><seemfa marker="stdlib:maps#from_list/1">maps:from_list/1</seemfa> is
      implemented in C.</p>
    </section>

    <section>
      <title>maps:from_keys/2</title>
      <p><seemfa marker="stdlib:maps#from_keys/2">maps:from_keys/2</seemfa> is
      implemented in C.</p>
    </section>

    <section>
      <title>maps:is_key/2</title>
      <p>As an optimization, the compiler rewrites calls to <seemfa
      marker="stdlib:maps#is_key/2">maps:is_key/2</seemfa> to calls
      to the guard BIF <seemfa
      marker="erts:erlang#is_map_key/2">is_map_key/2</seemfa>. A call
      to a guard BIF is more efficient than calls to other BIFs,
      making the performance similar to using the map matching syntax.</p>
    </section>

    <section>
      <title>maps:iterator/1</title>
      <p><seemfa marker="stdlib:maps#iterator/1">maps:iterator/1</seemfa>
      is efficiently implemented in C and Erlang.</p>
    </section>

    <section>
      <title>maps:keys/1</title>
      <p><seemfa marker="stdlib:maps#keys/1">maps:keys/1</seemfa> is
      implemented in C. If the resulting list needs to be ordered, use
      <seemfa marker="stdlib:lists#sort/1">lists:sort/1</seemfa> to
      sort the result.</p>
    </section>

    <section>
      <title>maps:map/2</title>
      <p><seemfa marker="stdlib:maps#map/2">maps:map/2</seemfa> is
      implemented in Erlang. It creates a new map using <seemfa
      marker="stdlib:maps#from_list/1">maps:from_list/1</seemfa>. If
      it is known that only a minority of the values will be updated,
      it can be more efficient to avoid <c>maps:map/2</c> and write a
      function that will call <seemfa
      marker="stdlib:maps#update/3">maps:update/3</seemfa> to update
      only the values that have changed.</p>
    </section>

    <section>
      <title>maps:merge/2</title>
      <p><seemfa marker="stdlib:maps#merge/2">maps:merge/2</seemfa>
      is implemented in C.</p>
    </section>

    <section>
      <title>maps:merge_with/3</title>
      <p><seemfa marker="stdlib:maps#merge_with/3">maps:merge_with/3</seemfa>
      is implemented in Erlang. It updates and returns the larger of the
      two maps.</p>
    </section>

    <section>
      <title>maps:new/0</title>
      <p>The compiler rewrites a call to <seemfa
      marker="stdlib:maps#new/0">maps:new/0</seemfa> to using the
      syntax <c>#{}</c> for constructing an empty map.</p>
    </section>

    <section>
      <title>maps:next/1</title>
      <p><seemfa marker="stdlib:maps#next/1">maps:next/1</seemfa>
      is efficiently implemented in C and Erlang.</p>
    </section>

    <section>
      <title>maps:put/3</title>
      <p><seemfa marker="stdlib:maps#put/3">maps:put/3</seemfa> is
      implemented in C.</p>
      <p>If the key is known to already exist in the map, <seemfa
      marker="stdlib:maps#update/3">maps:update/3</seemfa> is
      slightly more efficient than <c>maps:put/3</c>.</p>
      <p>If the keys are constants known at compile-time, using the
      map update syntax with the <c>=></c> operator is more
      efficient than multiple calls to <c>maps:put/3</c>,
      especially for small maps.</p>
    </section>

    <section>
      <title>maps:remove/2</title>
      <p><seemfa marker="stdlib:maps#remove/2">maps:remove/2</seemfa>
      is implemented in C.</p>
    </section>

    <section>
      <title>maps:size/1</title>
      <p>As an optimization, the compiler rewrites calls to
      <seemfa marker="stdlib:maps#size/1">maps:size/1</seemfa>
      to calls to the guard BIF <seemfa
      marker="erts:erlang#map_size/1">map_size/1</seemfa>.
      Calls to guard BIFs are more efficient than calls to other BIFs.</p>
    </section>

    <section>
      <title>maps:take/2</title>
      <p><seemfa marker="stdlib:maps#take/2">maps:take/2</seemfa>
      is implemented in C.</p>
    </section>

    <section>
      <title>maps:to_list/1</title>
      <p><seemfa
      marker="stdlib:maps#to_list/1">maps:to_list/1</seemfa> is
      efficiently implemented in C and Erlang. If the resulting list
      needs to be ordered, use <seemfa
      marker="stdlib:lists#sort/1">lists:sort/1</seemfa> to sort
      the result.</p>

      <note><p>Maps are usually more performant than <seeerl
      marker="stdlib:gb_trees">gb_trees</seeerl>, but if it is
      necessary to frequently convert to and from sorted lists,
      <c>gb_trees</c> can be a better choice.</p></note>
    </section>

    <section>
      <title>maps:update/3</title>
      <p><seemfa marker="stdlib:maps#update/3">maps:update/3</seemfa>
      is implemented in C.</p>

      <p>If the keys are constants known at compile-time, using the
      map update syntax with the <c>:=</c> operator is more
      efficient than multiple calls to <c>maps:update/3</c>,
      especially for small maps.</p>
    </section>

    <section>
      <title>maps:values/1</title>
      <p><seemfa marker="stdlib:maps#values/1">maps:values/1</seemfa>
      is implemented in C.</p>
    </section>

    <section>
      <title>maps:with/2</title>
      <p><seemfa marker="stdlib:maps#with/2">maps:with/2</seemfa>
      is implemented in Erlang. It creates a new map using <seemfa
      marker="stdlib:maps#from_list/1">maps:from_list/1</seemfa>.</p>
    </section>

    <section>
      <title>maps:without/2</title>
      <p><seemfa marker="stdlib:maps#without/2">maps:without/2</seemfa>
      is implemented in Erlang. It returns a modified copy of the
      input map.</p>
    </section>
  </section>
</chapter>
