Recipe3.9.Computing Combinatorial Functions

Recipe 3.9. Computing Combinatorial Functions

Problem

You need to compute the number of permutations or combinations of size r of a given set.

Solution

XSLT 1.0

If you know the formula for permutations of size r is N! / r! and you know that the formula for combinations is N! / r! * (N-r)!, then you might disregard this example; this book already gave an example for factorial. However, since factorials get very big quickly, you need to be a little crafty to get the best bang for your calculating buck:

<xsl:template name="ckbk:P">      <xsl:param name="n" select="1"/>      <xsl:param name="r" select="1"/>       <xsl:choose>         <xsl:when test="$n &lt; 0 or $r &lt; 0">NaN</xsl:when>         <xsl:when test="$n = 0">0</xsl:when>         <xsl:otherwise>                <xsl:call-template name="prod-range">                <xsl:with-param name="start" select="$r + 1"/>                <xsl:with-param name="end" select="$n"/>           </xsl:call-template>         </xsl:otherwise>       </xsl:choose> </xsl:template>     <xsl:template name="ckbk:C">      <xsl:param name="n" select="1"/>      <xsl:param name="r" select="1"/>       <xsl:choose>         <xsl:when test="$n &lt; 0 or $r &lt; 0">NaN</xsl:when>         <xsl:when test="$n = 0">0</xsl:when>         <xsl:otherwise>           <xsl:variable name="min"                 select="($r &lt;= $n - $r) * $r +  ($r > $n - $r) * $n - $r"/>           <xsl:variable name="max"                 select="($r >= $n - $r) * $r +  ($r &lt; $n - $r) * $n - $r"/>           <xsl:variable name="numerator">             <xsl:call-template name="prod-range">                  <xsl:with-param name="start" select="$max + 1"/>                  <xsl:with-param name="end" select="$n"/>             </xsl:call-template>           </xsl:variable>           <xsl:variable name="denominator">             <xsl:call-template name="ckbk:fact">               <xsl:with-param name="number" select="$min"/>             </xsl:call-template>           </xsl:variable>           <xsl:value-of select="$numerator div $denominator"/>         </xsl:otherwise>       </xsl:choose> </xsl:template>

XSLT 2.0

<xsl:function name="ckbk:P" as="xs:decimal">   <xsl:param name="r" as="xs:integer"/>   <xsl:param name="n" as="xs:integer"/>   <xsl:sequence select="if ($n eq 0) then 0 else ckbk:prod-range($r + 1, $n)"/> </xsl:function>     <xsl:function name="ckbk:C" as="xs:decimal">   <xsl:param name="r" as="xs:integer"/>   <xsl:param name="n" as="xs:integer"/>   <xsl:variable name="min" select="min( ($r, $n - $r) )" as="xs:integer"/>   <xsl:variable name="max" select="max( ($r, $n - $r) )" as="xs:integer"/>   <xsl:sequence select="if ($n eq 0) then 0                          else ckbk:prod-range($max + 1, $n) div                               ckbk:factorial($min)"/> </xsl:function>

Discussion

The solutions are designed to reduce the number of multiplications; if you divide one factorial by a smaller factorial, then the smaller factorial effectively cancels out that many multiplications from the larger. Hence, it is better to implement such functions using prod-range (Recipe 3.5) rather than factorial. The combinatorial is slightly more complex because you want to cancel out the large of r and (n - r).