A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1: n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i: j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O ( √ n log log n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O ( p n / log n) worst-case time. Furthermore, we present strong evidence that a query time significantly below √ n cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two √ n × √ n matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linear-space data structures for orthogonal range mode in higher dimensions (queries in near O(n 1−1/2d) time) and for halfspace range mode in higher dimensions (queries in O(n 1−1/d2) time).

Abstract. Many classic data structures have been proposed to support geometric queries, such as range search, point location and nearest neigh-bor search. For a two-dimensional geometric data set consisting of n ele-ments, 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 space-efficient geometric data structures. In particular, two different but closely related lines of research will be considered: succinct geometric data struc-tures 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 geo-metric 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 sup-port queries. The succinct and implicit data structures surveyed in this article support several fundamental geometric queries and their variants. 1