<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!--Converted with LaTeX2HTML 2008 (1.71) original version by: Nikos Drakos, CBLU, University of Leeds * revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan * with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --> <HTML> <HEAD> <TITLE>Sorting utilities</TITLE> <META NAME="description" CONTENT="Sorting utilities"> <META NAME="keywords" CONTENT="userhtml"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <META NAME="Generator" CONTENT="LaTeX2HTML v2008"> <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css"> <LINK REL="STYLESHEET" HREF="userhtml.css"> <LINK REL="previous" HREF="node69.html"> <LINK REL="up" HREF="node43.html"> <LINK REL="next" HREF="node71.html"> </HEAD> <BODY > <!--Navigation Panel--> <A NAME="tex2html1052" HREF="node71.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A> <A NAME="tex2html1048" HREF="node43.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A> <A NAME="tex2html1044" HREF="node69.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A> <A NAME="tex2html1050" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A> <BR> <B> Next:</B> <A NAME="tex2html1053" HREF="node71.html">Parallel environment routines</A> <B> Up:</B> <A NAME="tex2html1049" HREF="node43.html">Data management routines</A> <B> Previous:</B> <A NAME="tex2html1045" HREF="node69.html">psb_sizeof Memory</A> <B> <A NAME="tex2html1051" HREF="node1.html">Contents</A></B> <BR> <BR> <!--End of Navigation Panel--> <H2><A NAME="SECTION000727000000000000000"> Sorting utilities</A> </H2> <P> <FONT SIZE="+1"><B>psb_msort -- Sorting by the Merge-sort algorithm</B></FONT> <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="img110.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="img110.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="img111.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="13" ALIGN="BOTTOM" BORDER="0" SRC="img26.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="img112.png" ALT="$flag = psb\_sort\_ovw\_idx\_$"> then the entries in <IMG WIDTH="62" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="img113.png" ALT="$ix(1:n)$"> where <IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" SRC="img15.png" ALT="$n$"> is the size of <IMG WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" SRC="img26.png" ALT="$x$"> are initialized to <!-- MATH $ix(i) \leftarrow i$ --> <IMG WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="img114.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="img115.png" ALT="$ix(i)$"> the position that the item <IMG WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="img116.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="img117.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="img118.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="img119.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="img120.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="img119.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> <HR> <!--Navigation Panel--> <A NAME="tex2html1052" HREF="node71.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A> <A NAME="tex2html1048" HREF="node43.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A> <A NAME="tex2html1044" HREF="node69.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A> <A NAME="tex2html1050" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A> <BR> <B> Next:</B> <A NAME="tex2html1053" HREF="node71.html">Parallel environment routines</A> <B> Up:</B> <A NAME="tex2html1049" HREF="node43.html">Data management routines</A> <B> Previous:</B> <A NAME="tex2html1045" HREF="node69.html">psb_sizeof Memory</A> <B> <A NAME="tex2html1051" HREF="node1.html">Contents</A></B> <!--End of Navigation Panel--> </BODY> </HTML>