Nested Intervals AKA Matrix Encoding

SQL block database: about nested sets: https://bitcointalk.org/index.php?topic=89046.0

http://www.rubyflow.com/items/7310-acts-as-nested-interval-like-nested-set-only-faster

https://github.com/chinasaur/acts_as_nested_moebius

http://stackoverflow.com/questions/360738/are-nested-intervals-a-viable-solution-to-nested-set-modified-pre-order-travers/364215#364215

https://github.com/grondilu/libbitcoin-perl

mobeus encoding.. matrix encoding more advanced than the rational number?

https://vadimtropashko.wordpress.com/2008/07/12/matrix-encoding-growth/

http://en.wikipedia.org/wiki/Nested_intervals

http://en.wikipedia.org/wiki/Farey_sequence

https://github.com/pythonic/acts_as_nested_interval

https://github.com/kidlab/acts_as_nested_interval

https://github.com/clyfe/acts_as_nested_interval

http://stackoverflow.com/questions/7535062/interesting-tree-hierarchical-data-structure-problem

Example Code:

https://tabo.pe/projects/django-treebeard/

http://code.tabo.pe/django-treebeard/src/93b579395a9c/treebeard/mp_tree.py

http://drupal.org/files/issues/tree.inc__4.patch

[FORUM] Vadim Tropashko - (first mention of real or rational numbers?) Nested set model with large gaps and spreads in the numbering Sept 9th 2002 - https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/a7w6LzJrBbc/VQzL7sYyn_EJ

[PAPER] Vadim Tropashko - (binary fractions) (introducing nested intervals) Trees in SQL: Nested Sets and Materialized Path - 2003 - http://web.archive.org/web/20040117214039/http://www.dbazine.com/tropashko4.shtml https://communities.bmc.com/communities/docs/DOC-9902


 * The idea of generalizing nested integers to nested fractions is still valid, although the particular encoding schema with Binary Rationals proved to be not practical.

[PAPER] Vadim Tropashko - (binary fractions) Relocating Subtrees in Nested Intervals Model - Nov 18 2003: http://web.archive.org/web/20031118185410/http://www.dbazine.com/tropashko5.shtml


 * Uses the same Binary Rationals encoding schema solving tree relocation problem.

Using the node type to solve problems with hierarchies in DB2 Universal Database - Jacques Roy - Feb 26 2003 - http://www-106.ibm.com/developerworks/db2/library/techarticle/0302roy/0302roy.html http://www3.software.ibm.com/ibmdl/pub/software/dw/dm/db2/0302roy/0302roy.pdf

[FORUM] (Celko, Vadim Tropashko) - Nested Sets Insertion - May 10 2003: https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/fhWRecgKPSU/8cXQAR7IC5oJ

[FORUM] Tropashko nested sets and materialized path - great idea but how do I insert? Sept 11 2003: https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20sets/comp.databases.theory/nZqEkKJ1FeY/NCDJRzWu4VYJ

[FORUM] Vadim Tropashko - Questions about Nested Intervals by Vadim Tropashko one more time Dec 1 2003 - https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/A3N8sGyyzjQ/1cYFmoUjNpUJ

[FORUM] Vadim Tropashko - Making a tree with "millions and millions" of dynamic nodes - Dec 2 2003: https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/gGqjMniLAkg/hzlV4ec-GeoJ

[FORUM] Hierachical structures - an overview - Jan 4 2004: https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/yG1TyRTT10s/ozDnzAQkHXQJ

[PAPER] Vadim Tropashko - (farey fractions) Nested Intervals with Farey Fractions - Jan 18 2004 - http://arxiv.org/html/cs.DB/0401014

[PAPER] Vadim Tropashko - (continued fractions) Nested Intervals Tree Encoding with Continued Fractions - Feb 20 2004 - http://arxiv.org/abs/cs/0402051

[FORUM] Vadim Tropashko - (continued fractions, matrix encoding nice example) Comments: self referencing tables/ nested sets etc... Mar 29 2004 - https://groups.google.com/group/comp.databases.theory/msg/e9d611b60ffad1a2?hl=da

[FORUM] Vadim Tropashko - (matrix encoding) Typo in http://arxiv.org/.../0402051.pdf? June 22 2004: http://www.dbmonster.com/Uwe/Threads/List.aspx/db-theory/439/Typo-in-http-arxiv-org-0402051-pdf

[PAPER] Vadim Tropashko - Nested Intervals Tree Encoding in SQL - June 2005 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.643&rep=rep1&type=pdf

[PAPER] Vadim Tropashko - Nested Intervals Tree Encoding in SQL - Citations June 2005: http://academic.research.microsoft.com/Publication/1731838/nested-intervals-tree-encoding-in-sql

[PAPER] Vadim Tropashko - Nested Intervals Tree Encoding in SQL - old link June 2005 http://www.sigmod.org/sigmod/record/issues/0506/p47-article-tropashko.pdf

[FORUM] Vadim Tropashko - Questions about nested intervals. Help me please!!! - July 7 2005: https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/XIz6eGAAoXo/dgeNkQZ3HKwJ

[FORUM] Vadim Tropashko - matrix encoding IS adjacency list - Sept 9 2005: https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/_RmwdiacFmI/KfFFOSA1pwkJ

[FORUM] Vadim Tropashko - Overview of nesting methods - Nov 15 2005: https://groups.google.com/group/comp.databases.theory/msg/7b772060322df739

[FORUM] naive questions about nested intervals with Farey fractions - May 18 2007 https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/bRmQhswbu7E/-zuscKl3fs8J

[PAPER] Dan Hazel - Using rational numbers to key nested sets - June 19 2008: http://arxiv.org/abs/0806.3115

[BLOG] Vadim Tropashko - One more Nested Intervals encoding - Notes on Dan Haze's Method June 20 2008: http://vadimtropashko.wordpress.com/2008/06/20/one-more-nested-intervals-encoding/

[FORUM] Tegiri Nenashi - Nested interval tree encoding July 3 2008: http://www.dbmonster.com/Uwe/Forum.aspx/db-theory/1980/Nested-interval-tree-encoding

[BLOG] One more Nested Intervals vs. Adjacency List comparison August 9 2008: http://vadimtropashko.wordpress.com/2008/08/09/one-more-nested-intervals-vs-adjacency-list-comparison/

[FORUM] (Tegiri Nenashi) Adding history/versioning to a Nested Set model (is it possible?) Oct 28 2009: https://groups.google.com/forum/?hl=en&fromgroups#!searchin/comp.databases.theory/nested$20intervals/comp.databases.theory/4Rzdn_IQDPU/ZuqWpl0VXqcJ

[FORUM] Getting a grip on nested intervals and matrix encoding July 1 2010: http://groups.google.com/group/comp.databases.theory/browse_frm/thread/f40bd30a635ee7a5?hl=en#

[BLOG] More Errata: Ancestor Query in Matrix Encoding July 3 2010: http://vadimtropashko.wordpress.com/2010/07/03/more-errata-ancestor-query-in-matrix-encoding/

(from https://groups.google.com/forum/?hl=en&fromgroups#!topic/comp.databases.theory/9AvTCmNe56U )

Nested Sets SQL Tips by Burleson Consulting Interval Halving: http://www.dba-oracle.com/t_sql_patterns_nested_sets.htm

http://vadimtropashko.wordpress.com/

http://vadimtropashko.wordpress.com/2008/07/12/matrix-encoding-growth/

http://translate.google.com/translate?js=n&prev=_t&hl=en&ie=UTF-8&layout=2&eotf=1&sl=auto&tl=en&u=http%3A%2F%2Fhabrahabr.ru%2Fblogs%2Fmysql%2F67722%2F

http://en.wikipedia.org/wiki/Farey_sequence

http://en.wikipedia.org/wiki/Stern-Brocot_tree

http://129.81.170.14/~erowland/courses/2009-2/projects/Mills.pdf

http://www.progsoc.org/wiki/index.php/Euler_Solution_73

http://troels.arvin.dk/db/rdbms/links/

When questioned what the sum of ½ and ¼ is some suggest that the result is ½ + ¼ = (1+1)/(2+4) = 2/6 = 1/3. Ironically, their naïve approach is not without merit. The operation of adding fractions in this “wrong way” is called the mediant. The mediant is the simplest number between two fractions if using the smallness of a denominator as a measure of simplicity. Indeed, the average of ¼ and ½ has denominator 8, while the mediant has denominator 3.

If we start with the point 0 and 1 and continue on taking the mediant iteratively; then another famous set of numbers, Farey fractions, is produced.

http://www.dba-oracle.com/t_sql_patterns_nested_sets.htm

http://www.dbmonster.com/Uwe/Forum.aspx/db-theory/986/Nested-Sets-vs-Nested-Intervals

http://www.numbertheory.org/pdfs/decimal.pdf

http://www.scribd.com/doc/39475526/52/Types-of-Decimal-Expansions

http://drupal.org/node/344019

http://code.activestate.com/lists/python-tutor/10913/