\relax 
\citation{clrs2009}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Minimum Spanning Trees}{1}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{loa}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}Minimum Spanning Trees}{1}}
\newlabel{sec:intro}{{1.1}{1}}
\@writefile{loa}{\contentsline {algocf}{\numberline {1.1}{\ignorespaces Generic-MST}}{1}}
\newlabel{alg:generic-mst}{{1.1}{1}}
\newlabel{light-edges}{{1.1}{2}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces An example of Corollary 1.1\hbox {}. The edge $(u,v)$ connects $C$ to some other component of $G_A$ and is a light edge; it is therefore safe to add to the MST.}}{2}}
\newlabel{fig:cutthm}{{1.1}{2}}
\newlabel{light-edges-cor}{{1.1}{2}}
\citation{dijkstra1959}
\@writefile{toc}{\contentsline {section}{\numberline {1.2}Kruskal's Algorithm for MST}{3}}
\newlabel{sec:kruskal-mst}{{1.2}{3}}
\newlabel{thm:kruskal}{{1.2}{3}}
\@writefile{loa}{\contentsline {algocf}{\numberline {1.2}{\ignorespaces Kruskal-MST}}{4}}
\newlabel{alg:kruskal-mst}{{1.2}{4}}
\@writefile{toc}{\contentsline {section}{\numberline {1.3}Prim's MST algorithm}{4}}
\newlabel{sec:prim-mst}{{1.3}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces Example of Prim's Algorithm showing the set $A$ (encircled) and the least cost edges associated with each vertex in $V \setminus A$.}}{4}}
\newlabel{fig:prim}{{1.2}{4}}
\citation{motwani1995}
\citation{chan1998}
\citation{boruuvka1926a}
\citation{boruuvka1926b}
\@writefile{loa}{\contentsline {algocf}{\numberline {1.3}{\ignorespaces PRIM-MST}}{5}}
\newlabel{alg:prim-mst}{{1.3}{5}}
\@writefile{toc}{\contentsline {section}{\numberline {1.4}Randomized Algorithms for Minimum Spanning Trees}{5}}
\newlabel{sec:random-mst}{{1.4}{5}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4.1}Boruvka's Algorithm}{6}}
\newlabel{sec:boruvka}{{1.4.1}{6}}
\@writefile{loa}{\contentsline {algocf}{\numberline {1.4}{\ignorespaces Boruvka-MST}}{6}}
\newlabel{alg:boruvka}{{1.4}{6}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4.2}Heavy Edges}{6}}
\newlabel{sec:heavyedges}{{1.4.2}{6}}
\citation{chan1998}
\newlabel{lem:heavy-edge}{{1.1}{7}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4.3}Randomized Algorithm}{7}}
\newlabel{sec:random-mst-alg}{{1.4.3}{7}}
\@writefile{loa}{\contentsline {algocf}{\numberline {1.5}{\ignorespaces Randomized-MST}}{7}}
\newlabel{alg:rand-mst}{{1.5}{7}}
\newlabel{lem:observe-mst}{{1.2}{8}}
\newlabel{lem:sampling}{{1.3}{8}}
\@writefile{toc}{\contentsline {section}{\numberline {1.5}MST Verification}{8}}
\newlabel{sec:mstv}{{1.5}{8}}
\citation{tarjan1979}
\newlabel{lem:mst_if_all_heavy}{{1.4}{9}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5.1}Overview of verification algorithms}{9}}
\newlabel{prb:tpmaxima}{{1.1}{9}}
\citation{komlos1984}
\citation{komlos1985}
\citation{dixon1992}
\citation{karger1995}
\citation{klein1994}
\citation{king1995}
\citation{king1997}
\citation{hagerup2010}
\citation{komlos1984}
\citation{komlos1985}
\citation{harel1980}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5.2}Koml\'{o}s's Algorithm}{10}}
\newlabel{ss:komlos}{{1.5.2}{10}}
\citation{harel1984}
\citation{schieber1988}
\newlabel{prop:nd_weights}{{1.1}{11}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5.3}Dixon \emph  {et al.}'s Technique}{11}}
\newlabel{ss:dixon}{{1.5.3}{11}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces An example tree with marked vertices. The marked subtree defined by $u$ is enclosed in red.}}{12}}
\newlabel{fig:dixon_marked}{{1.3}{12}}
\citation{king1997}
\citation{king1995}
\citation{king1997}
\citation{dixon1992}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5.4}King's Method}{13}}
\newlabel{ss:kingsmethod}{{1.5.4}{13}}
\@writefile{loa}{\contentsline {algocf}{\numberline {1.6}{\ignorespaces FullBranchingTree}}{14}}
\newlabel{alg:fbt}{{1.6}{14}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.4}{\ignorespaces labelInTOC}}{15}}
\newlabel{fig:fbt}{{1.4}{15}}
\newlabel{lem:btp}{{1.5}{15}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5.5}Hagerup's Method}{16}}
\citation{tarjan1975}
\citation{boruuvka1926a}
\citation{boruuvka1926b}
\citation{nesetril2001}
\citation{tarjan1983}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5.6}Putting it all together}{17}}
\newlabel{ss:puttogether}{{1.5.6}{17}}
\@writefile{toc}{\contentsline {section}{\numberline {1.6}Bibliographic Notes}{17}}
\newlabel{sec:bibnotes}{{1.6}{17}}
\@writefile{toc}{\contentsline {section}{\numberline {1.7}Exercises}{17}}
\bibstyle{abbrv}
\bibdata{comp5703mstv}
\bibcite{boruuvka1926a}{1}
\bibcite{boruuvka1926b}{2}
\bibcite{chan1998}{3}
\bibcite{clrs2009}{4}
\bibcite{dijkstra1959}{5}
\bibcite{dixon1992}{6}
\bibcite{hagerup2010}{7}
\bibcite{harel1980}{8}
\bibcite{harel1984}{9}
\bibcite{karger1995}{10}
\bibcite{king1995}{11}
\bibcite{king1997}{12}
\bibcite{klein1994}{13}
\bibcite{komlos1984}{14}
\bibcite{komlos1985}{15}
\bibcite{motwani1995}{16}
\bibcite{nesetril2001}{17}
\bibcite{schieber1988}{18}
\bibcite{tarjan1983}{19}
\bibcite{tarjan1975}{20}
\bibcite{tarjan1979}{21}