Reverse k-nearest neighbor search in the presence of obstacles

Abstract

In this paper, we study a new form of reverse nearest neighbor (RNN) queries, i.e., obstructed reverse nearest neighbor (ORNN) search. It considers the impact of obstacles on the distance between objects, which is ignored by the existing work on RNN retrieval. Given a data set P, an obstacle set O, and a query point q in a two-dimensional space, an ORNN query finds from P, all the points/objects that have q as their nearest neighbor, according to the obstructed distance metric, i.e., the length of the shortest path between two points without crossing any obstacle. We formalize ORNN search, develop effective pruning heuristics (via introducing a novel concept of boundary region), and propose efficient algorithms for ORNN query processing assuming that both P and O are indexed by traditional data-partitioning indexes (e.g., R-trees). In addition, several interesting variations of ORNN queries, namely, obstructed reverse k-nearest neighbor (ORkNN) search, ORkNN search with maximum obstructed distance δ (δ-ORkNN), and constrained ORkNN (CORkNN) search, have been introduced, and they can be tackled by extending the ORNN query techniques, which demonstrates the flexibility of the proposed ORNN query algorithm. Extensive experimental evaluation using both real and synthetic data sets verifies the effectiveness of pruning heuristics and the performance of algorithms, respectively.

Introduction

Given a multi-dimensional data set P and a query point q, a reverse nearest neighbor (RNN) query retrieves all the points in P that have q as their nearest neighbor (NN). Due to its wide application base such as decision support [15], profile-based marketing [15], [26], and resource allocation [15], [35], RNN is one of the most popular variants of NN queries [7], [12], [14], [17], [20]. Formally, RNN(q) = {p ∈ P | q ∈ NN(p)}, in which RNN(q) represents the set of reverse nearest neighbors to q and NN(p) denotes the NN of a point p ∈ P. Consider an example in Fig. 1a, where the data set P consists of three data points (i.e., p₁, p₂, p₃) in a two-dimensional (2D) space. Each point p_i (1 ≤ i ≤ 3) is associated with a vicinity circle/arc cir(p_i, r) centered at p_i and having r = dist(p_i, NN(p_i)) as its radius, i.e., the vicinity circle/arc cir(p_i, r) covers the NN of p_i. Here, dist() refers to a specified distance metric. As shown in Fig. 1a, for the RNN query issued at a point q that uses the Euclidean distance as the distance metric, its result set RNN(q) = {p₁}, since q is only inside p₁'s vicinity arc cir(p₁, dist(p₁, p₂)).

RNN search has been well studied, and many efficient algorithms have been proposed to support RNN query and its variants. A short review of some representative algorithms will be presented in Section 2.1. Existing algorithms employ either the Euclidean distance in a Euclidean space or the network distance in a road network to measure the proximity between objects. To the best of our knowledge, all those algorithms do not take into account the existence of obstacles (e.g., buildings and blindage). However, obstacles are ubiquitous in the real world, and their existence may change the distance between objects, and hence affect the final query result. Recently, the impact of obstacles on various research problems has attracted much attention from academy [10], [11], [13], [19], [23], [32], [34], [39], and many work have been conducted, by taking the influence of obstacles into consideration. For example, the spatial clustering in the presence of obstacles (e.g., COD_CLARANS [29], DBRS_O [30], DBCLuC [38], etc.) is a new research direction for the data mining community formed by considering the impact of obstacles on spatial clustering.

In this paper, we study the impact of obstacles on RNN retrieval in a Euclidean space, and form a new type of RNN queries, namely, obstructed reverse nearest neighbor (ORNN) search. Given a data set P, an obstacle set O, and a query point q in a 2D space, an ORNN query finds from P, all the points that take q as their NN, according to the obstructed distance, i.e., the distance/length of the shortest path that connects two points without crossing any obstacle. An example is depicted in Fig. 1b, where P = {p₁, p₂, p₃} and O = {o₁, o₂}. To simplify the discussion in this paper, we assume that obstacles are in rectangular shapes, although they could be in any other shape as well.

Let ||p_i, q|| be the obstructed distance from a point p_i to q, and ONN(p_i) be the obstructed nearest neighbor (ONN) of p_i that has the smallest obstructed distance to p_i compared with other points. We associate each point p_i ∈ P with (i) an arc arc(p_i, ||p_i, ONN(p_i)||) centered at p_i and with radius ||p_i, ONN(p_i)||, and (ii) its obstructed path to q. For instance, the arc arc(p₃, ||p₃, p₂||) centered at p₃ and having ||p₃, p₂|| as the radius indicates that p₂ is the ONN to p₃, and the straight line from p₃ to q denotes the obstructed path between them without crossing any obstacle. It is observed that ||p₃, q|| < ||p₃, ONN(p₃) = p₂|| and ||p₂, q|| < ||p₂, ONN(p₂) = p₃||, and thus, q is the ONN to both p₂ and p₃, i.e., q's ORNN set ORNN(q) = {p₂, p₃}. Note that, p₁ is the RNN of q in a Euclidean space (see Fig. 1a), but it is not the ORNN of q in an obstructed space due to the block of obstacle o₁.

We focus on ORNN search because, it is not only a challenging problem from the research point of view, but also very useful in many applications. As an example, suppose KFC plans to open a new restaurant and wants to distribute coupons to its potential customers for promotion. Assume that there are some buildings and parks (i.e., obstacles) around the new restaurant, and customers who have the new restaurant as their obstructed nearest restaurant are more likely to visit. Consequently, in order to ensure the effectiveness of the promotion, KFC needs to identify the persons that take the new restaurant as their obstructed nearest restaurant, and distribute coupons to them. In addition, due to the ubiquity of obstacles, the ORNN query is obviously important, as a stand-alone tool or a stepping stone, in location-based services, geographic information systems, and complex spatial data analysis/mining involving obstacles.

In addition to the ORNN query, we also study several interesting variations, i.e., (1) obstructed reverse k-nearest neighbor (ORkNN) search, which retrieves all the points in the dataset P that take a given query point q as one of their obstructed k-nearest neighbors (OkNN); (2) ORkNN retrieval with an obstructed distance threshold δ (δ-ORkNN), which finds the ORkNN points that has the obstructed distances to q bounded by a pre-defined threshold δ; and (3) constrained ORkNN (CORkNN) search, which returns the ORkNN points in a specified restricted area (defined by the spatial region constraints).

In this paper, we present an efficient solution to tackle the ORNN query, which follows a filter-refinement framework and does not require any pre-processing. Moreover, we extend ORNN query algorithm to efficiently handle ORkNN, δ-ORkNN, and CORkNN queries, respectively. In brief, the key contributions of the paper are summarized as follows:

• We formalize ORNN search, a new addition to the family of spatial queries in the presence of obstacles.

• We introduce a new concept of boundary region to facilitate the pruning of unqualified data points and node entries.

• We develop efficient algorithms to answer exact or approximate ORNN retrieval.

• We extend ORNN query techniques to handle several variations of ORNN queries, i.e., ORkNN search, δ-ORkNN search, and CORkNN search.

• We conduct extensive experiments with both real and synthetic data sets to verify the effectiveness of the presented pruning heuristics and the performance of the proposed algorithms.

Note that, this paper extends our preliminary work [9] in several substantial ways. First, we investigate three new ORNN query variants, i.e., ORkNN, δ-ORkNN, and CORkNN queries. Second, we conduct a more comprehensive performance evaluation which incorporates the new classes of queries. Third, we present a more complete review of the related work and more illustrative examples, to make the paper self-contained.

The rest of this paper is organized as follows. Section 2 reviews related work. Section 3 formulates the ORNN query. Section 4 discusses pruning heuristics. Section 5 presents ODC and BRF Algorithms. Section 6 elaborates algorithms for processing ORNN search. Section 7 extends ORNN query solution to tackle several ORNN query variants. Considerable experimental results and our findings are reported in Section 8. Finally, Section 9 concludes the paper with some directions for future work.