Improved range searching lower bounds

Abstract

In this paper we present a number of improved lower bounds for range searching in the pointer machine and the group model. In the pointer machine, we prove lower bounds for the approximate simplex range reporting problem. In approximate simplex range reporting, points that lie within a distance of ε · diam(s) from the border of a query simplex s, are free to be included or excluded from the output, where ε ≥ 0 is an input parameter to the range searching problem. We prove our lower bounds by constructing a hard input set and query set, and then invoking Chazelle and Rosenberg's [CGTA'96] general theorem on the complexity of navigation in the pointer machine.For the group model, we show that input sets and query sets that are hard for range reporting in the pointer machine (i.e. by Chazelle and Rosenberg's theorem), are also hard for dynamic range searching in the group model. This theorem allows us to reuse decades of research on range reporting lower bounds to immediately obtain a range of new group model lower bounds. Amongst others, this includes an improved lower bound for the fundamental problem of dynamic d-dimensional orthogonal range searching, stating that tqtu = Ω((lg n/ lg lg n) d−1 ). Here tq denotes the query time and tu the update time of the data structure. This is an improvement of a lg 1−δ n factor over the recent lower bound of Larsen [FOCS'11], where δ > 0 is a small constant depending on the dimension.

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“…This example can be found in the work of Chazelle and Liu [7], where it is stated for R being lines (i. e., degenerate simplices). Note that the lower bound of Chazelle and Lui [7] is for range reporting in the pointer machine, but using the observations in Larsen [17] and Larsen and Nguyen [18] it is easily seen that the above properties hold. Even if these properties are not enough to obtain lower bounds for linear operators, we believe the geometric approach might be useful in its own right.…”

“…Furthermore, the research on data structure lower bounds also provides a lot of insight into which concrete sets P and R might be difficult. More specifically, polynomial lower bounds for simplex range searching have been proved for: range reporting in the pointer machine (Chazelle and Rosenberg [8], Afshani [1]) and I/O-model (Afshani [1]), range searching in the semigroup model (Chazelle [6]) and range searching in the group model (Larsen [17], Larsen and Nguyen [18]). The group model comes closest in spirit to linear data structures.…”

Adapt or Die: Polynomial Lower Bounds for Non-Adaptive Dynamic Data Structures

“…degenerate simplices). Note that the lower bound in [4] is for range reporting in the pointer machine, but using the observations in [11,14] it is easily seen that all the above properties hold.…”

“…Furthermore, the research on data structure lower bounds also provide a lot of insight into which concrete sets P and R that might be difficult. More specifically, polynomial lower bounds for simplex range searching has been proved for: range reporting in the pointer machine [5,1] and I/O-model [1], range searching in the semi-group model [3] and range searching in the group model [11,14]. The group model comes closest in spirit to linear data structures.…”