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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H3><A NAME="SECTION0017312000000000000000">Running Time Analysis</A></H3>
<P>
The running time of the depth-first traversal routine
depends on the graph representation scheme used.
The traversal visits each node in the graph at most once.
When a node is visited,
all the edges emanating from that node are considered.
During a complete traversal enumerates every edge in the graph.
<P>
Therefore, the worst-case running time for the depth-first traversal
of a graph is represented using an adjacency matrix is
<P> <IMG WIDTH=409 HEIGHT=19 ALIGN=BOTTOM ALT="displaymath71979" SRC="img2398.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2398.gif" ><P>
When adjacency lists are used,
the worst case running time for the depth-first traversal routine is
<P> <IMG WIDTH=424 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath71980" SRC="img2399.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2399.gif" ><P>
<P>
Recall that for a sparse graph graph <IMG WIDTH=82 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71739" SRC="img2360.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2360.gif" >.
If the sparse graph is represented using adjacency lists
and if <IMG WIDTH=137 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline71985" SRC="img2400.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2400.gif" > and <IMG WIDTH=144 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline71987" SRC="img2401.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2401.gif" >
the worst-case running time of the depth-first traversal
is simply <IMG WIDTH=43 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71805" SRC="img2371.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2371.gif" >.
<P>
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