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<b>Data Structures and Algorithms 
with Object-Oriented Design Patterns in C++</b><br>
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<H3><A NAME="SECTION009112000000000000000">Spreading Keys Evenly</A></H3>
<P>
Let  <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline58415" SRC="img85.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img85.gif"  > be the probability that the hash function  <IMG WIDTH=52 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline62210" SRC="img877.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img877.gif"  >.
A hash function which spreads keys evenly has the property that
for  <IMG WIDTH=71 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline62212" SRC="img878.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img878.gif"  >,  <IMG WIDTH=67 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline62214" SRC="img879.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img879.gif"  >
In other words,
the hash values computed by the function  <IMG WIDTH=24 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59195" SRC="img264.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img264.gif"  >
are <em>uniformly distributed</em><A NAME=11200>&#160;</A>.
Unfortunately,
in order to say something about the distribution of the hash values,
we need to know something about the distribution of the keys.
<P>
In the absence of any information to the contrary,
we assume that the keys are equiprobable.
Let  <IMG WIDTH=16 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline62218" SRC="img880.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img880.gif"  > be the set of keys that map to the value <I>i</I>.
I.e.,  <IMG WIDTH=167 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline62222" SRC="img881.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img881.gif"  >.
If this is the case,
the requirement to spread the keys uniformly implies that   <IMG WIDTH=94 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline62224" SRC="img882.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img882.gif"  >.
An equal number of keys should map into each array position.
<P>
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