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 <H1><A NAME="SECTION00010000000000000000">
   About this document ... </A>
 </H1> 
  <STRONG>Using Space-filling Curves for Multi-dimensional Indexing</STRONG><P>
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 <P>
 Copyright &#169; 1993, 1994, 1995, 1996, 1997,
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 The translation was initiated by Jonathan Lawder on 2000-06-28
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 <I>Jonathan Lawder</I>
 <BR><I>2000-06-28</I>
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 <H1 ALIGN="CENTER">Using Space-filling Curves for Multi-dimensional Indexing</H1>
 <P ALIGN="CENTER"><STRONG>by
 <BR>
 J K Lawder and P J H King</STRONG></P>
 <P ALIGN="CENTER"><STRONG>4th July 2000</STRONG></P>
 <P ALIGN="LEFT"></P>
 
 <P>
 <DIV ALIGN="LEFT">
 <P>
 
 <FONT SIZE=" 1"><B>Outline</B>
 </FONT><UL>
 <P>
 <LI>What is a space-filling curve?
 <LI>Summary of the work undertaken
 <LI>Why research into multi-dimensional indexing?
 <LI>Why use space-filling curves?
 <LI>Mapping techniques
 <LI>Querying : an example 
 
 <P>
 </UL>
 <P>
 
 <FONT SIZE=" 1"><B>What is a space-filling curve?</B>
 </FONT>
 <P>
 <UL>
 <LI>a line passing through <B>every</B> point in a space, in a particular <B>order</B>, according to some <B>algorithm</B>
 <LI>some curves pass through points <B>once</B> only and so each point lies a <B>unique distance</B> along the curve away from its beginning - these are the ones we are interested in
 <LI>examples include the <B>Hilbert Curve</B> and the <B>Z-Order Curve</B>
 </UL>
 <P>
 
 
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 <P>
 
 <FONT SIZE=" 1"><B>Summary of the work undertaken</B>
 </FONT><UL>
 <LI>designed and implemented a fully functioning application for the storage and retrieval of multi-dimensional data
 <LI>it uses space-filling curves to map multi-dimensional data to one dimensional values
 <LI>currently, it works in up to 16 dimensions but can be easily extended 
 </UL>
 <P>
 
 <FONT SIZE=" 1"><B>Why research into multi-dimensional indexing?</B>
 </FONT><UL>
 <LI>despite the volume of previous work, it's a problem waiting for a generally accepted <EM><B>good</B></EM> solution
 <LI>data is being gathered in ever increasing volume
 <LI>this data is increasingly higher dimensional in nature
 <LI>growing aspirations for sophisticated data processing to extract valuable information
 </UL>
 <P>
 
 <FONT SIZE=" 1"><B>Why use space-filling curves?</B>
 </FONT><UL>
 <LI>in mapping multi-dimensional space to one-dimensional values, they allow <EM><B>simple</B></EM> indexing methods to be used - like the B-tree
 <LI>unlike traditional hashing functions, they preserve in the one-dimensional values the proximity of points in space
 <LI>although of interest to the research community, most previous work relates to the theoretical clustering properties of the curves and little relates to practical implementation
 </UL>
 <P>
 
 <FONT SIZE=" 1"><B>What are the main problems associated with space-filling curves?</B>
 </FONT><UL>
 <LI>how to map between 1 and <I>n</I> dimensions
 	<UL>
 <LI>using state diagrams
 <LI>by calculation
 	</UL>
 <LI>how to execute queries
 <LI>what information to store and how to store it
 </UL>
 <P>
 
 
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 <P>
 
 
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 <FONT SIZE=" 1"><B>Conclusions</B>
 </FONT><UL>
 <LI>worth pursuing further
 </UL>
 <P>
 
 </DIV>
 <P>
 <BR><HR>
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