http:^^www.cs.washington.edu^education^courses^322^96w^hw1soln^hw1soln.html

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Date: Wed, 08 Jan 1997 20:46:43 GMTServer: NCSA/1.4.2Content-type: text/html<!DOCTYPE HTML PUBLIC "-//W3O//DTD W3 HTML 2.0//EN"><!Converted with LaTeX2HTML 95 (Thu Jan 19 1995) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds ><HEAD><TITLE>CSE 322  Assignment 1 Solution Set  Friday, January 12, 1996</TITLE></HEAD><BODY><meta name="description" value="CSE 322  Assignment 1 Solution Set  Friday, January 12, 1996"><meta name="keywords" value="hw1soln"><meta name="resource-type" value="document"><meta name="distribution" value="global"><P> <BR> <HR><A NAME=tex2html1 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="next" SRC="http://www.cs.washington.edu/general/latex2html_icons//next_motif.gif"></A>   <IMG ALIGN=BOTTOM ALT="up" SRC="http://www.cs.washington.edu/general/latex2html_icons//up_motif_gr.gif">   <IMG ALIGN=BOTTOM ALT="previous" SRC="http://www.cs.washington.edu/general/latex2html_icons//previous_motif_gr.gif">         <BR><B> Next:</B> <A NAME=tex2html2 HREF="node1.html">   About this document </A><BR> <HR> <P><P><H1>CSE 322  Assignment 1 Solution Set  Friday, January 12, 1996</H1><P><STRONG></STRONG><P><P><STRONG></STRONG><P><P><OL><LI> Prove that <IMG  ALIGN=MIDDLE ALT="" SRC="img1.gif"> for all <b>n &gt; 0</b>. (#27 p. 27)<P>We can prove this by induction on <b>n</b>:<P><b>Basis:</b> The statement is true for <b>n = 1</b> because<IMG  ALIGN=MIDDLE ALT="" SRC="img2.gif">.<P><b>Inductive hypothesis:</b> Assume that the statement is true for <b>n=k</b>:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img3.gif"><P><b>Inductive step:</b> From this we need to show it is true for <b>n=k+1</b>:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img4.gif"><P>By induction, the statement is true for all <IMG  ALIGN=MIDDLE ALT="" SRC="img5.gif">.<P><LI> Using the tree on p. 27:(#27 p. 27)<OL><LI> The depth of the tree is 4.  (The depth of the root is 0.)<LI> The ancestors of <IMG  ALIGN=MIDDLE ALT="" SRC="img6.gif"> are <IMG  ALIGN=MIDDLE ALT="" SRC="img7.gif">,<IMG  ALIGN=MIDDLE ALT="" SRC="img8.gif">,<IMG  ALIGN=MIDDLE ALT="" SRC="img9.gif">, and <IMG  ALIGN=MIDDLE ALT="" SRC="img10.gif">.(Every node is an ancestor of itself.)<LI> The minimal common ancestor of <IMG  ALIGN=MIDDLE ALT="" SRC="img11.gif"> and <IMG  ALIGN=MIDDLE ALT="" SRC="img12.gif"> is <IMG  ALIGN=MIDDLE ALT="" SRC="img13.gif"> andof <IMG  ALIGN=MIDDLE ALT="" SRC="img14.gif"> and <IMG  ALIGN=MIDDLE ALT="" SRC="img15.gif"> is <IMG  ALIGN=MIDDLE ALT="" SRC="img16.gif">.<LI> The subtree generated by <IMG  ALIGN=MIDDLE ALT="" SRC="img17.gif"> is the following:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img18.gif"><P><LI> The frontier of the tree is <IMG  ALIGN=MIDDLE ALT="" SRC="img19.gif">.</OL><LI> <OL><LI>The rank of <IMG  ALIGN=BOTTOM ALT="" SRC="img20.gif"> in the enumeration ordering of <IMG  ALIGN=MIDDLE ALT="" SRC="img21.gif"> is <P><IMG  ALIGN=BOTTOM ALT="" SRC="img22.gif"><P>This is because if we look at the strings of length <b>2n</b>, the first stringsin the enumeration are those that begin with <IMG  ALIGN=BOTTOM ALT="" SRC="img23.gif">.  Of the strings that begin with <IMG  ALIGN=BOTTOM ALT="" SRC="img24.gif">, <IMG  ALIGN=BOTTOM ALT="" SRC="img25.gif"> is the firstand <IMG  ALIGN=BOTTOM ALT="" SRC="img26.gif"> is the last of these.  This gives us the following:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img27.gif"><P>The rank of <IMG  ALIGN=BOTTOM ALT="" SRC="img28.gif"> is <IMG  ALIGN=BOTTOM ALT="" SRC="img29.gif">.The number of strings  of length <b>2n</b> that begin with <IMG  ALIGN=BOTTOM ALT="" SRC="img30.gif"> is justthe number of ways of writing the last <b>n</b> letters as a string in the alphabet.There are <IMG  ALIGN=BOTTOM ALT="" SRC="img31.gif"> such ending strings.  Combining these observations gives us:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img32.gif"><P><P><LI>The rank of <IMG  ALIGN=BOTTOM ALT="" SRC="img33.gif"> in the enumeration ordering of <IMG  ALIGN=MIDDLE ALT="" SRC="img34.gif"> is<P><IMG  ALIGN=BOTTOM ALT="" SRC="img35.gif"><P>In the enumeration ordering, the strings that come before <IMG  ALIGN=BOTTOM ALT="" SRC="img36.gif"> are exactlythe strings over <IMG  ALIGN=MIDDLE ALT="" SRC="img37.gif"> whose length is less than <b>n</b>.  Therefore,<P><IMG  ALIGN=BOTTOM ALT="" SRC="img38.gif"><P>For any <b>i</b>, the number of strings of length <b>i</b> is <IMG  ALIGN=BOTTOM ALT="" SRC="img39.gif">.  Thus the number of strings of length less than <b>n</b> is the sum of all these,so:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img40.gif"><P></OL></OL><BR> <HR><UL> <LI> <A NAME=tex2html3 HREF="node1.html#SECTION00010000000000000000">   About this document ... </A></UL><BR> <HR><A NAME=tex2html1 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="next" SRC="http://www.cs.washington.edu/general/latex2html_icons//next_motif.gif"></A>   <IMG ALIGN=BOTTOM ALT="up" SRC="http://www.cs.washington.edu/general/latex2html_icons//up_motif_gr.gif">   <IMG ALIGN=BOTTOM ALT="previous" SRC="http://www.cs.washington.edu/general/latex2html_icons//previous_motif_gr.gif">         <BR><B> Next:</B> <A NAME=tex2html2 HREF="node1.html">   About this document </A><BR> <HR> <P><BR> <HR><P><ADDRESS><I>James Fix <BR>Mon Jan 29 18:05:51 PST 1996</I></ADDRESS></BODY>

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