Optimal Algorithms for Range Searching over Multi-Armed Bandits

This paper studies a multi-armed bandit (MAB) version of the range-searching problem. In its basic form, range searching considers as input a set of points (on the real line) and a collection of (real) intervals. Here, with each specified point, we have an associated weight, and the problem objective is to find a maximum-weight point within every given interval. The current work addresses range searching with stochastic weights: each point corresponds to an arm (that admits sample access) and the point's weight is the (unknown) mean of the underlying distribution. In this MAB setup, we develop sample-efficient algorithms that find, with high probability, near-optimal arms within the given intervals, i.e., we obtain PAC (probably approximately correct) guarantees. We also provide an algorithm for a generalization wherein the weight of each point is a multi-dimensional vector. The sample complexities of our algorithms depend, in particular, on the size of the {optimal hitting set} of the given intervals. Finally, we establish lower bounds proving that the obtained sample complexities are essentially tight. Our results highlight the significance of geometric constructs (specifically, hitting sets) in our MAB setting.

Partial Enclosure Range Searching

A new type of range searching problem, called the partial enclosure range searching problem, is introduced in this paper. Given a set of geometric objects [Formula: see text] and a query region [Formula: see text], our goal is to identify those objects in [Formula: see text] which intersect the query region [Formula: see text] by at least a fixed proportion of their original size. Two variations of this problem are studied. In the first variation, the objects in [Formula: see text] are axis-parallel line segments and the goal is to count the total number of members of [Formula: see text] so that their intersection with [Formula: see text] is at least a given proportion of their size. Here, [Formula: see text] can be an axis-parallel rectangle or a parallelogram of arbitrary orientation. In the second variation, [Formula: see text] is a polygon and [Formula: see text] is an axis-parallel rectangle. The problem is to report the area of the intersection between the polygon [Formula: see text] and a query rectangle [Formula: see text].