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UB-tree
Two dimensional Z-order
Type	tree
Invented by	Rudolf Bayer and Volker Markl
Time complexity in big O notation Operation Average Worst case Space complexity

The UB-tree as proposed by Rudolf Bayer and Volker Markl is a balanced tree for storing and efficiently retrieving multidimensional data. Like a B+ tree, information is stored only in the leaves. Records are stored according to Z-order, also called Morton order. Z-order is calculated by bitwise interlacing of the keys.

Insertion, deletion, and point query are done as with ordinary B+ trees. To perform range searches in multidimensional point data, however, an algorithm must be provided for calculating, from a point encountered in the data base, the next Z-value which is in the multidimensional search range.

The original algorithm to solve this key problem was exponential with the dimensionality and thus not feasible ("GetNextZ-address"). A solution to this "crucial part of the UB-tree range query" has been described later. This method has already been described in an older paper where using Z-order with search trees has first been proposed.

References

1. Markl, V. (1999). "MISTRAL: Processing Relational Queries using a Multidimensional Access Technique". CiteSeerX 10.1.1.32.6487.
2. Ramsak, Frank; Markl, Volker; Fenk, Robert; Zirkel, Martin; Elhardt, Klaus; Bayer, Rudolf (September 10–14, 2000). Integrating the UB-tree into a Database System Kernel (PDF). 26th International Conference on Very Large Data Bases. pp. 263–272.
3. Tropf, H.; Herzog, H. "Multidimensional Range Search in Dynamically Balanced Trees" (PDF). Angewandte Informatik (Applied Informatics) (2/1981): 71–77. ISSN 0013-5704.

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