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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H1><A NAME="SECTION004500000000000000000">Exercises</A></H1>
<P>
<OL><LI>
Consider the function <IMG WIDTH=133 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60763" SRC="img549.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img549.gif" >.
Using Definition <A HREF="page57.html#defnbigoh" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page57.html#defnbigoh"><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> show that <IMG WIDTH=92 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59085" SRC="img244.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img244.gif" >.<LI>
Consider the function <IMG WIDTH=133 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60763" SRC="img549.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img549.gif" >.
Using Definition <A HREF="page66.html#defnomega" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page66.html#defnomega"><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> show that <IMG WIDTH=92 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59939" SRC="img411.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img411.gif" >.<LI>
Consider the functions <IMG WIDTH=133 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60763" SRC="img549.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img549.gif" > and <IMG WIDTH=123 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60773" SRC="img550.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img550.gif" >.
Using Theorem <A HREF="page60.html#theoremii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html#theoremii"><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> show that <IMG WIDTH=142 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60775" SRC="img551.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img551.gif" >.<LI>
Consider the functions <IMG WIDTH=75 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline60777" SRC="img552.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img552.gif" > and <IMG WIDTH=84 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60779" SRC="img553.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img553.gif" >.
Using Theorem <A HREF="page60.html#theoremii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html#theoremii"><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> show that <IMG WIDTH=148 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline60781" SRC="img554.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img554.gif" >.<LI>
For each pair of functions, <I>f</I>(<I>n</I>) and <I>g</I>(<I>n</I>),
in the following table,
indicate if <I>f</I>(<I>n</I>)=<I>O</I>(<I>g</I>(<I>n</I>)) and if <I>g</I>(<I>n</I>)=<I>O</I>(<I>f</I>(<I>n</I>)).
<DIV ALIGN=CENTER><P ALIGN=CENTER><TABLE COLS=2 BORDER FRAME=HSIDES RULES=GROUPS>
<COL ALIGN=CENTER><COL ALIGN=CENTER>
<TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<I>f</I>(<I>n</I>) </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <I>g</I>(<I>n</I>) </TD></TR>
</TBODY><TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>10<I>n</I> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=62 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60797" SRC="img555.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img555.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=16 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline60799" SRC="img556.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img556.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=52 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline60801" SRC="img557.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img557.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=44 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60803" SRC="img558.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img558.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=62 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60805" SRC="img559.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img559.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=33 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59565" SRC="img353.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img353.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=22 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline60809" SRC="img560.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img560.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=26 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline58535" SRC="img113.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img113.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=33 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59565" SRC="img353.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img353.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=69 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60815" SRC="img561.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img561.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=33 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59565" SRC="img353.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img353.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=55 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60819" SRC="img562.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img562.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=33 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59565" SRC="img353.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img353.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=14 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline60823" SRC="img563.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img563.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=21 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline60825" SRC="img564.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img564.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=20 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline60021" SRC="img418.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img418.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=20 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline60829" SRC="img565.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img565.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=68 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60831" SRC="img566.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img566.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=68 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60833" SRC="img567.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img567.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<IMG WIDTH=16 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline60835" SRC="img568.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img568.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=63 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60837" SRC="img569.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img569.gif" > </TD></TR>
</TBODY>
</TABLE>
</P></DIV><LI><A NAME="exerciseasymptoticfib"> </A>
Show that the Fibonacci numbers (see Equation <A HREF="page73.html#eqnasymptoticfibonacci" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page73.html#eqnasymptoticfibonacci"><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>)
satisfy the identities
<P> <IMG WIDTH=173 HEIGHT=43 ALIGN=BOTTOM ALT="gather2331" SRC="img570.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img570.gif" ><P>
for <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59533" SRC="img344.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img344.gif" >.<LI>
Prove each of the following formulas:
<OL><LI> <IMG WIDTH=92 HEIGHT=50 ALIGN=MIDDLE ALT="tex2html_wrap_inline60841" SRC="img571.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img571.gif" ><LI> <IMG WIDTH=99 HEIGHT=50 ALIGN=MIDDLE ALT="tex2html_wrap_inline60843" SRC="img572.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img572.gif" ><LI> <IMG WIDTH=100 HEIGHT=50 ALIGN=MIDDLE ALT="tex2html_wrap_inline60845" SRC="img573.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img573.gif" ></OL><LI>
Show that <IMG WIDTH=107 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline60847" SRC="img574.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img574.gif" >,
where <IMG WIDTH=65 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline58953" SRC="img219.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img219.gif" > and <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59063" SRC="img241.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img241.gif" >.<LI>
Show that <IMG WIDTH=128 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline60853" SRC="img575.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img575.gif" >.<LI>
Solve each of the following recurrences:
<OL><LI>
<IMG WIDTH=307 HEIGHT=56 ALIGN=MIDDLE ALT="tex2html_wrap_inline60855" SRC="img576.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img576.gif" ><LI>
<IMG WIDTH=309 HEIGHT=56 ALIGN=MIDDLE ALT="tex2html_wrap_inline60857" SRC="img577.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img577.gif" ><LI>
<IMG WIDTH=310 HEIGHT=56 ALIGN=MIDDLE ALT="tex2html_wrap_inline60859" SRC="img578.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img578.gif" ><LI>
<IMG WIDTH=312 HEIGHT=56 ALIGN=MIDDLE ALT="tex2html_wrap_inline60861" SRC="img579.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img579.gif" ></OL><LI>
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