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<h1><a name="assoc" id="assoc">Associative-Container
Performance Tests</a></h1>
<h2><a name="settings" id="settings">Settings</a></h2>
<p>This section describes performance tests and their results.
In the following, <a href="#gcc"><u>g++</u></a>, <a href="#msvc"><u>msvc++</u></a>, and <a href="#local"><u>local</u></a> (the build used for generating this
documentation) stand for three different builds:</p>
<div id="gcc_settings_div">
<div class="c1">
<h3><a name="gcc" id="gcc"><u>g++</u></a></h3>
<ul>
<li>CPU speed - cpu MHz : 2660.644</li>
<li>Memory - MemTotal: 484412 kB</li>
<li>Platform -
Linux-2.6.12-9-386-i686-with-debian-testing-unstable</li>
<li>Compiler - g++ (GCC) 4.0.2 20050808 (prerelease)
(Ubuntu 4.0.1-4ubuntu9) Copyright (C) 2005 Free Software
Foundation, Inc. This is free software; see the source
for copying conditions. There is NO warranty; not even
for MERCHANTABILITY or FITNESS FOR A PARTICULAR
PURPOSE.</li>
</ul>
</div>
<div class="c2"></div>
</div>
<div id="msvc_settings_div">
<div class="c1">
<h3><a name="msvc" id="msvc"><u>msvc++</u></a></h3>
<ul>
<li>CPU speed - cpu MHz : 2660.554</li>
<li>Memory - MemTotal: 484412 kB</li>
<li>Platform - Windows XP Pro</li>
<li>Compiler - Microsoft (R) 32-bit C/C++ Optimizing
Compiler Version 13.10.3077 for 80x86 Copyright (C)
Microsoft Corporation 1984-2002. All rights
reserved.</li>
</ul>
</div>
<div class="c2"></div>
</div>
<div id="local_settings_div"><div style = "border-style: dotted; border-width: 1px; border-color: lightgray"><h3><a name = "local"><u>local</u></a></h3><ul>
<li>CPU speed - cpu MHz : 2250.000</li>
<li>Memory - MemTotal: 2076248 kB</li>
<li>Platform - Linux-2.6.16-1.2133_FC5-i686-with-redhat-5-Bordeaux</li>
<li>Compiler - g++ (GCC) 4.1.1 20060525 (Red Hat 4.1.1-1)
Copyright (C) 2006 Free Software Foundation, Inc.
This is free software; see the source for copying conditions. There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
</li>
</ul>
</div><div style = "width: 100%; height: 20px"></div></div>
<h2><a name="assoc_tests" id="assoc_tests">Tests</a></h2>
<h3><a name="hash_based" id="hash_based">Hash-Based
Containers</a></h3>
<ol>
<li><a href="hash_text_find_find_timing_test.html">Hash-Based
Text <tt>find</tt> Find Timing Test</a></li>
<li><a href="hash_random_int_find_find_timing_test.html">Hash-Based
Random-Integer <tt>find</tt> Find Timing Test</a></li>
<li><a href="hash_random_int_subscript_find_timing_test.html">Hash-Based
Random-Integer Subscript Find Timing Test</a></li>
<li><a href="hash_random_int_subscript_insert_timing_test.html">Hash-Based
Random-Integer Subscript Insert Timing Test</a></li>
<li><a href="hash_zlob_random_int_find_find_timing_test.html">Hash-Based
Skewed-Distribution Random-Integer <tt>find</tt> Find Timing
Test</a></li>
<li><a href="hash_random_int_erase_mem_usage_test.html">Hash-Based Erase
Memory Use Test</a></li>
</ol>
<h3><a name="tree_like_based" id="tree_like_based">Tree-Like-Based Containers</a></h3>
<ol>
<li><a href="tree_text_insert_timing_test.html">Tree-Based
and Trie-Based Text Insert Timing Test</a></li>
<li><a href="tree_text_find_find_timing_test.html">Tree-Based
and Trie-Based Text <tt>find</tt> Find Timing Test</a></li>
<li><a href="tree_text_lor_find_find_timing_test.html">Tree-Based
Locality-of-Reference Text <tt>find</tt> Find Timing
Test</a></li>
<li><a href="tree_random_int_find_find_timing_test.html">Tree-Based
Random-Integer <tt>find</tt> Find Timing Test</a></li>
<li><a href="tree_split_join_timing_test.html">Tree Split and
Join Timing Test</a></li>
<li><a href="tree_order_statistics_timing_test.html">Tree
Order-Statistics Timing Test</a></li>
</ol>
<h3><a name="multimaps" id="multimaps">Multimaps</a></h3>
<ol>
<li><a href="multimap_text_find_timing_test_small.html">"Multimap"
Text Find Timing Test with <u>Small</u> Average Secondary-Key
to Primary-Key Ratio</a></li>
<li><a href="multimap_text_find_timing_test_large.html">"Multimap"
Text Find Timing Test with <u>Large</u> Average Secondary-Key
to Primary-Key Ratio</a></li>
<li><a href="multimap_text_insert_timing_test_small.html">"Multimap"
Text Insert Timing Test with <u>Small</u> Average
Secondary-Key to Primary-Key Ratio</a></li>
<li><a href="multimap_text_insert_timing_test_large.html">"Multimap"
Text Insert Timing Test with <u>Large</u> Average
Secondary-Key to Primary-Key Ratio</a></li>
<li><a href="multimap_text_insert_mem_usage_test_small.html">"Multimap"
Text Insert Memory-Use Test with <u>Small</u> Average
Secondary-Key to Primary-Key Ratio</a></li>
<li><a href="multimap_text_insert_mem_usage_test_large.html">"Multimap"
Text Insert Memory-Use Test with <u>Large</u> Average
Secondary-Key to Primary-Key Ratio</a></li>
</ol>
<h2><a name="assoc_observations" id="assoc_observations">Observations</a></h2>
<h3><a name="dss_family_choice" id="dss_family_choice">Underlying Data-Structure Families</a></h3>
<p>In general, hash-based containers (see <a href="hash_based_containers.html">Design::Associative
Containers::Hash-Based Containers</a>) have better timing
performance than containers based on different underlying-data
structures. The main reason to choose a tree-based (see
<a href="tree_based_containers.html">Design::Associative
Containers::Tree-Based Containers</a>) or trie-based container
(see <a href="trie_based_containers.html">Design::Associative
Containers::Trie-Based Containers</a>) is if a byproduct of the
tree-like structure is required: either order-preservation, or
the ability to utilize node invariants (see <a href="tree_based_containers.html#invariants">Design::Associative
Containers::Tree-Based Containers::Node Invariants</a> and
<a href="trie_based_containers.html#invariants">Design::Associative
Containers::Trie-Based Containers::Node Invariants</a>). If
memory-use is the major factor, an ordered-vector tree (see
<a href="tree_based_containers.html">Design::Associative
Containers::Tree-Based Containers</a>) gives optimal results
(albeit with high modificiation costs), and a list-based
container (see <a href="lu_based_containers.html">Design::Associative
Containers::List-Based Containers</a>) gives reasonable
results.</p>
<h3><a name="hash_based_types" id="hash_based_types">Hash-Based
Container Types</a></h3>
<p>Hash-based containers are typically either collision
chaining or probing (see <a href="hash_based_containers.html">Design::Associative
Containers::Hash-Based Containers</a>). Collision-chaining
containers are more flexible internally, and so offer better
timing performance. Probing containers, if used for simple
value-types, manage memory more efficiently (they perform far
fewer allocation-related calls). In general, therefore, a
collision-chaining table should be used. A probing container,
conversely, might be used efficiently for operations such as
eliminating duplicates in a sequence, or counting the number of
occurrences within a sequence. Probing containers might be more
useful also in multithreaded applications where each thread
manipulates a hash-based container: in the STL, allocators have
class-wise semantics (see [<a href="references.html#meyers96more">meyers96more</a>] - Item 10); a
probing container might incur less contention in this case.</p>
<h3><a name="hash_based_policies" id="hash_based_policies">Hash-Based Containers' Policies</a></h3>
<p>In hash-based containers, the range-hashing scheme (see
<a href="hash_based_containers.html#hash_policies">Design::Associative
Containers::Hash-Based Containers::Hash Policies</a>) seems to
affect performance more than other considerations. In most
settings, a mask-based scheme works well (or can be made to
work well). If the key-distribution can be estimated a-priori,
a simple hash function can produce nearly uniform hash-value
distribution. In many other cases (<i>e.g.</i>, text hashing,
floating-point hashing), the hash function is powerful enough
to generate hash values with good uniformity properties
[<a href="references.html#knuth98sorting">knuth98sorting</a>];
a modulo-based scheme, taking into account all bits of the hash
value, appears to overlap the hash function in its effort.</p>
<p>The range-hashing scheme determines many of the other
policies (see <a href="hash_based_containers.html#policy_interaction">Design::Hash-Based
Containers::Policy Interaction</a>). A mask-based scheme works
well with an exponential-size policy (see <a href="hash_based_containers.html#resize_policies">Design::Associative
Containers::Hash-Based Containers::Resize Policies</a>) ; for
probing-based containers, it goes well with a linear-probe
function (see <a href="hash_based_containers.html#hash_policies">Design::Associative
Containers::Hash-Based Containers::Hash Policies</a>).</p>
<p>An orthogonal consideration is the trigger policy (see
<a href="hash_based_containers.html#resize_policies">Design::Associative
Containers::Hash-Based Containers::Resize Policies</a>). This
presents difficult tradeoffs. <i>E.g.</i>, different load
factors in a load-check trigger policy yield a
space/amortized-cost tradeoff.</p>
<h3><a name="tree_like_based_types" id="tree_like_based_types">Tree-Like-Based Container
Types</a></h3>
<p>In general, there are several families of tree-based
underlying data structures: balanced node-based trees
(<i>e.g.</i>, red-black or AVL trees), high-probability
balanced node-based trees (<i>e.g.</i>, random treaps or
skip-lists), competitive node-based trees (<i>e.g.</i>, splay
trees), vector-based "trees", and tries. (Additionally, there
are disk-residing or network-residing trees, such as B-Trees
and their numerous variants. An interface for this would have
to deal with the execution model and ACID guarantees; this is
out of the scope of this library.) Following are some
observations on their application to different settings.</p>
<p>Of the balanced node-based trees, this library includes a
red-black tree (see <a href="tree_based_containers.html">Design::Associative
Containers::Tree-Based Containers</a>), as does STL (in
practice). This type of tree is the "workhorse" of tree-based
containers: it offers both reasonable modification and
reasonable lookup time. Unfortunately, this data structure
stores a huge amount of metadata. Each node must contain,
besides a value, three pointers and a boolean. This type might
be avoided if space is at a premium [<a href="references.html#austern00noset">austern00noset</a>].</p>
<p>High-probability balanced node-based trees suffer the
drawbacks of deterministic balanced trees. Although they are
fascinating data structures, preliminary tests with them showed
their performance was worse than red-black trees. The library
does not contain any such trees, therefore.</p>
<p>Competitive node-based trees have two drawbacks. They are
usually somewhat unbalanced, and they perform a large number of
comparisons. Balanced trees perform one comparison per each
node they encounter on a search path; a splay tree performs two
comparisons. If the keys are complex objects, <i>e.g.</i>,
<tt>std::string</tt>, this can increase the running time.
Conversely, such trees do well when there is much locality of
reference. It is difficult to determine in which case to prefer
such trees over balanced trees. This library includes a splay
tree (see <a href="tree_based_containers.html">Design::Associative
Containers::Tree-Based Containers</a>).</p>
<p>Ordered-vector trees (see <a href="tree_based_containers.html">Design::Associative
Containers::Tree-Based Containers</a>) use very little space
[<a href="references.html#austern00noset">austern00noset</a>].
They do not have any other advantages (at least in this
implementation).</p>
<p>Large-fan-out PATRICIA tries (see <a href="trie_based_containers.html">Design::Associative
Containers::Trie-Based Containers</a>) have excellent lookup
performance, but they do so through maintaining, for each node,
a miniature "hash-table". Their space efficiency is low, and
their modification performance is bad. These tries might be
used for semi-static settings, where order preservation is
important. Alternatively, red-black trees cross-referenced with
hash tables can be used. [<a href="references.html#okasaki98mereable">okasaki98mereable</a>]
discusses small-fan-out PATRICIA tries for integers, but the
cited results seem to indicate that the amortized cost of
maintaining such trees is higher than that of balanced trees.
Moderate-fan-out trees might be useful for sequences where each
element has a limited number of choices, <i>e.g.</i>, DNA
strings (see <a href="assoc_examples.html#trie_based">Examples::Associative
Containers::Trie-Based Containers</a>).</p>
<h3><a name="msc" id="msc">Mapping-Semantics
Considerations</a></h3>
<p>Different mapping semantics were discussed in <a href="motivation.html#assoc_mapping_semantics">Motivation::Associative
Containers::Alternative to Multiple Equivalent Keys</a> and
<a href="tutorial.html#assoc_ms">Tutorial::Associative
Containers::Associative Containers Others than Maps</a>. We
will focus here on the case where a keys can be composed into
primary keys and secondary keys. (In the case where some keys
are completely identical, it is trivial that one should use an
associative container mapping values to size types.) In this
case there are (at least) five possibilities:</p>
<ol>
<li>Use an associative container that allows equivalent-key
values (such as <tt>std::multimap</tt>)</li>
<li>Use a unique-key value associative container that maps
each primary key to some complex associative container of
secondary keys, say a tree-based or hash-based container (see
<a href="tree_based_containers.html">Design::Associative
Containers::Tree-Based Containers</a> and <a href="hash_based_containers.html">Design::Associative
Containers::Hash-Based Containers</a>)</li>
<li>Use a unique-key value associative container that maps
each primary key to some simple associative container of
secondary keys, say a list-based container (see <a href="lu_based_containers.html">Design::Associative
Containers::List-Based Containers</a>)</li>
<li>Use a unique-key value associative container that maps
each primary key to some non-associative container
(<i>e.g.</i>, <tt>std::vector</tt>)</li>
<li>Use a unique-key value associative container that takes
into account both primary and secondary keys.</li>
</ol>
<p>We do not think there is a simple answer for this (excluding
option 1, which we think should be avoided in all cases).</p>
<p>If the expected ratio of secondary keys to primary keys is
small, then 3 and 4 seem reasonable. Both types of secondary
containers are relatively lightweight (in terms of memory use
and construction time), and so creating an entire container
object for each primary key is not too expensive. Option 4
might be preferable to option 3 if changing the secondary key
of some primary key is frequent - one cannot modify an
associative container's key, and the only possibility,
therefore, is erasing the secondary key and inserting another
one instead; a non-associative container, conversely, can
support in-place modification. The actual cost of erasing a
secondary key and inserting another one depends also on the
allocator used for secondary associative-containers (The tests
above used the standard allocator, but in practice one might
choose to use, <i>e.g.</i>, [<a href="references.html#boost_pool">boost_pool</a>]). Option 2 is
definitely an overkill in this case. Option 1 loses out either
immediately (when there is one secondary key per primary key)
or almost immediately after that. Option 5 has the same
drawbacks as option 2, but it has the additional drawback that
finding all values whose primary key is equivalent to some key,
might be linear in the total number of values stored (for
example, if using a hash-based container).</p>
<p>If the expected ratio of secondary keys to primary keys is
large, then the answer is more complicated. It depends on the
distribution of secondary keys to primary keys, the
distribution of accesses according to primary keys, and the
types of operations most frequent.</p>
<p>To be more precise, assume there are <i>m</i> primary keys,
primary key <i>i</i> is mapped to <i>n<sub>i</sub></i>
secondary keys, and each primary key is mapped, on average, to
<i>n</i> secondary keys (<i>i.e.</i>,
<i><b>E</b>(n<sub>i</sub>) = n</i>).</p>
<p>Suppose one wants to find a specific pair of primary and
secondary keys. Using 1 with a tree based container
(<tt>std::multimap</tt>), the expected cost is
<i><b>E</b>(Θ(log(m) + n<sub>i</sub>)) = Θ(log(m) +
n)</i>; using 1 with a hash-based container
(<tt>std::tr1::unordered_multimap</tt>), the expected cost is
<i>Θ(n)</i>. Using 2 with a primary hash-based container
and secondary hash-based containers, the expected cost is
<i>O(1)</i>; using 2 with a primary tree-based container and
secondary tree-based containers, the expected cost is (using
the Jensen inequality [<a href="references.html#motwani95random">motwani95random</a>])
<i><b>E</b>(O(log(m) + log(n<sub>i</sub>)) = O(log(m)) +
<b>E</b>(O(log(n<sub>i</sub>)) = O(log(m)) + O(log(n))</i>,
assuming that primary keys are accessed equiprobably. 3 and 4
are similar to 1, but with lower constants. Using 5 with a
hash-based container, the expected cost is <i>O(1)</i>; using 5
with a tree based container, the cost is
<i><b>E</b>(Θ(log(mn))) = Θ(log(m) +
log(n))</i>.</p>
<p>Suppose one needs the values whose primary key matches some
given key. Using 1 with a hash-based container, the expected
cost is <i>Θ(n)</i>, but the values will not be ordered
by secondary keys (which may or may not be required); using 1
with a tree-based container, the expected cost is
<i>Θ(log(m) + n)</i>, but with high constants; again the
values will not be ordered by secondary keys. 2, 3, and 4 are
similar to 1, but typically with lower constants (and,
additionally, if one uses a tree-based container for secondary
keys, they will be ordered). Using 5 with a hash-based
container, the cost is <i>Θ(mn)</i>.</p>
<p>Suppose one wants to assign to a primary key all secondary
keys assigned to a different primary key. Using 1 with a
hash-based container, the expected cost is <i>Θ(n)</i>,
but with very high constants; using 1 with a tree-based
container, the cost is <i>Θ(nlog(mn))</i>. Using 2, 3,
and 4, the expected cost is <i>Θ(n)</i>, but typically
with far lower costs than 1. 5 is similar to 1.</p>
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