A New Lower Bound for Semigroup Orthogonal Range Searching

We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle's result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao's influential result had shown that the problem is already non-trivial in one dimension~\cite{Yao-1Dlb}: using $m$ units of space, the query time $Q(n)$ must be $\Omega( \alpha(m,n) + \frac{n}{m-n+1})$ where $\alpha(\cdot,\cdot)$ is the inverse Ackermann's function, a very slowly growing function. In $d$ dimensions, Bernard Chazelle~\cite{Chazelle.LB.II} proved that the query time must be $Q(n) = \Omega( (\log_\beta n)^{d-1})$ where $\beta = 2m/n$. Chazelle's lower bound is known to be tight for when space consumption is `high' i.e., $m = \Omega(n \log^{d+\varepsilon}n)$. We have two main results. The first is a lower bound that shows Chazelle's lower bound was not tight for `low space': we prove that we must have $m (n) = \Omega(n (\log n \log\log n)^{d-1})$. Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions

On the importance of idempotence

Answering range queries is a problem of fundamental importance in spatial information retrieval and computational geometry. The objective is to store a set of n points P in R d, each associated with a weight, so that it is possible to count, or more generally to compute some function of the weights of the points lying inside a given query range. Range searching is among the most heavily studied problems, and many search structures have been proposed and analyzed [1, 7]. There is a spectrum of space-time tradeoffs. The most relevant work to ours involves halfspace range counting queries, which Matouˇsek [6] has shown can be answered in n/m 1/d time from a data structure of space O(m). Nearly matching lower bounds were given by by Brönnimann, Chazelle and Pach [4] (or BCP). Given the relatively high complexity of range searching, it is natural to consider the problem in the context of approximation. We are given an approximation parameter ε> 0 and assume that ranges are bounded. Let η denote a range, and let diam(η) denote its diameter. All the points that lie in the range must be counted, and any of the points that lie within distance ε · diam(η) of the range’s boundary may be counted as well. Arya and Mount [3] showed that in any fixed dimension d with O(n log n) preprocessing time and O(n) space, ε-approximate range queries for any bounded convex range can be answered in time O(log n+1/ε d−1) [3]. Later, Chazelle, Liu, and Magen [5] considered approximate halfspace range and Euclidean ball searching in the high dimensional setting. Ignoring polylogarithmic factors, they showed that is possible to answer queries in O(d/ε 2) time with O(dn O(1/ε2) ) space

ABSTRACT On the Importance of Idempotence

Range searching is among the most fundamental problems in computational geometry. An n-element point set in R d is given along with an assignment of weights to these points from some commutative semigroup. Subject to a fixed space of possible range shapes, the problem is to preprocess the points so that the total semigroup sum of the points lying within a given query range η can be determined quickly. In the approximate version of the problem we assume that η is bounded, and we are given an approximation parameter ε> 0. We are to determine the semigroup sum of all the points contained within η and may additionally include any of the points lying within distance ε · diam(η) ofη’s boundary. In this paper we contrast the complexity of range searching based on semigroup properties. A semigroup (S, +) is idempotent if x + x = x for all x ∈ S, and it is integral if for all k ≥ 2, the k-fold sum x + ·· · + x is not equal to x. For example, (R, min) and ({0, 1}, ∨) are both idempotent, and (N, +) is integral. To date, all upper and lower bounds hold irrespective of the semigroup. We show that semigroup properties do indeed make a difference for both exact and approximate range searching, and in the case of approximate range searching the differences are dramatic. First, we consider exact halfspace range searching. The assumption that the semigroup is integral allows us to improve the best lower bounds in the semigroup arithmetic model. For example, assuming O(n) storage in the plane and ignoring polylog factors, we provide an Ω ∗ (n 2/5) lower bound for integral semigroups, improving upon the best lower bound of Ω ∗ (n 1/3), thus closing the gap with the O(n 1/2) upper bound. We also consider approximate range searching for Eu-∗ This author’s work was supported in part by the Researc

The effect of corners on the complexity of approximate range searching

Range searching is among the most fundamental problems in computational geometry. Given an n-element point set in R d, the problem is to preprocess the points so that the total weight (or generally semigroup sum) of the points lying within a given query range η can be determined quickly. In the ε-approximate version we assume that η is bounded and we are to determine the semigroup sum of all the points contained within η and may additionally include any of the points lying within distance ε · diam(η) ofη’s boundary. In this paper we contrast the complexity of approximate range searching based on properties of the semigroup and range space. A semigroup (S, +) is idempotent if x + x = x for all x ∈ S, and it is integral if for all k ≥ 2, the k-fold sum x + ···+x is not equal to x. Idempotence is important because points may be multiply counted, and this implies that generator subsets may overlap one another. Our recent results [Arya, Malamatos, Mount, “On the Importance of Idempotence, ” STOC 2006, to appear] imply that for approximate Euclidean-ball range searching, idempotence offers significant advantages. In particular, nearly matching upper and lower bounds show that the exponents in the εdependencies are roughly halved for idempotent semigroups. These prior results made critical use of two properties of Euclidean balls: smoothness and rotational symmetry. In this paper we consider two alternative formulations that arise from relaxing these properties. The first involves ranges with sharp corners and the second involves arbitrary smooth convex ranges. We show that, as with integrality, sharp corners have an adverse effect on the problem’s complexity. We consider d-dimensional unit hypercube ranges under rigid motions. Assuming linear space, we show here that in ∗ This author’s work was supported in part by the Researc