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<TITLE>Rules For Big Oh Analysis of Running Time</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="SECTION004410000000000000000">Rules For Big Oh Analysis of Running Time</A></H2>
<P>
In this section we present some simple rules for determining
a big-oh upper bound on the running time of the
basic compound statements<A NAME=1867> </A> in a C++ program.
<P>
<P><A NAME="rulei"> </A> <IMG WIDTH=575 HEIGHT=185 ALIGN=BOTTOM ALT="ruledef1868" SRC="img446.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img446.gif" ><P>
<P>
Rule <A HREF="page71.html#rulei" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page71.html#rulei"><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> follows directly from Theorem <A HREF="page60.html#theoremi" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html#theoremi"><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 total running time of a sequence of statements is equal to the
sum of the running times of the individual statements.
By Theorem <A HREF="page60.html#theoremi" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html#theoremi"><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>,
when computing the sum of a series of functions
it is the largest one (the <IMG WIDTH=30 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline59303" SRC="img293.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img293.gif" >) that determines the bound.
<P>
<P><A NAME="ruleii"> </A> <IMG WIDTH=579 HEIGHT=134 ALIGN=BOTTOM ALT="ruledef1881" SRC="img447.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img447.gif" ><P>
<P>
Rule <A HREF="page71.html#ruleii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page71.html#ruleii"><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> appears somewhat complicated due to the semantics
of the C++ <tt>for</tt> statement.
However, it follows directly from Theorem <A HREF="page60.html#theoremiv" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html#theoremiv"><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>.
Consider the following simple <em>counted do loop</em><A NAME=1897> </A>.
<PRE>for (int i = 0; i < n; ++i)
<IMG WIDTH=14 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap60263" SRC="img448.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img448.gif" >;
</PRE>
Here <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" > is <code>int i = 0</code>, so its running time is constant ( <IMG WIDTH=66 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60307" SRC="img450.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img450.gif" >);
<IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline60309" SRC="img451.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img451.gif" > is <code>i < n</code>, so its running time is constant ( <IMG WIDTH=66 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60311" SRC="img452.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img452.gif" >); and
<IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline60313" SRC="img453.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img453.gif" > is <code>++i</code>, so its running time is constant ( <IMG WIDTH=66 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60315" SRC="img454.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img454.gif" >).
Also, the number of iterations is <I>I</I>(<I>n</I>)=<I>n</I>.
According to Rule <A HREF="page71.html#ruleii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page71.html#ruleii"><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 running time of this is <IMG WIDTH=276 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60319" SRC="img455.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img455.gif" >,
which simplifies to <IMG WIDTH=149 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60321" SRC="img456.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img456.gif" >.
Furthermore, if the loop body <em>does anything at all</em>,
its running time must be <IMG WIDTH=91 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60323" SRC="img457.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img457.gif" >.
Hence, the loop body will dominate the calculation of the maximum,
and the running time of the loop is simply <IMG WIDTH=90 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60325" SRC="img458.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img458.gif" >.
<P>
If we don't know the exact number of iterations executed, <I>I</I>(<I>n</I>),
we can still use Rule <A HREF="page71.html#ruleii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page71.html#ruleii"><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> provided we have an upper bound,
<I>I</I>(<I>n</I>)=<I>O</I>(<I>f</I>(<I>n</I>)), on the number of iterations executed.
In this case, the running time is
<IMG WIDTH=441 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline60331" SRC="img459.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img459.gif" >.
<P>
<P><A NAME="ruleiii"> </A> <IMG WIDTH=576 HEIGHT=174 ALIGN=BOTTOM ALT="ruledef1904" SRC="img460.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img460.gif" ><P>
<P>
Rule <A HREF="page71.html#ruleiii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page71.html#ruleiii"><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> follows directly from the observation
that the total running time for an if-then-else statement
will never exceed the sum of the running time of the conditional test,
<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" >, plus the larger of the running times of the <em>then part</em>, <IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline60309" SRC="img451.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img451.gif" >,
and the <em>else part</em>, <IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline60313" SRC="img453.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img453.gif" >.
<P>
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