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Two dimensional Z-order
Invented byRudolf 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. It is basically a B+ tree (information only in the leaves) with records stored according to Z-order, also called Morton order. Z-order is simply calculated by bitwise interlacing 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[1] ("GetNextZ-address"). A solution to this "crucial part of the UB-tree range query" linear with the z-address bit length has been described later.[2] This method has already been described in an older paper[3] where using Z-order with search trees has first been proposed.


  1. ^ Markl, V. (1999). "MISTRAL: Processing Relational Queries using a Multidimensional Access Technique". CiteSeerX
  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.