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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H2><A NAME="SECTION003170000000000000000">Average Running Times</A></H2>
<A NAME="secmodelavg"> </A>
<P>
In the previous section,
we found the function, <IMG WIDTH=149 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58401" SRC="img82.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img82.gif" >,
which gives the running time of Program <A HREF="page42.html#progfindmaxc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page42.html#progfindmaxc"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>
as a function both of number of inputs, <I>n</I>,
and of the actual input values.
Suppose instead we are interested in a function <IMG WIDTH=73 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline58409" SRC="img83.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img83.gif" >
which gives the running time <em>on average</em>
for <I>n</I> inputs, regardless of the values of those inputs.
In other words, if we run Program <A HREF="page42.html#progfindmaxc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page42.html#progfindmaxc"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>,
a large number of times on a selection of random inputs of length <I>n</I>,
what will the average running time be?
<P>
We can write the sum of the running times given in Table <A HREF="page42.html#tblfindmaxc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page42.html#tblfindmaxc"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>
in the following form
<P><A NAME="eqnmodelavg"> </A> <IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation533" SRC="img84.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img84.gif" ><P>
where <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" > is the probability that line 6 of the program is executed.
The probability <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" > is given by
<P> <IMG WIDTH=337 HEIGHT=38 ALIGN=BOTTOM ALT="displaymath58403" SRC="img86.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img86.gif" ><P>
I.e., <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" > is the probability that the <IMG WIDTH=17 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58387" SRC="img77.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img77.gif" > array entry, <IMG WIDTH=11 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline58389" SRC="img78.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img78.gif" >,
is larger than the maximum of all the preceding array entries,
<IMG WIDTH=106 HEIGHT=13 ALIGN=MIDDLE ALT="tex2html_wrap_inline58425" SRC="img87.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img87.gif" >.
<P>
In order to determine <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" >,
we need to know (or to assume)
something about the distribution of input values.
For example, if we know <em>a priori</em> that the array passed to the function
<tt>FindMaximum</tt> is ordered from smallest to largest,
then we know that <IMG WIDTH=42 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58429" SRC="img88.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img88.gif" >.
Conversely, if we know that the array is ordered from largest to smallest,
then we know that <IMG WIDTH=43 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58431" SRC="img89.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img89.gif" >.
<P>
In the general case, we have no <em>a priori</em> knowledge of the
distribution of the values in the input array.
In this case, consider the <IMG WIDTH=17 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58387" SRC="img77.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img77.gif" > iteration of the loop.
In this iteration <IMG WIDTH=11 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline58389" SRC="img78.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img78.gif" > is compared with the maximum of the <I>i</I> values,
<IMG WIDTH=106 HEIGHT=13 ALIGN=MIDDLE ALT="tex2html_wrap_inline58425" SRC="img87.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img87.gif" > preceding it in the array.
Line 6 of Program <A HREF="page42.html#progfindmaxc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page42.html#progfindmaxc"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> is only executed if <IMG WIDTH=11 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline58389" SRC="img78.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img78.gif" > is the largest
of the <I>i</I>+1 values <IMG WIDTH=88 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline58445" SRC="img90.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img90.gif" >.
All things being equal,
we can say that this will happen with probability 1/(<I>i</I>+1).
Thus
<P><A NAME="eqnmodelprob"> </A> <IMG WIDTH=500 HEIGHT=79 ALIGN=BOTTOM ALT="eqnarray549" SRC="img91.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img91.gif" ><P>
<P>
Substituting this expression for <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" > in Equation <A HREF="page43.html#eqnmodelavg" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page43.html#eqnmodelavg"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>
and simplifying the result we get
<P><A NAME="eqnmodeltn"> </A> <IMG WIDTH=500 HEIGHT=179 ALIGN=BOTTOM ALT="eqnarray555" SRC="img92.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img92.gif" ><P>
where <IMG WIDTH=93 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline58451" SRC="img93.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img93.gif" >,
is the <IMG WIDTH=21 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58453" SRC="img94.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img94.gif" > <em>harmonic number</em><A NAME=574> </A>.
<P>
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