## Re: [xsl] Word Ladders as an example of a "Find shortest path between two nodes in a graph" problem

 Subject: Re: [xsl] Word Ladders as an example of a "Find shortest path between two nodes in a graph" problem From: Dimitre Novatchev Date: Thu, 6 Dec 2012 12:56:42 -0800
```On Thu, Dec 6, 2012 at 10:29 AM, Hermann Stamm-Wilbrandt
<STAMMW@xxxxxxxxxx> wrote:
> Hi Dimitre,
>
> you are right with O(|E|), but not with O(N^2).
> Implementation of "5.c" was a bit sloppy, initEdges() is indeed O(N^2).
>
> But if you have N real language words of length k, each single word can
> have at most  k*26 = O(1) neighbors.
>
> Testing whether a word "c_1 c_2 ... c_k" is among the "words" can be done
> in
> constant time, too, if your language allows for constant time array access
> (just use an array of size 26^k = O(1)).

Sure, if you would allocate 8GB for this array (for k = 7).

Also, calculating the index in the array requires k steps (and
probably multiplication and addition at each such step).

>
> Now with each node having constant number of neighbors O(|E|) = O(|V|) = O
> (N).
>

I was speaking about graph traversal in general. One may argue that
the big O notation isn't too useful when the value of N is limited and
not too-big.

For example 26*N in this particular case is  bigger than log2(N) * N
(unless we have more than 3 million words). So, this "O(N)"
implementation  may take more actual time than an "O(N*log(N)))

To put it in other words, the constants do matter when N is limited.

Cheers,

Dimitre

>
> Mit besten Gruessen / Best wishes,
>
> Hermann Stamm-Wilbrandt
> Level 3 support for XML Compiler team and Fixpack team lead
> WebSphere DataPower SOA Appliances
> https://www.ibm.com/developerworks/mydeveloperworks/blogs/HermannSW/
> ----------------------------------------------------------------------
> IBM Deutschland Research & Development GmbH
> Vorsitzende des Aufsichtsrats: Martina Koederitz
> Geschaeftsfuehrung: Dirk Wittkopp
> Sitz der Gesellschaft: Boeblingen
> Registergericht: Amtsgericht Stuttgart, HRB 243294
>
>
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>   |Dimitre Novatchev <dnovatchev@xxxxxxxxx>                                                                                                  |
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>   |xsl-list@xxxxxxxxxxxxxxxxxxxxxx,                                                                                                          |
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>   |12/06/2012 07:11 PM                                                                                                                       |
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>   |Re: [xsl] Word Ladders as an example of a "Find shortest path between two nodes in a graph" problem                                       |
>   >------------------------------------------------------------------------------------------------------------------------------------------|
>
>
>
>
>
> Hi Herman,
>
> I think that in a near-worst case the time-complexity is O(N^2),
> because every arc needs to be inspected and in a well-connected graph
> with N vertices there can be N*(N-1) vertices.
>
> This is also confirmed by Wikipedia and the 2-3 books on algoritms I have:
>
> "The time complexity can be expressed as O(|E|) since every vertex and
> every edge will be explored in the worst case. Note: O(|E|) may vary
> between O(|V|) and O(|V|^2), depending on how sparse the input graph
> is (assuming that the graph is connected)".
>
>
> To summarize, this is a property of the graph and not of the
> programming language that is used.
>
>
> Cheers,
> Dimitre
>
>
>
> On Thu, Dec 6, 2012 at 10:01 AM, Hermann Stamm-Wilbrandt
> <STAMMW@xxxxxxxxxx> wrote:
>> Hi Dimitre,
>>
>>> That I have initially used a not probably the most efficient algorithm
>>> / implementation, shouldn't be used to make general conclusions about
>>> the appropriateness of using XSLT in solving a particular class of
>>> problems.
>>>
>> I agree with you -- and your solution is nice.
>>
>> But breadth-first-search algorithm can be implemented as linear time
>> algorithm in C or C++ -- I doubt that you can do linear time
>> implementation in XSLT since constant time array access is missing ...
>>
>>
>> Mit besten Gruessen / Best wishes,
>>
>> Hermann Stamm-Wilbrandt
>> Level 3 support for XML Compiler team and Fixpack team lead
>> WebSphere DataPower SOA Appliances
>> https://www.ibm.com/developerworks/mydeveloperworks/blogs/HermannSW/
>> ----------------------------------------------------------------------
>> IBM Deutschland Research & Development GmbH
>> Vorsitzende des Aufsichtsrats: Martina Koederitz
>> Geschaeftsfuehrung: Dirk Wittkopp
>> Sitz der Gesellschaft: Boeblingen
>> Registergericht: Amtsgericht Stuttgart, HRB 243294
>>
>>
>> |------------>
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>>
>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>   |Dimitre Novatchev <dnovatchev@xxxxxxxxx>
> |
>>
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>>   |xsl-list@xxxxxxxxxxxxxxxxxxxxxx,
> |
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>>   |12/06/2012 05:50 PM
> |
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>>   |Re: [xsl] Word Ladders as an example of a "Find shortest path between
> two nodes in a graph" problem                                       |
>>
>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>
>>
>>
>>
>> Herman,
>>
>> That I have initially used a not probably the most efficient algorithm
>> / implementation, shouldn't be used to make general conclusions about
>> the appropriateness of using XSLT in solving a particular class of
>> problems.
>>
>> Cheers,
>>
>> Dimitre
>>
>> On Thu, Dec 6, 2012 at 8:15 AM, Hermann Stamm-Wilbrandt
>> <STAMMW@xxxxxxxxxx> wrote:
>>>> ... I think that this
>>>> isn't something that should be solved in XSLT at all, except as an
>>>>
>>> Agreed, nice XSLT solution, but not fast.
>>>
>>> This simple C program does find the longest path (35) to angry in a
>>> second (on a W520 Thinkpad) based on Dimitie's word list of length 5:
>>> http://www.stamm-wilbrandt.de/en/xsl-list/5.c
>>>
>>> \$ time ./5 angry
>>> yasht
>>> yacht
>>> pacht
>>> pecht
>>> wecht
>>> wicht
>>> wight
>>> dight
>>> digit
>>> dimit
>>> demit
>>> remit
>>> refit
>>> befit
>>> besit
>>> beset
>>> besee
>>> belee
>>> belve
>>> beeve
>>> breve
>>> brave
>>> brace
>>> braca
>>> araca
>>> arara
>>> amara
>>> amala
>>> alala
>>> alula
>>> aluta
>>> abuta
>>> abura
>>> anura
>>> anury
>>> angry
>>> 35
>>>
>>> real    0m1.046s
>>> user    0m1.039s
>>> sys     0m0.004s
>>> \$
>>>
>>>
>>> Mit besten Gruessen / Best wishes,
>>>
>>> Hermann Stamm-Wilbrandt
>>> Level 3 support for XML Compiler team and Fixpack team lead
>>> WebSphere DataPower SOA Appliances
>>> https://www.ibm.com/developerworks/mydeveloperworks/blogs/HermannSW/
>>> ----------------------------------------------------------------------
>>> IBM Deutschland Research & Development GmbH
>>> Vorsitzende des Aufsichtsrats: Martina Koederitz
>>> Geschaeftsfuehrung: Dirk Wittkopp
>>> Sitz der Gesellschaft: Boeblingen
>>> Registergericht: Amtsgericht Stuttgart, HRB 243294
>>>
>>>
>>> |------------>
>>> | From:      |
>>> |------------>
>>>
>>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>>   |Wolfgang Laun <wolfgang.laun@xxxxxxxxx>
>> |
>>>
>>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>> |------------>
>>> | To:        |
>>> |------------>
>>>
>>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>>   |xsl-list@xxxxxxxxxxxxxxxxxxxxxx,
>> |
>>>
>>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>> |------------>
>>> | Date:      |
>>> |------------>
>>>
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>
>>
>>>   |11/28/2012 07:15 PM
>> |
>>>
>>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>> |------------>
>>> | Subject:   |
>>> |------------>
>>>
>>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>>   |Re: [xsl] Word Ladders as an example of a "Find shortest path between
>> two nodes in a graph" problem                                       |
>>>
>>>------------------------------------------------------------------------------------------------------------------------------------------|
>
>>
>>>
>>>
>>>
>>>
>>>
>>> The fact that one XSLT program runs three times faster on one XSLT
>>> implementation
>>> than on another one is strange, *very* strange. But is Saxon 6.4 the
>>> "dernier cri"?
>>> I'd very much like to hear Michael Kay's opinion on this.
>>>
>>> With Saxon HE 9.2.0 running with the -t option, I compare execution
>> times:
>>>    1209 ms with 40065592 bytes for WL's solution
>>> to
>>>    2768 ms with 81184768 bytes for DN's solution.
>>>
>>> Note: DN's solution being the one *without* the optimizations!
>>>
>>> Not that this is conclusive. Algorithms like this one must be judged
>>> by more than a single run:
>>> they may behave well for small word lengths and small ladder sizes,
>>> the other way round. (Dimitre and I aren't even using the same word
>>> data, AFAIK.)
>>>
>>> As an aside, I'd like to say that neither DN's nor WL's solution is
>>> something that should
>>> be used if this problem (i.e., shortest path) should ever need a
>>> solution. I think that this
>>> isn't something that should be solved in XSLT at all, except as an
>>> (Feel free to disagree - I'll not reply to anything contradicting me.)
>>>
>>> Cheers
>>>
>>>>
>>>> Ok, I was running it with Saxon 6.4
>>>>
>>>> Now, the times are:
>>>>
>>>> With Saxon:
>>>>
>>>> Wolfgang's transformation: 25sec.
>>>>
>>>> Dimitre's :                            39sec.
>>>>
>>>>
>>>> However, with XQSharp:
>>>>
>>>> Wolfgang's transformation: 23sec.
>>>>
>>>> Dimitre's :                            14sec.
>>>>
>>>>
>>>> Therefore, one can't say wich transformation is faster -- it depends
>>>> on the XSLT processor being used.
>>>>
>>>>
>>>> Cheers,
>>>> Dimitre
>>>>
>>>>
>>>>
>>>> On Wed, Nov 28, 2012 at 7:27 AM, Dimitre Novatchev
>> <dnovatchev@xxxxxxxxx>
>>>> wrote:
>>>> > I get this error, trying to run your code:
>>>> >
>>>> > SAXON 6.5.4 from Michael Kay
>>>> > Java version 1.6.0_31
>>>> > Preparation time: 250 milliseconds
>>>> > Processing file:/C:\XSLT Projects\WordLadders\Ver 0.2\dictGraph4.xml
>>>> > Building tree for file:/C:\XSLT Projects\WordLadders\Ver
>>>> > 0.2\dictGraph4.xml using class com.icl.saxon.tinytree.TinyBuilder
>>>> > Tree built in 351 milliseconds
>>>> > Error at xsl:variable on line 23 of file:/(Untitled):
>>>> >   Error in expression key('kFindWord', \$pStartWord, \$vDictGraph)
>>>> >                     [count(../*)  lt  count(key('kFindWord',
>>>> > \$pTargetWord, \$vDictGraph)/../* )]                        |
>>>> >             key('kFindWord', \$pTargetWord, \$vDictGraph)
>>>> >            [count(../*) le count(key('kFindWord',  \$pStartWord,
>>>> > \$vDictGraph)/../*)]: expected "]", found "<name>"
>>>> >
>>>> >
>>>> > Cheers,
>>>> > Dimitre
>>>> >
>>>> > On Wed, Nov 28, 2012 at 5:40 AM, Wolfgang Laun
>>> <wolfgang.laun@xxxxxxxxx>
>>>> > wrote:
>>>> >> <xsl:stylesheet version="2.0"
>>>> >>    xmlns:xsl="http://www.w3.org/1999/XSL/Transform";
>>>> >>    xmlns:my="my:my"
>>>> >>    xmlns:xs="http://www.w3.org/2001/XMLSchema";
>>>> >>    exclude-result-prefixes="my xs">
>>>> >>
>>>> >>   <xsl:output method="text"/>
>>>> >>
>>>> >>   <xsl:variable name="vDictGraph" select="/"/>
>>>> >>   <xsl:key name="kFindWord" match="w" use="."/>
>>>> >>
>>>> >>   <xsl:param name="pStartWord"  select="'nice'" as="xs:string"/>
>>>> >>   <xsl:param name="pTargetWord" select="'evil'" as="xs:string"/>
>>>> >>
>>>> >>   <xsl:variable name="vStartWord" as="xs:string"
>>>> >>                 select="key('kFindWord', \$pStartWord, \$vDictGraph)
>>>> >>                          [count(../*)  lt  count(key('kFindWord',
>>>> >> \$pTargetWord, \$vDictGraph)/../* )]
>>>> >>                       |
>>>> >>                       key('kFindWord', \$pTargetWord, \$vDictGraph)
>>>> >>                          [count(../*) le count(key('kFindWord',
>>>> >> \$pStartWord, \$vDictGraph)/../*)]"/>
>>>> >>
>>>> >>   <xsl:variable name="vTargetWord" as="xs:string"
>>>> >>                 select="(\$pStartWord, \$pTargetWord)[not(. eq
>>>> >> \$vStartWord)]"/>
>>>> >>
>>>> >>   <!-- This function iterates over the temporary tree
>>>> >>
>>>
>>
> <result><arc level=".." from=".." to=".."/>...</result>
>>>> >>     to find
>>
> the
>>> ladder. It starts at a node matching @to with
>>>> >> \$vTargetWord
>>>>
>>>
>> and
>>> proceeds with decreasing @level. -->
>>>> >>   <xsl:function name="my:find-path" as="xs:string*">
>>>> >>     <xsl:param name="root"   as="node()"/>
>>>> >>     <xsl:param name="level"  as="xs:integer"/>
>>>> >>     <xsl:param name="start"  as="xs:string"/>
>>>> >>     <xsl:param name="target" as="xs:string"/>
>>>> >>     <xsl:param name="path"   as="xs:string"/>
>>>> >>
>>>> >>     <xsl:for-each select="\$root/result/arc[@level = \$level and @to =
>>>> >> \$target]">
>>>> >>       <xsl:variable name="from" select="./@from"/>
>>>> >>       <xsl:choose>
>>>> >>         <xsl:when test="\$start eq \$from">
>>>> >>           <xsl:value-of select="concat(\$from,'+',\$path)"/>
>>>> >>         </xsl:when>
>>>> >>         <xsl:otherwise>
>>>> >>           <xsl:value-of select="my:find-path(\$root,\$level
>>>> >> -1,\$start,\$from,concat(\$from,'+',\$path))"/>
>>>> >>         </xsl:otherwise>
>>>> >>       </xsl:choose>
>>>> >>     </xsl:for-each>
>>>> >>   </xsl:function>
>>>> >>
>>>> >>   <xsl:template match="/">
>>>> >>     <xsl:variable name='arcs'>
>>>> >>       <result>
>>>> >>       <xsl:call-template name="look-at-starts">
>>>> >>         <xsl:with-param name="level"  select="1"/>
>>>> >>         <xsl:with-param name="starts" select="\$vStartWord"/>
>>>> >>         <xsl:with-param name="target" select="\$vTargetWord"/>
>>>> >>         <xsl:with-param name="toskip" select="()"/>
>>>> >>       </xsl:call-template>
>>>> >>       </result>
>>>> >>     </xsl:variable>
>>>> >>
>>>> >>     <xsl:variable name="finalArcs" select="\$arcs/result/arc[@to =
>>>> >> \$vTargetWord]"/>
>>>> >>     <xsl:value-of select="my:find-path(\$arcs, \$finalArcs[1]/@level,
>>>> >> \$vStartWord, \$vTargetWord, \$vTargetWord)"/>
>>>> >>   </xsl:template>
>>>> >>
>>>> >>   <!-- Look at \$starters nodes obtained from the current set of
> words
>>>>
>>>>>      ending all incomplete ladders. Generate result/arc for
> each hop
>> to
>>>>
>>>>>      the next step. Recurse if none of the arc
> destinations is the
>>>> >>
>>> overall
>>>> >>      target word, otherwise
> return the last hop. -->
>>>> >>   <xsl:template name="look-at-starts">
>>>> >>     <xsl:param name="level"  as="xs:integer"/>
>>>> >>     <xsl:param name="starts" as="xs:string*"/>
>>>> >>     <xsl:param name="target" as="xs:string"/>
>>>> >>     <xsl:param name="toskip" as="node()*"/>
>>>> >>
>>>> >>     <xsl:variable name="starters" as="node()*"
>>>> >>                   select="key('kFindWord', \$starts, \$vDictGraph)/..
>>>> >> except \$toskip"/>
>>>> >>
>>>> >>     <xsl:for-each select="\$starters">
>>>> >>       <xsl:variable name="w" select="./w"/>
>>>> >>       <xsl:for-each select="./nb">
>>>> >>         <arc level="{\$level}" from="{\$w}" to="{.}"/>
>>>> >>       </xsl:for-each>
>>>> >>     </xsl:for-each>
>>>> >>
>>>> >>     <xsl:variable name="nbs" select="\$starters/nb"/>
>>>> >>
>>>> >>     <xsl:choose>
>>>> >>       <xsl:when test="\$target = \$nbs">
>>>> >>         <!--xsl:message select="'found a ladder'"/-->
>>>> >>       </xsl:when>
>>>> >>       <xsl:otherwise>
>>>> >>         <xsl:call-template name="look-at-starts">
>>>> >>           <xsl:with-param name="level"  select="\$level + 1"/>
>>>> >>           <xsl:with-param name="starts"
>>>> >> select="distinct-values(\$nbs)"/>
>>>> >>           <xsl:with-param name="target" select="\$target"/>
>>>> >>           <xsl:with-param name="toskip" select="\$toskip union
>>>> >> \$starters"/>
>>>> >>         </xsl:call-template>
>>>> >>       </xsl:otherwise>
>>>> >>     </xsl:choose>
>>>> >>   </xsl:template>
>>>> >> </xsl:stylesheet>

```