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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<meta name="generator" content=
"HTML Tidy for Linux/x86 (vers 12 April 2005), see www.w3.org" />
<title>Motivation</title>
<meta http-equiv="Content-Type" content=
"text/html; charset=us-ascii" />
</head>
<body>
<div id="page">
<h1>Motivation</h1>
<p>Many fine associative-container libraries were already
written, most notably, the STL's associative containers. Why
then write another library? This section shows some possible
advantages of this library, when considering the challenges in
<a href="introduction.html">Introduction</a>. Many of these
points stem from the fact that the STL introduced
associative-containers in a two-step process (first
standardizing tree-based containers, only then adding
hash-based containers, which are fundamentally different), did
not standardize priority queues as containers, and (in our
opinion) overloads the iterator concept.</p>
<h2><a name="assoc" id="assoc">Associative Containers</a></h2>
<h3><a name="assoc_policies" id="assoc_policies">More
Configuration Choices</a></h3>
<p>Associative containers require a relatively large number of
policies to function efficiently in various settings. In some
cases this is needed for making their common operations more
efficient, and in other cases this allows them to support a
larger set of operations</p>
<ol>
<li>Hash-based containers, for example, support look-up and
insertion methods (<i>e.g.</i>, <tt>find</tt> and
<tt>insert</tt>). In order to locate elements quickly, they
are supplied a hash functor, which instruct how to transform
a key object into some size type; <i>e.g.</i>, a hash functor
might transform <tt>"hello"</tt> into <tt>1123002298</tt>. A
hash table, though, requires transforming each key object
into some size-type type in some specific domain;
<i>e.g.</i>, a hash table with a 128-long table might
transform <tt>"hello"</tt> into position 63. The policy by
which the hash value is transformed into a position within
the table can dramatically affect performance (see <a href=
"hash_based_containers.html#hash_policies">Design::Associative
Containers::Hash-Based Containers::Hash Policies</a>).
Hash-based containers also do not resize naturally (as
opposed to tree-based containers, for example). The
appropriate resize policy is unfortunately intertwined with
the policy that transforms hash value into a position within
the table (see <a href=
"hash_based_containers.html#resize_policies">Design::Associative
Containers::Hash-Based Containers::Resize Policies</a>).
<p><a href=
"assoc_performance_tests.html#hash_based">Associative-Container
Performance Tests::Hash-Based Containers</a> quantifies
some of these points.</p>
</li>
<li>Tree-based containers, for example, also support look-up
and insertion methods, and are primarily useful when
maintaining order between elements is important. In some
cases, though, one can utilize their balancing algorithms for
completely different purposes.
<p>Figure <a href="#node_invariants">Metadata for
order-statistics and interval intersections</a>-A, for
example, shows a tree whose each node contains two entries:
a floating-point key, and some size-type <i>metadata</i>
(in bold beneath it) that is the number of nodes in the
sub-tree. (<i>E.g.</i>, the root has key 0.99, and has 5
nodes (including itself) in its sub-tree.) A container based
on this data structure can obviously answer efficiently
whether 0.3 is in the container object, but it can also
answer what is the order of 0.3 among all those in the
container object [<a href=
"references.html#clrs2001">clrs2001</a>] (see <a href=
"assoc_examples.html#tree_like_based">Associative Container
Examples::Tree-Like-Based Containers (Trees and
Tries)</a>).</p>
<p>As another example, Figure <a href=
"#node_invariants">Metadata for order-statistics and
interval intersections</a>-B shows a tree whose each node
contains two entries: a half-open geometric line interval,
and a number <i>metadata</i> (in bold beneath it) that is
the largest endpoint of all intervals in its sub-tree.
(<i>E.g.</i>, the root describes the interval <i>[20,
36)</i>, and the largest endpoint in its sub-tree is 99.) A
container based on this data structure can obviously answer
efficiently whether <i>[3, 41)</i> is in the container
object, but it can also answer efficiently whether the
container object has intervals that intersect <i>[3,
41)</i> (see <a href=
"assoc_examples.html#tree_like_based">Associative Container
Examples::Tree-Like-Based Containers (Trees and
Tries)</a>). These types of queries are very useful in
geometric algorithms and lease-management algorithms.</p>
<p>It is important to note, however, that as the trees are
modified, their internal structure changes. To maintain
these invariants, one must supply some policy that is aware
of these changes (see <a href=
"tree_based_containers.html#invariants">Design::Associative
Containers::Tree-Based Containers::Node Invariants</a>);
without this, it would be better to use a linked list (in
itself very efficient for these purposes).</p>
<p><a href=
"assoc_performance_tests.html#tree_like_based">Associative-Container
Performance Tests::Tree-Like-Based Containers</a>
quantifies some of these points.</p>
</li>
</ol>
<h6 class="c1"><a name="node_invariants" id=
"node_invariants"><img src="node_invariants.png" alt=
"no image" /></a></h6>
<h6 class="c1">Metadata for order-statistics and interval
intersections.</h6>
<h3><a name="assoc_ds_genericity" id="assoc_ds_genericity">More
Data Structures and Traits</a></h3>
<p>The STL contains associative containers based on red-black
trees and collision-chaining hash tables. These are obviously
very useful, but they are not ideal for all types of
settings.</p>
<p>Figure <a href=
"#different_underlying_data_structures">Different underlying
data structures</a> shows different underlying data structures
(the ones currently supported in <tt>pb_ds</tt>). A shows a
collision-chaining hash-table, B shows a probing hash-table, C
shows a red-black tree, D shows a splay tree, E shows a tree
based on an ordered vector(implicit in the order of the
elements), F shows a PATRICIA trie, and G shows a list-based
container with update policies.</p>
<p>Each of these data structures has some performance benefits,
in terms of speed, size or both (see <a href=
"assoc_performance_tests.html">Associative-Container
Performance Tests</a> and <a href=
"assoc_performance_tests.html#dss_family_choice">Associative-Container
Performance Tests::Observations::Underlying Data-Structure
Families</a>). For now, though, note that <i>e.g.</i>,
vector-based trees and probing hash tables manipulate memory
more efficiently than red-black trees and collision-chaining
hash tables, and that list-based associative containers are
very useful for constructing "multimaps" (see <a href=
"#assoc_mapping_semantics">Alternative to Multiple Equivalent
Keys</a>, <a href=
"assoc_performance_tests.html#multimaps">Associative Container
Performance Tests::Multimaps</a>, and <a href=
"assoc_performance_tests.html#msc">Associative Container
Performance Tests::Observations::Mapping-Semantics
Considerations</a>).</p>
<h6 class="c1"><a name="different_underlying_data_structures"
id="different_underlying_data_structures"><img src=
"different_underlying_dss.png" alt="no image" /></a></h6>
<h6 class="c1">Different underlying data structures.</h6>
<p>Now consider a function manipulating a generic associative
container, <i>e.g.</i>,</p>
<pre>
<b>template</b><
<b>class</b> Cntnr>
<b>int</b>
some_op_sequence
(Cntnr &r_cnt)
{
...
}
</pre>
<p>Ideally, the underlying data structure of <tt>Cntnr</tt>
would not affect what can be done with <tt>r_cnt</tt>.
Unfortunately, this is not the case.</p>
<p>For example, if <tt>Cntnr</tt> is <tt>std::map</tt>, then
the function can use <tt>std::for_each(r_cnt.find(foo),
r_cnt.find(bar), foobar)</tt> in order to apply <tt>foobar</tt>
to all elements between <tt>foo</tt> and <tt>bar</tt>. If
<tt>Cntnr</tt> is a hash-based container, then this call's
results are undefined.</p>
<p>Also, if <tt>Cntnr</tt> is tree-based, the type and object
of the comparison functor can be accessed. If <tt>Cntnr</tt> is
hash based, these queries are nonsensical.</p>
<p>There are various other differences based on the container's
underlying data structure. For one, they can be constructed by,
and queried for, different policies. Furthermore:</p>
<ol>
<li>Containers based on C, D, E and F store elements in a
meaningful order; the others store elements in a meaningless
(and probably time-varying) order. By implication, only
containers based on C, D, E and F can support erase
operations taking an iterator and returning an iterator to
the following element without performance loss (see <a href=
"#assoc_ers_methods">Slightly Different Methods::Methods
Related to Erase</a>).</li>
<li>Containers based on C, D, E, and F can be split and
joined efficiently, while the others cannot. Containers based
on C and D, furthermore, can guarantee that this is
exception-free; containers based on E cannot guarantee
this.</li>
<li>Containers based on all but E can guarantee that erasing
an element is exception free; containers based on E cannot
guarantee this. Containers based on all but B and E can
guarantee that modifying an object of their type does not
invalidate iterators or references to their elements, while
containers based on B and E cannot. Containers based on C, D,
and E can furthermore make a stronger guarantee, namely that
modifying an object of their type does not affect the order
of iterators.</li>
</ol>
<p>A unified tag and traits system (as used for the STL's
iterators, for example) can ease generic manipulation of
associative containers based on different underlying
data structures (see <a href=
"tutorial.html#assoc_ds_gen">Tutorial::Associative
Containers::Determining Containers' Attributes</a> and <a href=
"ds_gen.html#container_traits">Design::Associative
Containers::Data-Structure Genericity::Data-Structure Tags and
Traits</a>).</p>
<h3><a name="assoc_diff_it" id="assoc_diff_it">Differentiating
between Iterator Types</a></h3>
<p>Iterators are centric to the STL's design, because of the
container/algorithm/iterator decomposition that allows an
algorithm to operate on a range through iterators of some
sequence (<i>e.g.</i>, one originating from a container).
Iterators, then, are useful because they allow going over a
<u>sequence</u>. The STL also uses iterators for accessing a
<u>specific</u> element - <i>e.g.</i>, when an associative
container returns one through <tt>find</tt>. The STL, however,
consistently uses the same types of iterators for both
purposes: going over a range, and accessing a specific found
element. Before the introduction of hash-based containers to
the STL, this made sense (with the exception of priority
queues, which are discussed in <a href="#pq">Priority
Queues</a>).</p>
<p>Using the STL's associative containers together with
non-order-preserving associative containers (and also because
of priority-queues container), there is a possible need for
different types of iterators for self-organizing containers -
the iterator concept seems overloaded to mean two different
things (in some cases). The following subsections explain this;
<a href="tutorial.html#assoc_find_range">Tutorial::Associative
Containers::Point-Type and Range-Type Methods</a> explains an
alternative design which does not complicate the use of
order-preserving containers, but is better for unordered
containers; <a href=
"ds_gen.html#find_range">Design::Associative
Containers::Data-Structure Genericity::Point-Type and
Range-Type Methods</a> explains the design further.</p>
<h4><a name="assoc_find_it_range_it" id=
"assoc_find_it_range_it">Using Point-Type Iterators for
Range-Type Operations</a></h4>
<p>Suppose <tt>cntnr</tt> is some associative container, and
say <tt>c</tt> is an object of type <tt>cntnr</tt>. Then what
will be the outcome of</p>
<pre>
std::for_each(c.find(1), c.find(5), foo);
</pre>
<p>If <tt>cntnr</tt> is a tree-based container object, then an
in-order walk will apply <tt>foo</tt> to the relevant elements,
<i>e.g.</i>, as in Figure <a href="#range_it_in_hts">Range
iteration in different data structures</a> -A. If <tt>c</tt> is
a hash-based container, then the order of elements between any
two elements is undefined (and probably time-varying); there is
no guarantee that the elements traversed will coincide with the
<i>logical</i> elements between 1 and 5, <i>e.g.</i>, as in
Figure <a href="#range_it_in_hts">Range iteration in different
data structures</a>-B.</p>
<h6 class="c1"><a name="range_it_in_hts" id=
"range_it_in_hts"><img src="point_iterators_range_ops_1.png"
alt="no image" /></a></h6>
<h6 class="c1">Range iteration in different
data structures.</h6>
<p>In our opinion, this problem is not caused just because
red-black trees are order preserving while collision-chaining
hash tables are (generally) not - it is more fundamental. Most
of the STL's containers order sequences in a well-defined
manner that is determined by their <u>interface</u>: calling
<tt>insert</tt> on a tree-based container modifies its sequence
in a predictable way, as does calling <tt>push_back</tt> on a
list or a vector. Conversely, collision-chaining hash tables,
probing hash tables, priority queues, and list-based containers
(which are very useful for "multimaps") are self-organizing
data structures; the effect of each operation modifies their
sequences in a manner that is (practically) determined by their
<u>implementation</u>.</p>
<p>Consequently, applying an algorithm to a sequence obtained
from most containers <u>may or may not</u> make sense, but
applying it to a sub-sequence of a self-organizing container
<u>does not</u>.</p>
<h4><a name="assoc_range_it_for_find_it" id=
"assoc_range_it_for_find_it">The Cost of Enabling Range
Capabilities to Point-Type Iterators</a></h4>
<p>Suppose <tt>c</tt> is some collision-chaining hash-based
container object, and one calls <tt>c.find(3)</tt>. Then what
composes the returned iterator?</p>
<p>Figure <a href="#find_its_in_hash_tables">Point-type
iterators in hash tables</a>-A shows the simplest (and most
efficient) implementation of a collision-chaining hash table.
The little box marked <tt>point_iterator</tt> shows an object
that contains a pointer to the element's node. Note that this
"iterator" has no way to move to the next element (<i>i.e.</i>,
it cannot support <tt><b>operator</b>++</tt>). Conversely, the
little box marked <tt>iterator</tt> stores both a pointer to
the element, as well as some other information (<i>e.g.</i>,
the bucket number of the element). the second iterator, then,
is "heavier" than the first one- it requires more time and
space. If we were to use a different container to
cross-reference into this hash-table using these iterators - it
would take much more space. As noted in <a href=
"#assoc_find_it_range_it">Using Point-Type Iterators for
Range-Type Operations</a>, nothing much can be done by
incrementing these iterators, so why is this extra information
needed?</p>
<p>Alternatively, one might create a collision-chaining
hash-table where the lists might be linked, forming a
monolithic total-element list, as in Figure <a href=
"#find_its_in_hash_tables">Point-type iterators in hash
tables</a>-B (this seems similar to the Dinkumware design
[<a href="references.html#dinkumware_stl">dinkumware_stl</a>]).
Here the iterators are as light as can be, but the hash-table's
operations are more complicated.</p>
<h6 class="c1"><a name="find_its_in_hash_tables" id=
"find_its_in_hash_tables"><img src=
"point_iterators_range_ops_2.png" alt="no image" /></a></h6>
<h6 class="c1">Point-type iterators in hash tables.</h6>
<p>It should be noted that containers based on
collision-chaining hash-tables are not the only ones with this
type of behavior; many other self-organizing data structures
display it as well.</p>
<h4><a name="assoc_inv_guar" id="assoc_inv_guar">Invalidation
Guarantees</a></h4>
<p>Consider the following snippet:</p>
<pre>
it = c.find(3);
c.erase(5);
</pre>
<p>Following the call to <tt>erase</tt>, what is the validity
of <tt>it</tt>: can it be de-referenced? can it be
incremented?</p>
<p>The answer depends on the underlying data structure of the
container. Figure <a href=
"#invalidation_guarantee_erase">Effect of erase in different
underlying data structures</a> shows three cases: A1 and A2
show a red-black tree; B1 and B2 show a probing hash-table; C1
and C2 show a collision-chaining hash table.</p>
<h6 class="c1"><a name="invalidation_guarantee_erase" id=
"invalidation_guarantee_erase"><img src=
"invalidation_guarantee_erase.png" alt="no image" /></a></h6>
<h6 class="c1">Effect of erase in different underlying
data structures.</h6>
<ol>
<li>Erasing 5 from A1 yields A2. Clearly, an iterator to 3
can be de-referenced and incremented. The sequence of
iterators changed, but in a way that is well-defined by the
<u>interface</u>.</li>
<li>Erasing 5 from B1 yields B2. Clearly, an iterator to 3 is
not valid at all - it cannot be de-referenced or incremented;
the order of iterators changed in a way that is (practically)
determined by the <u>implementation</u> and not by the
<u>interface</u>.</li>
<li>Erasing 5 from C1 yields C2. Here the situation is more
complicated. On the one hand, there is no problem in
de-referencing <tt>it</tt>. On the other hand, the order of
iterators changed in a way that is (practically) determined
by the <u>implementation</u> and not by the
<u>interface</u>.</li>
</ol>
<p>So in classic STL, it is not always possible to express
whether <tt>it</tt> is valid or not. This is true also for
<tt>insert</tt>, <i>e.g.</i>. Again, the iterator concept seems
overloaded.</p>
<h3><a name="assoc_methods" id="assoc_methods">Slightly
Different Methods</a></h3>
<p>[<a href="references.html#meyers02both">meyers02both</a>]
points out that a class's methods should comprise only
operations which depend on the class's internal structure;
other operations are best designed as external functions.
Possibly, therefore, the STL's associative containers lack some
useful methods, and provide some other methods which would be
better left out (<i>e.g.</i>, [<a href=
"references.html#sgi_stl">sgi_stl</a>] ).</p>
<h4><a name="assoc_ers_methods" id="assoc_ers_methods">Methods
Related to Erase</a></h4>
<ol>
<li>Order-preserving STL associative containers provide the
method
<pre>
iterator
erase
(iterator it)
</pre>which takes an iterator, erases the corresponding element,
and returns an iterator to the following element. Also hash-based
STL associative containers provide this method. This <u>seemingly
increases</u> genericity between associative containers, since, <i>
e.g.</i>, it is possible to use
<pre>
<b>typename</b> C::iterator it = c.begin();
<b>typename</b> C::iterator e_it = c.end();
<b>while</b>(it != e_it)
it = pred(*it)? c.erase(it) : ++it;
</pre>in order to erase from a container object <tt>
c</tt> all element which match a predicate <tt>pred</tt>.
However, in a different sense this actually
<u>decreases</u> genericity: an integral implication of
this method is that tree-based associative containers'
memory use is linear in the total number of elements they
store, while hash-based containers' memory use is unbounded
in the total number of elements they store. Assume a
hash-based container is allowed to decrease its size when
an element is erased. Then the elements might be rehashed,
which means that there is no "next" element - it is simply
undefined. Consequently, it is possible to infer from the
fact that STL hash-based containers provide this method
that they cannot downsize when elements are erased
(<a href="assoc_performance_tests.html#hash_based">Performance
Tests::Hash-Based Container Tests</a> quantifies this.) As
a consequence, different code is needed to manipulate
different containers, assuming that memory should be
conserved. <tt>pb_ds</tt>'s non-order preserving
associative containers omit this method.
</li>
<li>All of <tt>pb_ds</tt>'s associative containers include a
conditional-erase method
<pre>
<b>template</b><
<b>class</b> Pred>
size_type
erase_if
(Pred pred)
</pre>which erases all elements matching a predicate. This is
probably the only way to ensure linear-time multiple-item erase
which can actually downsize a container.
</li>
<li>STL associative containers provide methods for
multiple-item erase of the form
<pre>
size_type
erase
(It b,
It e)
</pre>erasing a range of elements given by a pair of iterators. For
tree-based or trie-based containers, this can implemented more
efficiently as a (small) sequence of split and join operations. For
other, unordered, containers, this method isn't much better than an
external loop. Moreover, if <tt>c</tt> is a hash-based container,
then, <i>e.g.</i>, <tt>c.erase(c.find(2), c.find(5))</tt> is almost
certain to do something different than erasing all elements whose
keys are between 2 and 5, and is likely to produce other undefined
behavior.
</li>
</ol>
<h4><a name="assoc_split_join_methods" id=
"assoc_split_join_methods">Methods Related to Split and
Join</a></h4>
<p>It is well-known that tree-based and trie-based container
objects can be efficiently split or joined [<a href=
"references.html#clrs2001">clrs2001</a>]. Externally splitting
or joining trees is super-linear, and, furthermore, can throw
exceptions. Split and join methods, consequently, seem good
choices for tree-based container methods, especially, since as
noted just before, they are efficient replacements for erasing
sub-sequences. <a href=
"assoc_performance_tests.html#tree_like_based">Performance
Tests::Tree-Like-Based Container Tests</a> quantifies this.</p>
<h4><a name="assoc_insert_methods" id=
"assoc_insert_methods">Methods Related to Insert</a></h4>
<p>STL associative containers provide methods of the form</p>
<pre>
<b>template</b><
<b>class</b> It>
size_type
insert
(It b,
It e);
</pre>for inserting a range of elements given by a pair of
iterators. At best, this can be implemented as an external loop,
or, even more efficiently, as a join operation (for the case of
tree-based or trie-based containers). Moreover, these methods seem
similar to constructors taking a range given by a pair of
iterators; the constructors, however, are transactional, whereas
the insert methods are not; this is possibly confusing.
<h4><a name="assoc_equiv_comp_methods" id=
"assoc_equiv_comp_methods">Functions Related to
Comparison</a></h4>
<p>Associative containers are parametrized by policies
allowing to test key equivalence; <i>e.g.</i> a hash-based
container can do this through its equivalence functor, and a
tree-based container can do this through its comparison
functor. In addition, some STL associative containers have
global function operators, <i>e.g.</i>,
<tt><b>operator</b>==</tt> and <tt><b>operator</b><=</tt>,
that allow comparing entire associative containers.</p>
<p>In our opinion, these functions are better left out. To
begin with, they do not significantly improve over an external
loop. More importantly, however, they are possibly misleading -
<tt><b>operator</b>==</tt>, for example, usually checks for
equivalence, or interchangeability, but the associative
container cannot check for values' equivalence, only keys'
equivalence; also, are two containers considered equivalent if
they store the same values in different order? this is an
arbitrary decision.</p>
<h3><a name="assoc_mapping_semantics" id=
"assoc_mapping_semantics">Alternative to Multiple Equivalent
Keys</a></h3>
<p>Maps (or sets) allow mapping (or storing) unique-key values.
The STL, however, also supplies associative containers which
map (or store) multiple values with equivalent keys:
<tt>std::multimap</tt>, <tt>std::multiset</tt>,
<tt>std::tr1::unordered_multimap</tt>, and
<tt>unordered_multiset</tt>. We first discuss how these might
be used, then why we think it is best to avoid them.</p>
<p>Suppose one builds a simple bank-account application that
records for each client (identified by an <tt>std::string</tt>)
and account-id (marked by an <tt><b>unsigned long</b></tt>) -
the balance in the account (described by a
<tt><b>float</b></tt>). Suppose further that ordering this
information is not useful, so a hash-based container is
preferable to a tree based container. Then one can use</p>
<pre>
std::tr1::unordered_map<std::pair<std::string, <b>unsigned long</b>>, <b>float</b>, ...>
</pre>which <u>hashes every combination of client and
account-id</u>. This might work well, except for the fact that it
is now impossible to efficiently list all of the accounts of a
specific client (this would practically require iterating over all
entries). Instead, one can use
<pre>
std::tr1::unordered_multimap<std::pair<std::string, <tt><b>unsigned long</b></tt>>, <b>float</b>, ...>
</pre>which <u>hashes every client</u>, and <u>decides equivalence
based on client</u> only. This will ensure that all accounts
belonging to a specific user are stored consecutively.
<p>Also, suppose one wants an integers' priority queue
(<i>i.e.,</i> a container that supports <tt>push</tt>,
<tt>pop</tt>, and <tt>top</tt> operations, the last of which
returns the largest <tt><b>int</b></tt>) that also supports
operations such as <tt>find</tt> and <tt>lower_bound</tt>. A
reasonable solution is to build an adapter over
<tt>std::set<<b>int</b>></tt>. In this adapter,
<i>e.g.</i>, <tt>push</tt> will just call the tree-based
associative container's <tt>insert</tt> method; <tt>pop</tt>
will call its <tt>end</tt> method, and use it to return the
preceding element (which must be the largest). Then this might
work well, except that the container object cannot hold
multiple instances of the same integer (<tt>push(4)</tt>,
<i>e.g.</i>, will be a no-op if <tt>4</tt> is already in the
container object). If multiple keys are necessary, then one
might build the adapter over an
<tt>std::multiset<<b>int</b>></tt>.</p>
<p class="c1">STL non-unique-mapping containers, then, are
useful when (1) a key can be decomposed in to a primary key and
a secondary key, (2) a key is needed multiple times, or (3) any
combination of (1) and (2).</p>
<p>Figure <a href="#embedded_lists_1">Non-unique mapping
containers in the STL's design</a> shows how the STL's design
works internally; in this figure nodes shaded equally represent
equivalent-key values. Equivalent keys are stored consecutively
using the properties of the underlying data structure: binary
search trees (Figure <a href="#embedded_lists_1">Non-unique
mapping containers in the STL's design</a>-A) store
equivalent-key values consecutively (in the sense of an
in-order walk) naturally; collision-chaining hash tables
(Figure <a href="#embedded_lists_1">Non-unique mapping
containers in the STL's design</a>-B) store equivalent-key
values in the same bucket, the bucket can be arranged so that
equivalent-key values are consecutive.</p>
<h6 class="c1"><a name="embedded_lists_1" id=
"embedded_lists_1"><img src="embedded_lists_1.png" alt=
"no image" /></a></h6>
<h6 class="c1">Non-unique mapping containers in the STL's
design.</h6>
<p>Put differently, STL non-unique mapping
associative-containers are associative containers that map
primary keys to linked lists that are embedded into the
container. Figure <a href="#embedded_lists_2">Effect of
embedded lists in STL multimaps</a> shows again the two
containers from Figure <a href="#embedded_lists_1">Non-unique
mapping containers in the STL's design</a>, this time with the
embedded linked lists of the grayed nodes marked
explicitly.</p>
<h6 class="c1"><a name="embedded_lists_2" id=
"embedded_lists_2"><img src="embedded_lists_2.png" alt=
"no image" /></a></h6>
<h6 class="c1">Effect of embedded lists in STL multimaps.</h6>
<p>These embedded linked lists have several disadvantages.</p>
<ol>
<li>The underlying data structure embeds the linked lists
according to its own consideration, which means that the
search path for a value might include several different
equivalent-key values. For example, the search path for the
the black node in either of Figures <a href=
"#embedded_lists_1">Non-unique mapping containers in the
STL's design</a> A or B, includes more than a single gray
node.</li>
<li>The links of the linked lists are the underlying
data structures' nodes, which typically are quite structured.
<i>E.g.</i>, in the case of tree-based containers (Figure
<a href="#embedded_lists_2">Effect of embedded lists in STL
multimaps</a>-B), each "link" is actually a node with three
pointers (one to a parent and two to children), and a
relatively-complicated iteration algorithm. The linked lists,
therefore, can take up quite a lot of memory, and iterating
over all values equal to a given key (<i>e.g.</i>, through
the return value of the STL's <tt>equal_range</tt>) can be
expensive.</li>
<li>The primary key is stored multiply; this uses more
memory.</li>
<li>Finally, the interface of this design excludes several
useful underlying data structures. <i>E.g.</i>, of all the
unordered self-organizing data structures, practically only
collision-chaining hash tables can (efficiently) guarantee
that equivalent-key values are stored consecutively.</li>
</ol>
<p>The above reasons hold even when the ratio of secondary keys
to primary keys (or average number of identical keys) is small,
but when it is large, there are more severe problems:</p>
<ol>
<li>The underlying data structures order the links inside
each embedded linked-lists according to their internal
considerations, which effectively means that each of the
links is unordered. Irrespective of the underlying
data structure, searching for a specific value can degrade to
linear complexity.</li>
<li>Similarly to the above point, it is impossible to apply
to the secondary keys considerations that apply to primary
keys. For example, it is not possible to maintain secondary
keys by sorted order.</li>
<li>While the interface "understands" that all equivalent-key
values constitute a distinct list (<i>e.g.</i>, through
<tt>equal_range</tt>), the underlying data structure
typically does not. This means, <i>e.g.</i>, that operations
such as erasing from a tree-based container all values whose
keys are equivalent to a a given key can be super-linear in
the size of the tree; this is also true also for several
other operations that target a specific list.</li>
</ol>
<p>In <tt>pb_ds</tt>, therefore, all associative containers map
(or store) unique-key values. One can (1) map primary keys to
secondary associative-containers (<i>i.e.</i>, containers of
secondary keys) or non-associative containers (2) map identical
keys to a size-type representing the number of times they
occur, or (3) any combination of (1) and (2). Instead of
allowing multiple equivalent-key values, <tt>pb_ds</tt>
supplies associative containers based on underlying
data structures that are suitable as secondary
associative-containers (see <a href=
"assoc_performance_tests.html#msc">Associative-Container
Performance Tests::Observations::Mapping-Semantics
Considerations</a>).</p>
<p>Figures <a href="#embedded_lists_3">Non-unique mapping
containers in <tt>pb_ds</tt></a> A and B show the equivalent
structures in <tt>pb_ds</tt>'s design, to those in Figures
<a href="#embedded_lists_1">Non-unique mapping containers in
the STL's design</a> A and B, respectively. Each shaded box
represents some size-type or secondary
associative-container.</p>
<h6 class="c1"><a name="embedded_lists_3" id=
"embedded_lists_3"><img src="embedded_lists_3.png" alt=
"no image" /></a></h6>
<h6 class="c1">Non-unique mapping containers in the
<tt>pb_ds</tt>.</h6>
<p>In the first example above, then, one would use an
associative container mapping each user to an associative
container which maps each application id to a start time (see
<a href=
"http://gcc.gnu.org/viewcvs/*checkout*/trunk/libstdc%2B%2B-v3/testsuite/ext/pb_ds/example/basic_multimap.cc"><tt>basic_multimap.cc</tt></a>);
in the second example, one would use an associative container
mapping each <tt><b>int</b></tt> to some size-type indicating
the number of times it logically occurs (see <a href=
"http://gcc.gnu.org/viewcvs/*checkout*/trunk/libstdc%2B%2B-v3/testsuite/ext/pb_ds/example/basic_multiset.cc"><tt>basic_multiset.cc</tt></a>).</p>
<p><a href=
"assoc_performance_tests.html#multimaps">Associative-Container
Performance Tests::Multimaps</a> quantifies some of these
points, and <a href=
"assoc_performance_tests.html#msc">Associative-Container
Performance Tests::Observations::Mapping-Semantics
Considerations</a> shows some simple calculations.</p>
<p><a href="assoc_examples.html#mmaps">Associative-Container
Examples::Multimaps</a> shows some simple examples of using
"multimaps".</p>
<p><a href="lu_based_containers.html">Design::Associative
Containers::List-Based Containers</a> discusses types of
containers especially suited as secondary
associative-containers.</p>
<h2><a name="pq" id="pq">Priority Queues</a></h2>
<h3><a name="pq_more_ops" id="pq_more_ops">Slightly Different
Methods</a></h3>
<p>Priority queues are containers that allow efficiently
inserting values and accessing the maximal value (in the sense
of the container's comparison functor); <i>i.e.</i>, their
interface supports <tt>push</tt> and <tt>pop</tt>. The STL's
priority queues indeed support these methods, but they support
little else. For algorithmic and software-engineering purposes,
other methods are needed:</p>
<ol>
<li>Many graph algorithms [<a href=
"references.html#clrs2001">clrs2001</a>] require increasing a
value in a priority queue (again, in the sense of the
container's comparison functor), or joining two
priority-queue objects.</li>
<li>It is sometimes necessary to erase an arbitrary value in
a priority queue. For example, consider the <tt>select</tt>
function for monitoring file descriptors:
<pre>
<b>int</b>
select
(<b>int</b> nfds,
fd_set *readfds,
fd_set *writefds,
fd_set *errorfds,
<b>struct</b> timeval *timeout);
</pre>then, as the <tt>select</tt> manual page [<a href=
"references.html#select_man">select_man</a>] states:
<p><q>The nfds argument specifies the range of file
descriptors to be tested. The select() function tests file
descriptors in the range of 0 to nfds-1.</q></p>
<p>It stands to reason, therefore, that we might wish to
maintain a minimal value for <tt>nfds</tt>, and priority
queues immediately come to mind. Note, though, that when a
socket is closed, the minimal file description might
change; in the absence of an efficient means to erase an
arbitrary value from a priority queue, we might as well
avoid its use altogether.</p>
<p><a href="pq_examples.html#xref">Priority-Queue
Examples::Cross-Referencing</a> shows examples for these
types of operations.</p>
</li>
<li>STL containers typically support iterators. It is
somewhat unusual for <tt>std::priority_queue</tt> to omit
them (see, <i>e.g.</i>, [<a href=
"references.html#meyers01stl">meyers01stl</a>]). One might
ask why do priority queues need to support iterators, since
they are self-organizing containers with a different purpose
than abstracting sequences. There are several reasons:
<ol>
<li>Iterators (even in self-organizing containers) are
useful for many purposes, <i>e.g.</i>, cross-referencing
containers, serialization, and debugging code that uses
these containers.</li>
<li>The STL's hash-based containers support iterators,
even though they too are self-organizing containers with
a different purpose than abstracting sequences.</li>
<li>In STL-like containers, it is natural to specify the
interface of operations for modifying a value or erasing
a value (discussed previously) in terms of a iterators.
This is discussed further in <a href=
"pq_design.html#pq_it">Design::Priority
Queues::Iterators</a>. It should be noted that the STL's
containers also use iterators for accessing and
manipulating a specific value. <i>E.g.</i>, in hash-based
containers, one checks the existence of a key by
comparing the iterator returned by <tt>find</tt> to the
iterator returned by <tt>end</tt>, and not by comparing a
pointer returned by <tt>find</tt> to <tt>NULL</tt>.</li>
</ol>
</li>
</ol>
<p><a href="pq_performance_tests.html">Performance
Tests::Priority Queues</a> quantifies some of these points.</p>
<h3><a name="pq_ds_genericity" id="pq_ds_genericity">More Data
Structures and Traits</a></h3>
<p>There are three main implementations of priority queues: the
first employs a binary heap, typically one which uses a
sequence; the second uses a tree (or forest of trees), which is
typically less structured than an associative container's tree;
the third simply uses an associative container. These are
shown, respectively, in Figures <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> A1 and A2, B, and C.</p>
<h6 class="c1"><a name="pq_different_underlying_dss" id=
"pq_different_underlying_dss"><img src=
"pq_different_underlying_dss.png" alt="no image" /></a></h6>
<h6 class="c1">Underlying Priority-Queue Data-Structures.</h6>
<p>No single implementation can completely replace any of the
others. Some have better <tt>push</tt> and <tt>pop</tt>
amortized performance, some have better bounded (worst case)
response time than others, some optimize a single method at the
expense of others, <i>etc.</i>. In general the "best"
implementation is dictated by the problem (see <a href=
"pq_performance_tests.html#pq_observations">Performance
Tests::Priority Queues::Observations</a>).</p>
<p>As with associative containers (see <a href=
"#assoc_ds_genericity">Associative Containers::Traits for
Underlying Data-Structures</a>), the more implementations
co-exist, the more necessary a traits mechanism is for handling
generic containers safely and efficiently. This is especially
important for priority queues, since the invalidation
guarantees of one of the most useful data structures - binary
heaps - is markedly different than those of most of the
others.</p>
<p><a href="pq_design.html#pq_traits">Design::Priority
Queues::Traits</a> discusses this further.</p>
<h3><a name="pq_binary_heap" id="pq_binary_heap">Binary Heap
Implementation</a></h3>
<p>Binary heaps are one of the most useful underlying
data structures for priority queues. They are very efficient in
terms of memory (since they don't require per-value structure
metadata), and have the best amortized <tt>push</tt> and
<tt>pop</tt> performance for primitive types (<i>e.g.</i>,
<tt><b>int</b></tt>s).</p>
<p>The STL's <tt>priority_queue</tt> implements this data
structure as an adapter over a sequence, typically
<tt>std::vector</tt> or <tt>std::deque</tt>, which correspond
to Figures <a href="#pq_different_underlying_dss">Underlying
Priority-Queue Data-Structures</a> A1 and A2, respectively.</p>
<p>This is indeed an elegant example of the adapter concept and
the algorithm/container/iterator decomposition (see [<a href=
"references.html#nelson96stlpq">nelson96stlpql</a>]). There are
possibly reasons, however, why a binary-heap priority queue
would be better implemented as a container instead of a
sequence adapter:</p>
<ol>
<li><tt>std::priority_queue</tt> cannot erase values from its
adapted sequence (irrespective of the sequence type). This
means that the memory use of an <tt>std::priority_queue</tt>
object is always proportional to the maximal number of values
it ever contained, and not to the number of values that it
currently contains (see <a href=
"priority_queue_text_pop_mem_usage_test.html">Priority Queue
Text <tt>pop</tt> Memory Use Test</a>); this implementation
of binary heaps acts very differently than other underlying
data structures (<i>e.g.</i>, pairing heaps).</li>
<li>Some combinations of adapted sequences and value types
are very inefficient or just don't make sense. If one uses
<tt>std::priority_queue<std::vector<std::string>
> ></tt>, for example, then not only will each
operation perform a logarithmic number of
<tt>std::string</tt> assignments, but, furthermore, any
operation (including <tt>pop</tt>) can render the container
useless due to exceptions. Conversely, if one uses
<tt>std::priority_queue<std::deque<<b>int</b>> >
></tt>, then each operation uses incurs a logarithmic
number of indirect accesses (through pointers) unnecessarily.
It might be better to let the container make a conservative
deduction whether to use the structure in Figures <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> A1 or A2.</li>
<li>There does not seem to be a systematic way to determine
what exactly can be done with the priority queue.
<ol>
<li>If <tt>p</tt> is a priority queue adapting an
<tt>std::vector</tt>, then it is possible to iterate over
all values by using <tt>&p.top()</tt> and
<tt>&p.top() + p.size()</tt>, but this will not work
if <tt>p</tt> is adapting an <tt>std::deque</tt>; in any
case, one cannot use <tt>p.begin()</tt> and
<tt>p.end()</tt>. If a different sequence is adapted, it
is even more difficult to determine what can be
done.</li>
<li>If <tt>p</tt> is a priority queue adapting an
<tt>std::deque</tt>, then the reference return by
<tt>p.top()</tt> will remain valid until it is popped,
but if <tt>p</tt> adapts an <tt>std::vector</tt>, the
next <tt>push</tt> will invalidate it. If a different
sequence is adapted, it is even more difficult to
determine what can be done.</li>
</ol>
</li>
<li>Sequence-based binary heaps can still implement
linear-time <tt>erase</tt> and <tt>modify</tt> operations.
This means that if one needs, <i>e.g.</i>, to erase a small
(say logarithmic) number of values, then one might still
choose this underlying data structure. Using
<tt>std::priority_queue</tt>, however, this will generally
change the order of growth of the entire sequence of
operations.</li>
</ol>
</div>
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