From d461a7211f0df57f26735939703c7466e3274ccd Mon Sep 17 00:00:00 2001
From: Martin Goik <goik@hdm-stuttgart.de>
Date: Wed, 18 Jan 2023 15:31:46 +0100
Subject: [PATCH] Hash value clashes
---
Doc/Sd1/CoreClasses/coreClasses.xml | 107 ++++++++++++++++++++--------
1 file changed, 79 insertions(+), 28 deletions(-)
diff --git a/Doc/Sd1/CoreClasses/coreClasses.xml b/Doc/Sd1/CoreClasses/coreClasses.xml
index 9ae0e8da4..de66a8e2e 100644
--- a/Doc/Sd1/CoreClasses/coreClasses.xml
+++ b/Doc/Sd1/CoreClasses/coreClasses.xml
@@ -526,17 +526,18 @@ public boolean equals(Object anObject) {
<para><code language="java">true == a.equals(b)</code> ⟹ <code
language="java">a.hashCode() == b.hashCode()</code>.</para>
- <para>An ideal/perfect <methodname>hashCode()</methodname> method
- in addition will return different values whenever two instances
- <code>a</code> and <code>b</code> differ in value with respect to
- the underlying <methodname>equals()</methodname> method:</para>
+ <para>A so called <emphasis>perfect</emphasis>
+ <methodname>hashCode()</methodname> method in addition will return
+ different values whenever two instances <code>a</code> and
+ <code>b</code> differ in value with respect to the underlying
+ <methodname>equals()</methodname> method:</para>
<para><code language="java">false == a.equals(b)</code> ⟺ <code
language="java">a.hashCode() != b.hashCode()</code>.</para>
- <para>Combining these two statements a perfect hashCode() method
- will have the following property with respect to its corresponding
- <methodname>equals()</methodname> method:</para>
+ <para>Combining these two statements a
+ <emphasis>perfect</emphasis> hashCode() method will have the
+ following property:</para>
<para><code language="java">a.equals(b) == (a.hashCode() ==
b.hashCode())</code></para>
@@ -565,6 +566,23 @@ public boolean equals(Object anObject) {
So method 2 requiring just two additions offers (slightly) better
runtime performance at the expense of a higher hash value
collision rate.</para>
+
+ <note>
+ <para>Perfect hash functions are rare with respect to real world
+ modeling problems. In the current example
+ <classname>Timeperiod</classname> instances are limited by 24
+ hours, 59 minutes and 59 seconds. This limit is equal to 89999
+ seconds fitting well into the count of <inlineequation>
+ <m:math display="inline">
+ <m:msup>
+ <m:mi>2</m:mi>
+
+ <m:mi>32</m:mi>
+ </m:msup>
+ </m:math>
+ </inlineequation> different <code language="java">int</code>
+ values.</para>
+ </note>
</answer>
</qandaentry>
</qandadiv>
@@ -577,7 +595,7 @@ public boolean equals(Object anObject) {
<qandadiv>
<qandaentry>
<question>
- <para>In the previous exercise we found an ideal
+ <para>In the previous exercise we found a perfect
<methodname>hashCode()</methodname> implementation:</para>
<programlisting language="java">public class TimePeriod {
@@ -588,7 +606,9 @@ public boolean equals(Object anObject) {
}
}</programlisting>
- <para>Is this possible for instances of String as well?</para>
+ <para>Does a perfect hash function exist for <link
+ xlink:href="https://docs.oracle.com/en/java/javase/17/docs/api/java.base/java/lang/String.html">String</link>
+ instances as well?</para>
<tip>
<para>Consider the possible number of different strings.</para>
@@ -596,19 +616,27 @@ public boolean equals(Object anObject) {
</question>
<answer>
- <para>It is not possible to construct a perfect
- <methodname>hashCode()</methodname> method acting on arbitrary
- strings. A Java <classname
+ <para>It is not possible to construct a perfect <link
+ xlink:href="https://docs.oracle.com/en/java/javase/17/docs/api/java.base/java/lang/Object.html#hashCode()">hashCode()</link>
+ method acting on strings. A Java <classname
xlink:href="https://docs.oracle.com/en/java/javase/17/docs/api/java.base/java/lang/String.html">String</classname>
consists of individual <code language="java">char</code> elements
- each requiring two bytes. Considering strings of fixed length we
- have the following number of different strings:</para>
+ each requiring two bytes representing <inlineequation>
+ <m:math display="inline">
+ <m:msup>
+ <m:mi>2</m:mi>
+
+ <m:mi>16</m:mi>
+ </m:msup>
+ </m:math>
+ </inlineequation> different characters. Depending on a string's
+ length we have:</para>
<informaltable border="1">
<tr>
- <th>Number of chars</th>
+ <th>String length (number of characters)</th>
- <th>Number of possible strings</th>
+ <th>Number of different strings</th>
</tr>
<tr>
@@ -666,22 +694,45 @@ public boolean equals(Object anObject) {
</tr>
</informaltable>
- <para>A four byte <code language="java">int</code> only offers
- <inlineequation>
+ <para>Thus considering just the union of zero (empty), one- and
+ two character strings we have <inlineequation>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>1</m:mi>
+
+ <m:mo>+</m:mo>
+
+ <m:msup>
+ <m:mi>2</m:mi>
+
+ <m:mi>16</m:mi>
+ </m:msup>
+
+ <m:mo>+</m:mo>
+
+ <m:msup>
+ <m:mi>2</m:mi>
+
+ <m:mi>32</m:mi>
+ </m:msup>
+ </m:mrow>
+ </m:math>
+ </inlineequation> possibilities exceeding the <inlineequation>
<m:math display="inline">
<m:msup>
<m:mi>2</m:mi>
- <m:mrow>
- <m:mi>32</m:mi>
- </m:mrow>
+ <m:mi>32</m:mi>
</m:msup>
</m:math>
- </inlineequation> different values. Thus even mapping just one-
- and two-<xref linkend="glo_unicode"/> character strings exceeds
- the number of different <code language="java">int</code> values
- thus requiring different string instances being mapped to
- identical hash values. Consider for example:</para>
+ </inlineequation> count of different <code
+ language="java">int</code> values. Thus hash value clashes are
+ inevitable.</para>
+
+ <para>The <xref linkend="glo_JDK"/>'s <link
+ xlink:href="https://docs.oracle.com/en/java/javase/17/docs/api/java.base/java/lang/String.html#hashCode()">String.hashCode()</link>
+ implementation already reveals conflicts for <xref
+ linkend="glo_ASCII"/> strings of length 2:</para>
<informaltable border="1">
<tr>
@@ -691,10 +742,10 @@ public boolean equals(Object anObject) {
</tr>
<tr>
- <td valign="top"><programlisting language="java">System.out.println("hashcode of AA: " + "Aa".hashCode());
+ <td valign="top"><programlisting language="java">System.out.println("hashcode of Aa: " + "Aa".hashCode());
System.out.println("hashcode of BB: " + "BB".hashCode());</programlisting></td>
- <td valign="top"><screen>hashcode of AA: 2112
+ <td valign="top"><screen>hashcode of Aa: 2112
hashcode of BB: 2112</screen></td>
</tr>
</informaltable>
--
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