Subject: RE: [xsl] 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: [xsl] 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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