Subject: RE: How efficient is DVC? - A grouping example
From: "Michael Kay" <mhk@xxxxxxxxx>
Date: Sat, 22 Mar 2003 22:38:36 -0000
|
This is fascinating stuff, but the proof of the pudding is in the
eating: have you made any comparative performance measurements, using a
non-trivial input file?
Michael Kay
Software AG
home: Michael.H.Kay@xxxxxxxxxxxx
work: Michael.Kay@xxxxxxxxxxxxxx
> -----Original Message-----
> From: owner-xsl-list@xxxxxxxxxxxxxxxxxxxxxx
> [mailto:owner-xsl-list@xxxxxxxxxxxxxxxxxxxxxx] On Behalf Of
> Robbert van Dalen
> Sent: 22 March 2003 21:07
> To: xsl-list@xxxxxxxxxxxxxxxxxxxxxx
> Subject: How efficient is DVC? - A grouping example
>
>
> Hello all,
>
> Everyone interested in efficient algorithms might be
> interested in this 'article'. Note that I'm not used to write
> articles so excuse me if I'm not making a point clearly.
>
> All the code is free - copy it if you like.
>
> Cheers,
>
> RvD
>
> _____________________________________________________________
> ABSTRACT
>
> Grouping without using the key() function is not very
> difficult in practice. But to implement it efficiently is not
> easy task. This article presents a modified Divide And
> Conquer (DVC) algorithm to implement grouping. The algorithm
> is not only useful for grouping, but may be generalised to
> help improve other DVC algorithms.
>
>
> INTRODUCTION
>
> Muenchian grouping is by far the most efficient method to
> group nodes with XSLT. But there are some situations when
> Muenchian grouping can't be used, for example if you want to
> group tree-fragments Tree-fragments are heavily used when
> multiple passes are needed to compute a result with only one
> stylesheet. However, you cannot use the key() function on
> tree-fragments because XSLT doesn't allow you to (there is no
> nodeset parameter)
>
> PREREQUISITES
> All examples are tested against XALAN. Make sure you always
> include the following stylesheet header:
>
> <xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
> xmlns:xalan="http://xml.apache.org/xalan" version="1.1"
> exclude-result-prefixes="xalan">
>
>
> GROUPING EXAMPLE
> The following example will give you an idea of grouping
> tree-fragments without using the key() function.
>
> Input XML (taken from Michael Kay's book)
>
> <cities>
> <city name="Barcelona" country="Espana"/>
> <city name="Paris" country="France"/>
> <city name="Roma" country="Italia"/>
> <city name="Madrid" country="Espana"/>
> <city name="Milano" country="Italia"/>
> <city name="Firenze" country="Italia"/>
> <city name="Napoli" country="Italia"/>
> <city name="Lyon" country="France"/>
> </cities>
>
> Stylesheet (partly copied from Michael Kay's book)
>
> <xsl:template match="cities">
> <xsl:variable name="sorted">
> <xsl:for-each select="./city">
> <xsl:sort select="@country"/>
> <xsl:copy-of select="."/>
> </xsl:for-each>
> </xsl:copy-of>
>
> <xsl:variable name="sorted-tree-fragment"
> select="xalan:nodeset($sorted)/*"/>
>
> <!-- Gets the groups -->
> <xsl:variable name="groups">
> <xsl:apply-templates select="$sorted-tree-fragment"/>
> </xsl:variable>
>
> <!-- Iterate through all the groups -->
> <xsl:for-each select="xalan:nodeset($groups)/*">
> <xsl:variable name="country" select="."/>
> <xsl:copy>
> <!-- Copy the nodes with the same country -->
> <xsl:copy-of select="$sorted-tree-fragment[country =
> $country]"/>
> <xsl:copy-of select="@*"/>
> </xsl:copy>
> </xsl:for-each>
> </xsl:template>
>
> <xsl:template match="city">
> <variable name="preceding" select="./preceding-sibling::*[1]"/>
> <xsl:if test="not(./@country = $preceding/@country)">
> <group id="{$preceding/@country}"/>
> </xsl:if>
> </xsl:template>
>
> Output:
>
> <group id="Espana">
> <city name="Barcelona" country="Espana"/>
> <city name="Madrid" country="Espana"/>
> </group>
> <group id="France">
> <city name="Paris" country="France"/>
> <city name="Lyon" country="France"/>
> </group>
> <group id="Italia">
> <city name="Roma" country="Italia"/>
> <city name="Milano" country="Italia"/>
> <city name="Firenze" country="Italia"/>
> <city name="Napoli" country="Italia"/>
> </group>
>
> This looks like a OK solution, but let's get a closer look on
> what is going on.
>
> 1) First the cities are sorted on the @country attribute.
> After this, cities that share the same @country value will be
> following each other, which is a property we can exploit in step 2.
> 2) Then the template that matches city nodes will be called N
> times if there are N cities to be grouped. For each city node
> in the sorted set the 'following-sibling::*[1]' node(s) are
> matched. If they're not equal, the city node will mark a new
> group. As Michael Kay already pointed out in his book, the
> efficiency of this approach depends on the implementation of
> 'following-sibling::*[1]'. If this expression has time
> complexity O(1) then the overall time complexity of getting
> all the groups will be O(N) (leaving sorting out of the equation).
> 3) Strangely enough, the last step is actually the most
> problematic. Let's say the second step gave us 3 groups.
> Then, for each group, the expression
> '$sorted-tree-fragment[country = $country] will be evaluated
> with time complexity O(N).
>
> So, does this mean the overall time complexity will be 3*N =
> O(N)? The answer is definitely no! It does hold for a small
> number of groups, but if we have N/2 groups then time
> complexity will be O(N^2). Selecting nodes with XPATH
> expressions is usually OK, but in this example we want to
> select the K cities that share the same @country value in
> O(K) time, not
> O(N) time.
> So the question we really want to anwer is: 'how can we
> efficiently select a subset of nodes without traversing them
> all?'. The anwser is: 'this all depends on the selection
> criterium.' Still, if the selection criterium isn't too
> complex, we can still hope for a better solution. One
> solution is that we don't use XPATH expressions to select
> nodes, but rather walk through the nodes by using recursive calls.
>
>
> GROUPING USING RECURSION
>
> One idea to reduce time complexity of the previous example is
> by slightly modifying the match='city' template:
>
> <xsl:template match="city">
> <variable name="preceding" select="./preceding-sibling::*[1]"/>
> <xsl:choose>
> <xsl:when test="not(./@country = $preceding/@country)">
> <group id="{./@country}">
> <xsl:copy-of select="."/>
> <xsl:apply-templates name="./following-sibling::*[1]"/>
> </group>
> </xsl:when>
> <xsl:otherwise>
> <xsl:copy-of select="."/>
> <xsl:apply-templates name="./following-sibling::*[1]"/>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:template>
>
> If we take the same input and use the following 'apply-templates'
>
> <xsl:apply-templates match="xalan:nodeset($sorted)/*[1]"/>
>
> ...we get the following result.
>
> <group id="Espana">
> <city name="Barcelona" country="Espana"/>
> <city name="Madrid" country="Espana"/>
> <group id="France">
> <city name="Paris" country="France"/>
> <city name="Lyon" country="France"/>
> <group id="Italia">
> <city name="Roma" country="Italia"/>
> <city name="Milano" country="Italia"/>
> <city name="Firenze" country="Italia"/>
> <city name="Napoli" country="Italia"/>
> </group>
> </group>
> </group>
>
> This is almost what we want. The following 'apply-templates'
> will flatten the tree structure to return the same result as
> the previous example.
>
> <xsl:apply-templates select="xalan:nodeset($groups)"/>
>
> <xsl:template match="group">
> <xsl:copy>
> <xsl:apply-templates select="./group"/>
> <xsl:copy-of select="./city"/>
> <xsl:copy-of select="@*"/>
> </xsl:copy>
> </xsl:template>
>
> The time complexity of the recursive solution can be proven
> to be O(N) but with the recursion depth also to be O(N).
> Unfortunately, most XSLT implementations have a maximum
> recursion depth (~1000) so this is not a general solution.
>
>
> DVC AND THE BINARY TREE
> Dimitre Novatchev was one of the first to mention Divide and
> Conquer (DVC) algorithms to reduce recursion depth. Because
> most XSLT implementations out there still do not support
> tail-recursion elimination, DVC is the way to go if you want
> to process a lot of nodes. The idea behind DVC is that to
> attack a big problem, you should divide it into a number of
> smaller problems. Not surprisingly, dividing a problem into
> just 2 subproblems is enough to reduce recursion depth to be
> O(log2(N)). The following example will give you an idea of
> how this works:
>
>
> The XML input:
>
> <nodes>
> <node v="1"/>
> <node v="2"/>
> <node v="3"/>
> <node v="4"/>
> <node v="5"/>
> <node v="6"/>
> <node v="7"/>
> <node v="8"/>
> </nodes>
>
> The Stylesheet:
>
> <xsl:template match="/">
> <xsl:call-template name="partition">
> <xsl:with-param name="nodes" select="//node"/>
> </xsl:call-template>
> </xsl:template>
>
>
> <xsl:template name="partition">
> <xsl:param name="nodes"/>
>
> <xsl:variable name="half" select="floor(count($nodes) div 2)"/>
>
> <b>
> <xsl:choose>
> <xsl:when test="count($nodes) <= 1">
> <!-- There is only one node left: stop dividing problem -->
> <xsl:copy-of select="$nodes"/>
> </xsl:when>
> <xsl:otherwise>
> <!-- divide in first half of nodes (left) -->
> <xsl:call-template name="partition">
> <xsl:with-param name="nodes"
> select="$nodes[position() <= $half]"/>
> </xsl:call-template>
> <!-- divide in second half of nodes (right) -->
> <xsl:call-template name="partition">
> <xsl:with-param name="nodes"
> select="$nodes[position() > $half]"/>
> </xsl:call-template>
> </xsl:otherwise>
> </xsl:choose>
> </b>
> </xsl:template>
>
> The output:
>
> <b>
> <b>
> <b>
> <b>
> <node v="1"/>
> </b>
> <b>
> <node v="2"/>
> </b>
> </b>
> <b>
> <b>
> <node v="3"/>
> </b>
> <b>
> <node v="4"/>
> </b>
> </b>
> </b>
> <b>
> <b>
> <b>
> <node v="5"/>
> </b>
> <b>
> <node v="6"/>
> </b>
> </b>
> <b>
> <b>
> <node v="7"/>
> </b>
> <b>
> <node v="8"/>
> </b>
> </b>
> </b>
> </b>
>
> The result is what is called a binary tree representation. At
> first this representation doesn't seem all that useful. Later
> we will see that specialised binary trees can be (re-)used to
> implement almost any recursive function without exceeding the
> maximum recursion depth.
>
> Let's sum all the @v values with the use of the binary
> (fragment) tree:
>
> <xsl:template match="/"/>
> <xsl:variable name="btree">
> <xsl:call-template name="partition">
> <xsl:with-param name="nodes" select="//node"/>
> </xsl:call-template>
> </xsl:variable>
>
> <xsl:call-template name="sum-binary-tree">
> <xsl:with-param name="bnode" select="xalan:nodeset($btree)/*"/>
> </xsl:call-template>
> </xsl:template>
>
> <xsl:template name="sum-binary-tree">
> <xsl:param name="bnode"/>
>
> <xsl:choose>
> <xsl:when test="$bnode/node">
> <xsl:value-of select="$bnode/node/@v"/>
> </xsl:when>
> <xsl:otherwise>
> <xsl:variable name="first">
> <xsl:call-template name="sum-binary-tree">
> <xsl:with-param name="bnode" select="$bnode/b[1]"/>
> </xsl:call-template>
> </xsl:variable>
> <xsl:variable name="second">
> <xsl:call-template name="sum-binary-tree">
> <xsl:with-param name="bnode" select="$bnode/b[2]"/>
> </xsl:call-template>
> </xsl:variable>
> <xsl:value-of select="$first + $second"/>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:template>
>
> This gives the result: 36
>
> Let's analyse the partition template in terms of time
> complexity. It's easy to prove that it is equal to the number
> nodes being copied. The partition algorithm uses the XPATH
> expression '$nodes[count() > $half]' to split the nodes in
> half. This construction is almost exclusively used by all DVC
> or 'chunk' algorithms including many of Dimitre Novatchev's
> examples. But how about the number of nodes being copied? The
> following table lists the number of nodes being copied for
> each partition.
>
> partition(1)
> number of copies 1
>
> partition(2)
> number of copies
> l: 1 + partition(1)
> r: 1 + partition(1)
>
> partition(4)
> number of copies
> l: 2 + partition(2)
> r: 2 + partition(2)
>
> partition(8)
> number of copies
> l: 4 + partition(4)
> r: 4 + partition(4)
>
> etc.
>
> The number of copies when calling partition(4) is:
> (2 + (1 + 1) + 2 + (1 + 1)) = 2*4
>
> The number of copies when calling partition(8) is:
> 4 + (2 + (1 + 1) + 2 + (1 + 1)) + 4 + (2 + (1 + 1) + 2 + (1 +
> 1)) = 3*8
>
> So the overall 'copy' complexity is O(log2(N)*N).
> Although the number of recursive calls is O(N) the XSLT
> processor still spends at least O(log2(N)*N) time because it
> must copy (and select) half of the nodes for the each
> recursive call (twice). Copying nodes should be avoided as
> much as possible because it slows down recursion considerably.
>
>
> MODIFIED DVC ALGORITHM: RANGE PARTITIONING
>
> The following implementation of a binary partition doesn't
> copy a list of nodes but just one node at each recursive
> call. It uses the so called 'sibling' axis to walk through
> the list. Because there are O(N) recursive calls, this means
> that O(N) nodes are copied. Does this mean that the overall
> time complexity will be O(N) too? The answer is: probably
> yes, but at worst it will be O(N^2).
>
> Input XML:
>
> <nodes>
> <node v="1"/>
> <node v="2"/>
> <node v="3"/>
> <node v="4"/>
> <node v="5"/>
> <node v="6"/>
> <node v="7"/>
> <node v="8"/>
> </nodes>
>
> The Stylesheet:
>
> <xsl:template match="/">
> <xsl:call-template name="partition-ranges">
> <xsl:with-param name="node" select="//node[1]"/>
> </xsl:call-template>
> </xsl:template>
>
> <xsl:template name="partition-ranges">
> <xsl:param name="node"/>
> <xsl:param name="s"
> select="(count($node/preceding-sibling::*)) + 1"/>
> <xsl:param name="e"
> select="(count($node/following-sibling::*)) + $s"/>
>
> <xsl:if test="$node">
> <xsl:element name="r">
> <xsl:attribute name="s">
> <xsl:value-of select="$s"/>
> </xsl:attribute>
> <xsl:attribute name="e">
> <xsl:value-of select="$e"/>
> </xsl:attribute>
> <xsl:choose>
> <xsl:when test="$s = $e">
> <xsl:copy-of select="$node"/>
> </xsl:when>
> <xsl:otherwise>
> <xsl:variable name="w" select="floor(($e - $s + 1) div 2)"/>
> <xsl:variable name="m" select="$s + $w"/>
> <xsl:call-template name="partition-ranges">
> <xsl:with-param name="node" select="$node"/>
> <xsl:with-param name="s" select="$s"/>
> <xsl:with-param name="e" select="$m - 1"/>
> </xsl:call-template>
> <xsl:call-template name="partition-ranges">
> <xsl:with-param name="node"
> select="$node/following-sibling::*[$w]"/>
> <xsl:with-param name="s" select="$m"/>
> <xsl:with-param name="e" select="$e"/>
> </xsl:call-template>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:element>
> </xsl:if>
> </xsl:template>
>
> The output:
>
> <r s="1" e="8">
> <r s="1" e="4">
> <r s="1" e="2">
> <r s="1" e="1">
> <node v="1"/>
> </r>
> <r s="2" e="2">
> <node v="2"/>
> </r>
> </r>
> <r s="3" e="4">
> <r s="3" e="3">
> <node v="3"/>
> </r>
> <r s="4" e="4">
> <node v="4"/>
> </r>
> </r>
> </r>
> <r s="5" e="8">
> <r s="5" e="6">
> <r s="5" e="5">
> <node v="5"/>
> </r>
> <r s="6" e="6">
> <node v="6"/>
> </r>
> </r>
> <r s="7" e="8">
> <r s="7" e="7">
> <node v="7"/>
> </r>
> <r s="8" e="8">
> <node v="8"/>
> </r>
> </r>
> </r>
> </r>
>
> Note that the output resembles the previous example but
> instead of <b> nodes, <r> (Range) nodes are used. This just
> makes it more convenient to select ranges of nodes later on.
> The actual 'splitting' is done through the following
> expression '[$node/following-sibling::*[$w]' with $w being
> the lenght of the list divided by 2. Let's compare overall
> time complexity with the possible implementations of
> 'following-sibling::[w]'
>
> following-sibling::*[w] | total time
> _____________________________________
> O(1) | O(N)
> O(w) | O(log2(N)*N)
> O(N) | O(N^2)
>
> So at worst it will be quadratic. So the question still
> remains if it is theoretically possible to do binary
> partitioning without copying to much nodes. Nevertheless,
> experiments with XALAN have shown that the implementation is
> not quadratic.
>
>
> GROUPING WITH A BINARY TREE
>
> The new and improved grouping algorithm is more or less the
> same as the first one except where using ranges to select
> nodes which are in the same group.
> Thus:
>
> 1) we sort the nodes for a given key
> 2) then compute the ranges of nodes which have the same key
> 3) and then select the (sorted) nodes for each range.
>
> To efficiently select a range of nodes we will be using the
> binary tree.
>
> Here's the whole solution:
>
> Input XML:
>
> <cities>
> <city name="Barcelona" country="Espana"/>
> <city name="Paris" country="France"/>
> <city name="Roma" country="Italia"/>
> <city name="Madrid" country="Espana"/>
> <city name="Milano" country="Italia"/>
> <city name="Firenze" country="Italia"/>
> <city name="Napoli" country="Italia"/>
> <city name="Lyon" country="France"/>
> </cities>
>
>
> The stylesheet (WARNING, THIS IS A BIT LENGHTY):
>
> <!-- Group cities on country -->
> <xsl:template match="/">
> <xsl:call-template name="group-on-key">
> <xsl:with-param name="nodes" select="//city"/>
> <xsl:with-param name="key" select="'country'"/>
> </xsl:call-template>
> </xsl:template>
>
> <!--
> Template: group-on-key
> Use this template to group <nodes> which share a common
> attribute <key>
> The result will be sub-sets of <nodes> surrounded by <group/> tags
> -->
>
>
> <xsl:template name="group-on-key">
> <xsl:param name="nodes"/>
> <xsl:param name="key"/>
>
> <xsl:variable name="items">
> <xsl:for-each select="$nodes">
> <item>
> <key>
> <xsl:value-of select="./@*[name() = $key]"/>
> </key>
> <value>
> <xsl:copy-of select="."/>
> </value>
> </item>
> </xsl:for-each>
> </xsl:variable>
>
> <xsl:variable name="grouped-items">
> <xsl:call-template name="group-on-item">
> <xsl:with-param name="nodes" select="xalan:nodeset($items)/*"/>
> <xsl:with-param name="key" select="$key"/>
> </xsl:call-template>
> </xsl:variable>
>
> <xsl:for-each select="xalan:nodeset($grouped-items)/*">
> <xsl:copy>
> <xsl:for-each select="./*">
> <xsl:copy-of select="./value/*[1]"/>
> </xsl:for-each>
> </xsl:copy>
> </xsl:for-each>
> </xsl:template>
>
> <!--
> Template: group-on-item
> Use this template to group <nodes> which share a common
> structure. You can build this structure yourself if you want
> to group on something else
>
> The structure is the <item> structure and has the following
> layout <item>
> <key>
> aKey (can be anything, preferrably a string)
> </key>
> <value>
> aValue (can be anything, probably a node(set))
> </value>
> </item>
>
> <items> will we grouped on the string value of <key>
> The result will be sub-sets of <items> surrounded by <group/> tags
> -->
>
> <xsl:template name="group-on-item">
> <xsl:param name="nodes"/>
>
> <!-- Step 1 -->
> <xsl:variable name="sorted">
> <xsl:for-each select="$nodes">
> <xsl:sort select="./key[1]/"/>
> <xsl:copy-of select="."/>
> </xsl:for-each>
> </xsl:variable>
>
> <xsl:variable name="sorted-tree" select="xalan:nodeset($sorted)/*"/>
>
> <!-- Step 2.1 -->
> <xsl:variable name="pivots">
> <xsl:call-template name="pivots">
> <xsl:with-param name="nodes" select="$sorted-tree"/>
> </xsl:call-template>
> </xsl:variable>
>
> <!-- Step 2.2 -->
> <xsl:variable name="ranges">
> <xsl:call-template name="ranges">
> <xsl:with-param name="pivots"
> select="xalan:nodeset($pivots)/*"/>
> <xsl:with-param name="length" select="count($sorted-tree)"/>
> </xsl:call-template>
> </xsl:variable>
>
> <!-- Step 3.1 -->
> <xsl:variable name="partition-ranges">
> <xsl:call-template name="partition-ranges">
> <xsl:with-param name="node" select="$sorted-tree[1]"/>
> </xsl:call-template>
> </xsl:variable>
>
> <xsl:variable name="root-partition"
> select="xalan:nodeset($partition-ranges)/*[1]"/>
>
> <!-- Step 3.2 -->
> <xsl:for-each select="xalan:nodeset($ranges)/r">
> <xsl:variable name="s" select="./@s"/>
> <xsl:variable name="e" select="./@e"/>
>
> <group>
> <xsl:call-template name="range-in-partition">
> <xsl:with-param name="s" select="$s"/>
> <xsl:with-param name="e" select="$e"/>
> <xsl:with-param name="p" select="$root-partition"/>
> </xsl:call-template>
> </group>
> </xsl:for-each>
>
> </xsl:template>
>
> <xsl:template name="pivots">
> <xsl:param name="nodes"/>
> <xsl:param name="key"/>
>
> <xsl:for-each select="$nodes">
> <xsl:if test="not(string(./key[1]/) =
> string(./following-sibling::*[1]/key[1]/))">
> <pivot>
> <xsl:value-of select="position()"/>
> </pivot>
> </xsl:if>
> </xsl:for-each>
> </xsl:template>
>
> <xsl:template name="ranges">
> <xsl:param name="pivots" select="../"/>
> <xsl:param name="length" select="0"/>
>
> <xsl:choose>
> <xsl:when test="count($pivots) >= 1">
> <xsl:for-each select="$pivots">
> <xsl:variable name="p" select="./preceding-sibling::*[1]"/>
> <r>
> <xsl:attribute name="s">
> <xsl:choose>
> <xsl:when test="$p">
> <xsl:value-of select="$p + 1"/>
> </xsl:when>
> <xsl:otherwise>
> <xsl:value-of select="1"/>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:attribute>
> <xsl:attribute name="e">
> <xsl:value-of select="string(.)"/>
> </xsl:attribute>
> </r>
> </xsl:for-each>
> </xsl:when>
> <xsl:otherwise>
> <r>
> <xsl:attribute name="s">
> <xsl:value-of select="1"/>
> </xsl:attribute>
> <xsl:attribute name="e">
> <xsl:value-of select="$length"/>
> </xsl:attribute>
> </r>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:template>
>
> <!--
> Template: range-in-partition
> Selects a RANGE of nodes using a binary tree
>
> XSLT isn't really helping to make things easy but try to do
> this in a DVC style directly without the help of a binary tree.
> -->
>
> <xsl:template name="range-in-partition">
> <xsl:param name="p"/>
> <xsl:param name="s" select="$p/@s"/>
> <xsl:param name="e" select="$p/@e"/>
>
> <xsl:variable name="ps" select="number($p/@s)"/>
> <xsl:variable name="pe" select="number($p/@e)"/>
>
> <xsl:if test="$s <= $pe and $e >= $ps">
> <xsl:if test="$ps = $pe">
> <xsl:copy-of select="$p/*[1]"/>
> </xsl:if>
> <xsl:choose>
> <xsl:when test="$s > $ps">
> <xsl:variable name="s1" select="$s"/>
> <xsl:choose>
> <xsl:when test="$e < $pe">
> <xsl:variable name="e1" select="$e"/>
> <xsl:for-each select="$p/*">
> <xsl:call-template name="range-in-partition">
> <xsl:with-param name="s" select="$s1"/>
> <xsl:with-param name="e" select="$e1"/>
> <xsl:with-param name="p" select="."/>
> </xsl:call-template>
> </xsl:for-each>
> </xsl:when>
> <xsl:otherwise>
> <xsl:variable name="e1" select="$pe"/>
> <xsl:for-each select="$p/*">
> <xsl:call-template name="range-in-partition">
> <xsl:with-param name="s" select="$s1"/>
> <xsl:with-param name="e" select="$e1"/>
> <xsl:with-param name="p" select="."/>
> </xsl:call-template>
> </xsl:for-each>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:when>
> <xsl:otherwise>
> <xsl:variable name="s1" select="$ps"/>
> <xsl:choose>
> <xsl:when test="$e < $pe">
> <xsl:variable name="e1" select="$e"/>
> <xsl:for-each select="$p/*">
> <xsl:call-template name="range-in-partition">
> <xsl:with-param name="s" select="$s1"/>
> <xsl:with-param name="e" select="$e1"/>
> <xsl:with-param name="p" select="."/>
> </xsl:call-template>
> </xsl:for-each>
> </xsl:when>
> <xsl:otherwise>
> <xsl:variable name="e1" select="$pe"/>
> <xsl:for-each select="$p/*">
> <xsl:call-template name="range-in-partition">
> <xsl:with-param name="s" select="$s1"/>
> <xsl:with-param name="e" select="$e1"/>
> <xsl:with-param name="p" select="."/>
> </xsl:call-template>
> </xsl:for-each>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:otherwise>
> </xsl:choose>
> </xsl:if>
> </xsl:template>
>
> Output XML:
>
> <group>
> <city name="Barcelona" country="Espana"/>
> <city name="Madrid" country="Espana"/>
> </group>
> <group>
> <city name="Paris" country="France"/>
> <city name="Lyon" country="France"/>
> </group>
> <group>
> <city name="Roma" country="Italia"/>
> <city name="Milano" country="Italia"/>
> <city name="Firenze" country="Italia"/>
> <city name="Napoli" country="Italia"/>
> </group>
>
>
> CONCLUSION
>
> An efficient DVC algorithm is given for grouping using a
> binary tree. That binary trees can be build with time
> complexity O(N) and 'copy' complexity O(N) - without relying
> to much on implementations - is still an open question.
>
>
>
> XSL-List info and archive: http://www.mulberrytech.com/xsl/xsl-list
>
XSL-List info and archive: http://www.mulberrytech.com/xsl/xsl-list
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