cKd-tree: A Compact Kd-tree

Abstract

In the context of Big Data scenarios, the presence of extensive static datasets is not uncommon. To facilitate efficient queries on such datasets, the utilization of multiple indexes, such as the Kd-tree, becomes imperative. The current scale of managed points may, however, exceed the capacity of primary memory, posing a significant challenge. In this article we introduce cKd-tree, a compact data structure designed to represent a Kd-tree efficiently. The structure cKd-tree is essentially an encoding of the spiral code sequence of points within an implicit Kd-tree (iKd-tree) using Directly Addressable Codes (DACs). The unique feature of cKd-tree lies in its ability to perform spiral encoding and decoding of points by relying solely on knowledge of their parent points within the iKd-tree. This inherent property, combined with DACs’ direct access capability to sequence elements, enables cKd-tree to traverse and explore the tree while decoding only the nodes relevant to queries. The article details the algorithms necessary for creating and manipulating a cKd-tree, as well as algorithms for evaluating two fundamental queries over points: the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">point query</i> and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">range query</i> . To assess the performance of cKd-tree, a series of experiments are conducted, comparing it with iKd-tree and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -tree data structures. The evaluation metrics include compression efficiency and execution time of queries. cKd-tree achieves a compression ratio comparable to that of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -tree, approximately 70%, demonstrating heightened efficiency, particularly in scenarios characterized by sparse data. Additionally, consistent with expectations, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -tree exhibits superior performance in querying individual points, whereas cKd-tree outperforms in the context of aggregate data queries, such as range queries.

Topics & Concepts