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Salvatore Filippone
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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION000727000000000000000">
Sorting utilities</A>
</H2>
<H1><A NAME="SECTION00080000000000000000"></A>
<A NAME="sec:parenv"></A>
<BR>
Parallel environment routines
</H1>
<P>
<FONT SIZE="+1"><B>psb_msort -- Sorting by the Merge-sort
algorithm</B></FONT>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
<P>
<FONT SIZE="+1"><B>psb_qsort -- Sorting by the Quicksort
algorithm</B></FONT>
<P>
<FONT SIZE="+1"><B>psb_hsort -- Sorting by the Heapsort algorithm</B></FONT>
<PRE>
call psb_msort(x,ix,dir,flag)
call psb_qsort(x,ix,dir,flag)
call psb_hsort(x,ix,dir,flag)
</PRE>
<P>
These serial routines sort a sequence <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img112.png"
ALT="$X$"> into ascending or
descending order. The argument meaning is identical for the three
calls; the only difference is the algorithm used to accomplish the
task (see Usage Notes below).
<DL>
<DT><STRONG>Type:</STRONG></DT>
<DD>Asynchronous.
</DD>
<DT><STRONG><B>On Entry </B></STRONG></DT>
<DD>
</DD>
<DT><STRONG>x</STRONG></DT>
<DD>The sequence to be sorted.
<BR>
Type:<B>required</B>.
<BR>
Specified as: an integer, real or complex array of rank 1.
</DD>
<DT><STRONG>ix</STRONG></DT>
<DD>A vector of indices.
<BR>
Type:<B>optional</B>.
<BR>
Specified as: an integer array of (at least) the same size as <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img112.png"
ALT="$X$">.
</DD>
<DT><STRONG>dir</STRONG></DT>
<DD>The desired ordering.
<BR>
Type:<B>optional</B>.
<BR>
Specified as: an integer value: <DL>
<DT><STRONG>Integer and real data:</STRONG></DT>
<DD><code>psb_sort_up_</code>,
<code>psb_sort_down_</code>, <code>psb_asort_up_</code>, <code>psb_asort_down_</code>;
default <code>psb_sort_up_</code>.
</DD>
<DT><STRONG>Complex data:</STRONG></DT>
<DD><code>psb_lsort_up_</code>,
<code>psb_lsort_down_</code>, <code>psb_asort_up_</code>, <code>psb_asort_down_</code>;
default <code>psb_lsort_up_</code>.
</DD>
</DL>
</DD>
<DT><STRONG>flag</STRONG></DT>
<DD>Whether to keep the original values in <IMG
WIDTH="27" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img113.png"
ALT="$IX$">.
<BR>
Type:<B>optional</B>.
<BR>
Specified as: an integer value <code>psb_sort_ovw_idx_</code> or
<code>psb_sort_keep_idx_</code>; default <code>psb_sort_ovw_idx_</code>.
<P>
</DD>
</DL>
<P>
<DL>
<DT><STRONG><B>On Return</B></STRONG></DT>
<DD>
</DD>
<DT><STRONG>x</STRONG></DT>
<DD>The sequence of values, in the chosen ordering.
<BR>
Type:<B>required</B>.
<BR>
Specified as: an integer, real or complex array of rank 1.
</DD>
<DT><STRONG>ix</STRONG></DT>
<DD>A vector of indices.
<BR>
Type: <B>Optional</B>
<BR>
An integer array of rank 1, whose entries are moved to the same
position as the corresponding entries in <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$x$">.
</DD>
</DL>
<P>
<P>
<FONT SIZE="+1"><B>Notes</B></FONT>
<OL>
<LI>For integer or real data the sorting can be performed in the up/down direction, on the
natural or absolute values;
</LI>
<LI>For complex data the sorting can be done in a lexicographic
order (i.e.: sort on the real part with ties broken according to
the imaginary part) or on the absolute values;
</LI>
<LI>The routines return the items in the chosen ordering; the
output difference is the handling of ties (i.e. items with an
equal value) in the original input. With the merge-sort algorithm
ties are preserved in the same relative order as they had in the
original sequence, while this is not guaranteed for quicksort or
heapsort;
</LI>
<LI>If <!-- MATH
$flag = psb\_sort\_ovw\_idx\_$
-->
<IMG
WIDTH="180" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img114.png"
ALT="$flag = psb\_sort\_ovw\_idx\_$"> then the entries in <IMG
WIDTH="62" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img115.png"
ALT="$ix(1:n)$">
where <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img16.png"
ALT="$n$"> is the size of <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$x$"> are initialized to <!-- MATH
$ix(i) \leftarrow
i$
-->
<IMG
WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img116.png"
ALT="$ix(i) \leftarrow
i$">; thus, upon return from the subroutine, for each
index <IMG
WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$i$"> we have in <IMG
WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img117.png"
ALT="$ix(i)$"> the position that the item <IMG
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img118.png"
ALT="$x(i)$">
occupied in the original data sequence;
</LI>
<LI>If <!-- MATH
$flag = psb\_sort\_keep\_idx\_$
-->
<IMG
WIDTH="185" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img119.png"
ALT="$flag = psb\_sort\_keep\_idx\_$"> the routine will assume that
the entries in <IMG
WIDTH="35" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img120.png"
ALT="$ix(:)$"> have already been initialized by the user;
</LI>
<LI>The three sorting algorithms have a similar <IMG
WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img121.png"
ALT="$O(n \log n)$"> expected
running time; in the average case quicksort will be the
fastest and merge-sort the slowest. However note that:
<OL>
<LI>The worst case running time for quicksort is <IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img122.png"
ALT="$O(n^2)$">; the algorithm
implemented here follows the well-known median-of-three heuristics,
but the worst case may still apply;
</LI>
<LI>The worst case running time for merge-sort and heap-sort is
<IMG
WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img121.png"
ALT="$O(n \log n)$"> as the average case;
</LI>
<LI>The merge-sort algorithm is implemented to take advantage of
subsequences that may be already in the desired ordering prior to
the subroutine call; this situation is relatively common when
dealing with groups of indices of sparse matrix entries, thus
merge-sort is often the preferred choice when a sorting is needed
by other routines in the library.
</LI>
</OL>
</LI>
</OL>
<P>
<P>
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<UL>
<LI><A NAME="tex2html1081"
HREF="node73.html">psb_init -- Initializes PSBLAS parallel environment</A>
<LI><A NAME="tex2html1082"
HREF="node74.html">psb_info -- Return information about PSBLAS parallel
environment</A>
<LI><A NAME="tex2html1083"
HREF="node75.html">psb_exit -- Exit from PSBLAS parallel environment</A>
<LI><A NAME="tex2html1084"
HREF="node76.html">psb_get_mpicomm -- Get the MPI communicator</A>
<LI><A NAME="tex2html1085"
HREF="node77.html">psb_get_rank -- Get the MPI rank</A>
<LI><A NAME="tex2html1086"
HREF="node78.html">psb_wtime -- Wall clock timing</A>
<LI><A NAME="tex2html1087"
HREF="node79.html">psb_barrier -- Sinchronization point parallel
environment</A>
<LI><A NAME="tex2html1088"
HREF="node80.html">psb_abort -- Abort a computation</A>
<LI><A NAME="tex2html1089"
HREF="node81.html">psb_bcast -- Broadcast data</A>
<LI><A NAME="tex2html1090"
HREF="node82.html">psb_sum -- Global sum</A>
<LI><A NAME="tex2html1091"
HREF="node83.html">psb_max -- Global maximum</A>
<LI><A NAME="tex2html1092"
HREF="node84.html">psb_min -- Global minimum</A>
<LI><A NAME="tex2html1093"
HREF="node85.html">psb_amx -- Global maximum absolute value</A>
<LI><A NAME="tex2html1094"
HREF="node86.html">psb_amn -- Global minimum absolute value</A>
<LI><A NAME="tex2html1095"
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<LI><A NAME="tex2html1096"
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