The representation of binary relations has been intensively studied and many different theoretical and practical representations have been proposed to answer the usual queries in multiple domains. However, ternary relations have not received as much attention, even though many real-world applications require the processing of ternary relations. In this paper we present a new compressed and self-indexed data structure that we call Interleaved $K^2$-tree (I$K^2$-tree), designed to compactly represent and efficiently query general ternary relations. The I$K^2$-tree is an evolution of an existing data structure, the $K^2$-tree, initially designed to represent Web graphs and later applied to other domains. The I$K^2$-tree is able to extend the $K^2$-tree to represent a ternary relation, based on the idea of decomposing it into a collection of binary relations but providing indexing capabilities in all the three dimensions. We present different ways to use I$K^2$-tree to model different types of ternary relations using as reference two typical domains: RDF and Temporal Graphs. We also experimentally evaluate our representations comparing them in space usage and performance with other solutions of the state of the art.

This research has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie Actions H2020-MSCA-RISE-2015 BIRDS GA No. 690941, Journal of Discrete Algorithms (2017)