pq_performance_tests.html [plain text]
<!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>Priority-Queue Performance Tests</title>
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii" />
</head>
<body>
<div id="page">
<h1>Priority-Queue Performance Tests</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="pq_tests" id="pq_tests">Tests</a></h2>
<ol>
<li><a href="priority_queue_text_push_timing_test.html">Priority Queue
Text <tt>push</tt> Timing Test</a></li>
<li><a href="priority_queue_text_push_pop_timing_test.html">Priority
Queue Text <tt>push</tt> and <tt>pop</tt> Timing
Test</a></li>
<li><a href="priority_queue_random_int_push_timing_test.html">Priority
Queue Random Integer <tt>push</tt> Timing Test</a></li>
<li><a href="priority_queue_random_int_push_pop_timing_test.html">Priority
Queue Random Integer <tt>push</tt> and <tt>pop</tt> Timing
Test</a></li>
<li><a href="priority_queue_text_pop_mem_usage_test.html">Priority Queue
Text <tt>pop</tt> Memory Use Test</a></li>
<li><a href="priority_queue_text_join_timing_test.html">Priority Queue
Text <tt>join</tt> Timing Test</a></li>
<li><a href="priority_queue_text_modify_up_timing_test.html">Priority
Queue Text <tt>modify</tt> Timing Test - I</a></li>
<li><a href="priority_queue_text_modify_down_timing_test.html">Priority
Queue Text <tt>modify</tt> Timing Test - II</a></li>
</ol>
<h2><a name="pq_observations" id="pq_observations">Observations</a></h2>
<h3><a name="pq_observations_cplx" id="pq_observations_cplx">Underlying Data Structures
Complexity</a></h3>
<p>The following table shows the complexities of the different
underlying data structures in terms of orders of growth. It is
interesting to note that this table implies something about the
constants of the operations as well (see <a href="#pq_observations_amortized_push_pop">Amortized <tt>push</tt>
and <tt>pop</tt> operations</a>).</p>
<table class="c1" width="100%" border="1" summary="pq complexities">
<tr>
<td align="left"></td>
<td align="left"><tt>push</tt></td>
<td align="left"><tt>pop</tt></td>
<td align="left"><tt>modify</tt></td>
<td align="left"><tt>erase</tt></td>
<td align="left"><tt>join</tt></td>
</tr>
<tr>
<td align="left">
<p><tt>std::priority_queue</tt></p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p class="c1">Θ(log(n)) Worst</p>
</td>
<td align="left">
<p><i>Theta;(n log(n))</i> Worst</p>
<p><sub><a href="#std_mod1">[std note 1]</a></sub></p>
</td>
<td align="left">
<p class="c3">Θ(n log(n))</p>
<p><sub><a href="#std_mod2">[std note 2]</a></sub></p>
</td>
<td align="left">
<p class="c3">Θ(n log(n))</p>
<p><sub><a href="#std_mod1">[std note 1]</a></sub></p>
</td>
</tr>
<tr>
<td align="left">
<p><a href="priority_queue.html"><tt>priority_queue</tt></a></p>
<p>with <tt>Tag</tt> =</p>
<p><a href="pairing_heap_tag.html"><tt>pairing_heap_tag</tt></a></p>
</td>
<td align="left">
<p class="c1">O(1)</p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p class="c1">O(1)</p>
</td>
</tr>
<tr>
<td align="left">
<p><a href="priority_queue.html"><tt>priority_queue</tt></a></p>
<p>with <tt>Tag</tt> =</p>
<p><a href="binary_heap_tag.html"><tt>binary_heap_tag</tt></a></p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p class="c1">Θ(n)</p>
</td>
<td align="left">
<p class="c1">Θ(n)</p>
</td>
<td align="left">
<p class="c1">Θ(n)</p>
</td>
</tr>
<tr>
<td align="left">
<p><a href="priority_queue.html"><tt>priority_queue</tt></a></p>
<p>with <tt>Tag</tt> =</p>
<p><a href="binomial_heap_tag.html"><tt>binomial_heap_tag</tt></a></p>
</td>
<td align="left">
<p><i>Θ(log(n))</i> worst</p>
<p><i>O(1)</i> amortized</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
</tr>
<tr>
<td align="left">
<p><a href="priority_queue.html"><tt>priority_queue</tt></a></p>
<p>with <tt>Tag</tt> =</p>
<p><a href="rc_binomial_heap_tag.html"><tt>rc_binomial_heap_tag</tt></a></p>
</td>
<td align="left">
<p class="c1">O(1)</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
<td align="left">
<p class="c1">Θ(log(n))</p>
</td>
</tr>
<tr>
<td align="left">
<p><a href="priority_queue.html"><tt>priority_queue</tt></a></p>
<p>with <tt>Tag</tt> =</p>
<p><a href="thin_heap_tag.html"><tt>thin_heap_tag</tt></a></p>
</td>
<td align="left">
<p class="c1">O(1)</p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p><i>Θ(log(n))</i> worst</p>
<p><i>O(1)</i> amortized,</p>or
<p><i>Θ(log(n))</i> amortized</p>
<p><sub><a href="#thin_heap_note">[thin_heap_note]</a></sub></p>
</td>
<td align="left">
<p><i>Θ(n)</i> worst</p>
<p><i>Θ(log(n))</i> amortized</p>
</td>
<td align="left">
<p class="c1">Θ(n)</p>
</td>
</tr>
</table>
<p><sub><a name="std_mod1" id="std_mod1">[std note 1]</a> This
is not a property of the algorithm, but rather due to the fact
that the STL's priority queue implementation does not support
iterators (and consequently the ability to access a specific
value inside it). If the priority queue is adapting an
<tt>std::vector</tt>, then it is still possible to reduce this
to <i>Θ(n)</i> by adapting over the STL's adapter and
using the fact that <tt>top</tt> returns a reference to the
first value; if, however, it is adapting an
<tt>std::deque</tt>, then this is impossible.</sub></p>
<p><sub><a name="std_mod2" id="std_mod2">[std note 2]</a> As
with <a href="#std_mod1">[std note 1]</a>, this is not a
property of the algorithm, but rather the STL's implementation.
Again, if the priority queue is adapting an
<tt>std::vector</tt> then it is possible to reduce this to
<i>Θ(n)</i>, but with a very high constant (one must call
<tt>std::make_heap</tt> which is an expensive linear
operation); if the priority queue is adapting an
<tt>std::dequeu</tt>, then this is impossible.</sub></p>
<p><sub><a name="thin_heap_note" id="thin_heap_note">[thin_heap_note]</a> A thin heap has
<i>&Theta(log(n))</i> worst case <tt>modify</tt> time
always, but the amortized time depends on the nature of the
operation: I) if the operation increases the key (in the sense
of the priority queue's comparison functor), then the amortized
time is <i>O(1)</i>, but if II) it decreases it, then the
amortized time is the same as the worst case time. Note that
for most algorithms, I) is important and II) is not.</sub></p>
<h3><a name="pq_observations_amortized_push_pop" id="pq_observations_amortized_push_pop">Amortized <tt>push</tt>
and <tt>pop</tt> operations</a></h3>
<p>In many cases, a priority queue is needed primarily for
sequences of <tt>push</tt> and <tt>pop</tt> operations. All of
the underlying data structures have the same amortized
logarithmic complexity, but they differ in terms of
constants.</p>
<p>The table above shows that the different data structures are
"constrained" in some respects. In general, if a data structure
has lower worst-case complexity than another, then it will
perform slower in the amortized sense. Thus, for example a
redundant-counter binomial heap (<a href="priority_queue.html"><tt>priority_queue</tt></a> with
<tt>Tag</tt> = <a href="rc_binomial_heap_tag.html"><tt>rc_binomial_heap_tag</tt></a>)
has lower worst-case <tt>push</tt> performance than a binomial
heap (<a href="priority_queue.html"><tt>priority_queue</tt></a>
with <tt>Tag</tt> = <a href="binomial_heap_tag.html"><tt>binomial_heap_tag</tt></a>),
and so its amortized <tt>push</tt> performance is slower in
terms of constants.</p>
<p>As the table shows, the "least constrained" underlying
data structures are binary heaps and pairing heaps.
Consequently, it is not surprising that they perform best in
terms of amortized constants.</p>
<ol>
<li>Pairing heaps seem to perform best for non-primitive
types (<i>e.g.</i>, <tt>std::string</tt>s), as shown by
<a href="priority_queue_text_push_timing_test.html">Priority
Queue Text <tt>push</tt> Timing Test</a> and <a href="priority_queue_text_push_pop_timing_test.html">Priority
Queue Text <tt>push</tt> and <tt>pop</tt> Timing
Test</a></li>
<li>binary heaps seem to perform best for primitive types
(<i>e.g.</i>, <tt><b>int</b></tt>s), as shown by <a href="priority_queue_random_int_push_timing_test.html">Priority
Queue Random Integer <tt>push</tt> Timing Test</a> and
<a href="priority_queue_random_int_push_pop_timing_test.html">Priority
Queue Random Integer <tt>push</tt> and <tt>pop</tt> Timing
Test</a>.</li>
</ol>
<h3><a name="pq_observations_graph" id="pq_observations_graph">Graph Algorithms</a></h3>
<p>In some graph algorithms, a decrease-key operation is
required [<a href="references.html#clrs2001">clrs2001</a>];
this operation is identical to <tt>modify</tt> if a value is
increased (in the sense of the priority queue's comparison
functor). The table above and <a href="priority_queue_text_modify_up_timing_test.html">Priority Queue
Text <tt>modify</tt> Timing Test - I</a> show that a thin heap
(<a href="priority_queue.html"><tt>priority_queue</tt></a> with
<tt>Tag</tt> = <a href="thin_heap_tag.html"><tt>thin_heap_tag</tt></a>)
outperforms a pairing heap (<a href="priority_queue.html"><tt>priority_queue</tt></a> with
<tt>Tag</tt> =<tt>Tag</tt> = <a href="pairing_heap_tag.html"><tt>pairing_heap_tag</tt></a>),
but the rest of the tests show otherwise.</p>
<p>This makes it difficult to decide which implementation to
use in this case. Dijkstra's shortest-path algorithm, for
example, requires <i>Θ(n)</i> <tt>push</tt> and
<tt>pop</tt> operations (in the number of vertices), but
<i>O(n<sup>2</sup>)</i> <tt>modify</tt> operations, which can
be in practice <i>Θ(n)</i> as well. It is difficult to
find an <i>a-priori</i> characterization of graphs in which the
<u>actual</u> number of <tt>modify</tt> operations will dwarf
the number of <tt>push</tt> and <tt>pop</tt> operations.</p>
</div>
</body>
</html>