Succinct and Implicit Data Structures for Computational Geometry

Many classic data structures have been proposed to support geometric queries, such as range search, point location and nearest neighbor search. For a two-dimensional geometric data set consisting of n elements, these structures typically require O(n), close to O(n) or O(n lg n) words of space; while they support efficient queries, their storage costs are often much larger than the space required to encode the given data. As modern applications often process very large geometric data sets, it is often not practical to construct and store these data structures. This article surveys research that addresses this issue by designing spaceefficient geometric data structures. In particular, two different but closely related lines of research will be considered: succinct geometric data structures and implicit geometric data structures. The space usage of succinct geometric data structures is equal to the information-theoretic minimum space required to encode the given geometric data set plus a lower order term, and these structures also answer queries efficiently. Implicit geometric data structures are encoded as permutations of elements in the data sets, and only zero or O(1) words of extra space is required to support queries. The succinct and implicit data structures surveyed in this article support several fundamental geometric queries and their variants.