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<TITLE>Example-Prefix Sums</TITLE>
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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="SECTION004420000000000000000">Example-Prefix Sums</A></H2>
<P>
In this section, we will determine a tight big-oh bound
on the running time of a program to compute the series of sums
<IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline60349" SRC="img461.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img461.gif" >, <IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline60305" SRC="img449.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img449.gif" >, ..., <IMG WIDTH=33 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline58691" SRC="img151.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img151.gif" >, where
<P> <IMG WIDTH=289 HEIGHT=47 ALIGN=BOTTOM ALT="displaymath60347" SRC="img462.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img462.gif" ><P>
An algorithm to compute this series of summations
is given in Program <A HREF="page72.html#progprefixsumsc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page72.html#progprefixsumsc"><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>.
Table <A HREF="page72.html#tblprefixsumsc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page72.html#tblprefixsumsc"><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> summarizes the running time calculation.
<P>
<P><A NAME="2165"> </A><A NAME="progprefixsumsc"> </A> <IMG WIDTH=575 HEIGHT=200 ALIGN=BOTTOM ALT="program1924" SRC="img465.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img465.gif" ><BR>
<STRONG>Program:</STRONG> Program to compute <IMG WIDTH=52 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline60355" SRC="img463.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img463.gif" > for <IMG WIDTH=66 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60357" SRC="img464.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img464.gif" ><BR>
<P>
<P>
<P><A NAME="2167"> </A>
<P>
<A NAME="tblprefixsumsc"> </A>
<DIV ALIGN=CENTER><P ALIGN=CENTER><TABLE COLS=2 BORDER FRAME=HSIDES RULES=GROUPS>
<COL ALIGN=CENTER><COL ALIGN=LEFT>
<TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
statement </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> time </TD></TR>
</TBODY><TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>3a </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <I>O</I>(1) </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
3b </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=156 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60361" SRC="img466.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img466.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
3c </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=156 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60361" SRC="img466.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img466.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
5 </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=156 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60361" SRC="img466.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img466.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
6a </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=156 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60361" SRC="img466.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img466.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
6b </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=164 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60369" SRC="img467.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img467.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
6c </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=164 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60369" SRC="img467.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img467.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
7 </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=164 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60369" SRC="img467.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img467.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
8 </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=156 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60361" SRC="img466.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img466.gif" > </TD></TR>
</TBODY><TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>TOTAL </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif" > </TD></TR>
</TBODY>
<CAPTION ALIGN=BOTTOM><STRONG>Table:</STRONG> Computing the running time
of Program <A HREF="page72.html#progprefixsumsc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page72.html#progprefixsumsc"><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></CAPTION></TABLE>
</P></DIV><P>
<P>
Usually the easiest way to analyze program which contains nested loops
is to start with the body of the inner-most loop.
In Program <A HREF="page72.html#progprefixsumsc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page72.html#progprefixsumsc"><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>,
the inner-most loop comprises lines 6 and 7.
In all, a constant amount of work is done--this includes the loop body (line 7),
the conditional test (line 6b) and
the incrementing of the loop index (line 6c).
<P>
For a given value of <I>j</I>,
the inner-most loop is done a total <I>j</I>+1 times.
And since the outer loop is done for <IMG WIDTH=155 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline60383" SRC="img468.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img468.gif" >,
in the worst case, the inner-most loop is done <I>n</I> times.
Therefore, the contribution of the inner loop to the running
time of one iteration of the outer loop is <I>O</I>(<I>n</I>).
<P>
The rest of the outer loop (lines 3, 5 and 8)
does a constant amount of work in each iteration.
This constant work is dominated by the <I>O</I>(<I>n</I>) of the inner loop.
The outer loop is does exactly <I>n</I> iterations.
Therefore, the total running time of the program is <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif" >.
<P>
But is this a tight big oh bound?
We might suspect that it is not,
because of the worst-case assumption we made
in the analysis concerning the number of times the inner loop is executed.
The inner-most loop is done exactly <I>j</I>+1 times
for <IMG WIDTH=155 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline60383" SRC="img468.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img468.gif" >.
However, we did the calculation assuming the inner loop is done <I>O</I>(<I>n</I>) times,
in each iteration of the outer loop.
Unfortunately, in order to determine whether our answer is a tight bound,
we must determine more precisely the actual running time of the program.
<P>
However, there is one approximate calculation that we can easily make.
If we observe that the running time will be dominated by the
work done in the inner-most loop,
and that the work done in one iteration of the inner-most loop is constant,
then all we need to do is to determine exactly the number of times
the inner loop is actually executed.
This is given by:
<P> <IMG WIDTH=500 HEIGHT=113 ALIGN=BOTTOM ALT="eqnarray1950" SRC="img469.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img469.gif" ><P>
Therefore, the result <IMG WIDTH=95 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60401" SRC="img470.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img470.gif" > is a tight, big-oh bound
on the running time of Program <A HREF="page72.html#progprefixsumsc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page72.html#progprefixsumsc"><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>.
<P>
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