Wanbin Son

Spatial Skyline Queries: An Efficient Geometric Algorithm

As more data-intensive applications emerge, advanced retrieval semantics, such as ranking and skylines, have attracted attention. Geographic information systems are such an application with massive spatial data. Our goal is to efficiently support skyline queries over massive spatial data. To achieve this goal, we first observe that the best known algorithm VS 2, despite its claim, may fail to deliver correct results. In contrast, we present a simple and efficient algorithm that computes the correct results. To validate the effectiveness and efficiency of our algorithm, we provide an extensive empirical comparison of our algorithm and VS 2 in several aspects.

Covering points by disjoint boxes with outliers

For a set of n points in the plane, we consider the axis-aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain at least n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+klogk) time for p=1, and O(nlogn+k^plog^pk) time for p=2,3. For rectangles we get O(n+k^3) for p=1 and O(nlogn+k^2^+^plog^p^-^1k) time for p=2,3. In all cases, our algorithms use O(n) space.

MSSQ: manhattan spatial skyline queries

Skyline queries have gained attention lately for supporting effective retrieval over massive spatial data. While efficient algorithms have been studied for spatial skyline queries using Euclidean distance, or, L2 norm, these algorithms are (1) still quite computationally intensive and (2) unaware of the road constraints. Our goal is to develop a more efficient algorithm for L1 norm, also known as Manhattan distance, which closely reflects road network distance for metro areas with well-connected road networks. Towards this goal, we present a simple and efficient algorithm which, given a set P of data points and a set Q of query points in the plane, returns the set of spatial skyline points in just O(|P| log |P|) time, assuming that |Q| = |P|. This is significantly lower in complexity than the best known method. In addition to efficiency and applicability, our proposed algorithm has another desirable property of independent computation and extensibility to L∞ norm, which naturally invites parallelism and widens applicability. Our extensive empirical results suggest that our algorithm outperforms the state-of-the-art approaches by orders of magnitude.

Spatial skyline queries: exact and approximation algorithms

As more data-intensive applications emerge, advanced retrieval semantics, such as ranking and skylines, have attracted the attention of researchers. Geographic information systems are a good example of an application using a massive amount of spatial data. Our goal is to efficiently support exact and approximate skyline queries over massive spatial datasets. A spatial skyline query, consisting of multiple query points, retrieves data points that are not father than any other data points, from all query points. To achieve this goal, we present a simple and efficient algorithm that computes the correct results, also propose a fast approximation algorithm that returns a desirable subset of the skyline results. In addition, we propose a continuous query algorithm to trace changes of skyline points while a query point moves. To validate the effectiveness and efficiency of our algorithm, we provide an extensive empirical comparison between our algorithms and the best known spatial skyline algorithms from several perspectives.

Top-k Manhattan Spatial Skyline Queries

Efficiently retrieving relevant data from a huge spatial database is and has been the subject of research in fields like database systems, geographic information systems and also computational geometry for many years. In this context, we study the retrieval of relevant points with respect to a query and a scoring function: let P and Q be point sets in the plane, the skyline of P with respect to Q consists of points P for which no other point of P is closer to all points of Q. A skyline of a point set P with respect to a query set Q can be seen as the most “relevant” or “desirable” subset of P with respect to Q. As the skyline of a set P can be as large as the set P itself, it is reasonable to filter the skyline using a scoring function f, only reporting the k best skyline points with respect to f.

Group Nearest Neighbor Queries in the L 1 Plane

Let P be a set of n points in the plane. The k-nearest neighbor (k-NN) query problem is to preprocess P into a data structure that quickly reports k closest points in P for a query point q. This paper addresses a generalization of the k-NN query problem to a query set Q of points, namely, the group nearest neighbor problem, in the L 1 plane. More precisely, a query is assigned with a set Q of at most m points and a positive integer k with k ≤ n, and the distance between a point p and a query set Q is determined as the sum of L 1 distances from p to all q ∈ Q. The maximum number m of query points Q is assumed to be known in advance and to be at most n; that is, m ≤ n. In this paper, we propose two methods, one based on the range tree and the other based on the segment dragging query, obtaining the following complexity bounds: (1) a group k-NN query can be handled in O(m 2logn + k(loglogn + logm)) time after preprocessing P in O(m 2 n log2 n) time and space, or (2) a query can be handled in O(m 2logn + (k + m)log2 n) time after preprocessing P in O(m 2 nlogn) time using O(m 2 n) space. We also show that our approach can be applied to the group k-farthest neighbor query problem.

Square and Rectangle Covering with Outliers

For a set of n points in the plane, we consider the axis---aligned (p ,k ) -Box Covering problem: Find p axis-aligned, pairwise disjoint boxes that together contain exactly n *** k points. Here, our boxes are either squares or rectangles, and we want to minimize the area of the largest box. For squares, we present algorithms that find the solution in O (n + k logk ) time for p = 1, and in O (n logn + k p log p k ) time for p = 2,3. For rectangles we have running times of O (n + k 3) for p = 1 and O (n logn + k 2 + p log p *** 1 k ) time for p = 2,3. In all cases, our algorithms use O (n ) space.

Computing k-center over Streaming Data for Small k

The Euclidean k-center problem is to compute k congruent balls covering a given set of points in ℝ d such that the radius is minimized. We consider the k-center problem in ℝ d for k = 2,3 in a single-pass streaming model, where data is allowed to be examined once and only a small amount of information can be stored in a device. We present two approximation algorithms whose space complexity does not depend on the size of the input data. The first algorithm guarantees a (2 + e)-factor using O(d/e) space in arbitrary dimensions, and the second algorithm guarantees a (1 + e)-factor using O(1/e d ) space in constant dimensions. The same algorithms can be used to compute a k-center under any L p metric for k = 2,3.

Constrained Geodesic Centers of a Simple Polygon

For any two points in a simple polygon P, the geodesic distance between them is the length of the shortest path contained in P that connects them. A geodesic center of a set S of sites (points) with respect to P is a point in P that minimizes the geodesic distance to its farthest site. In many realistic facility location problems, however, the facilities are constrained to lie in feasible regions. In this paper, we show how to compute the geodesic centers constrained to a set of line segments or simple polygonal regions contained in P. Our results provide substantial improvements over previous algorithms.

Geometric Matching Algorithms for Two Realistic Terrains

We consider a geometric matching of two realistic terrains, each of which is modeled as a piecewise-linear bivariate function. For two realistic terrains f and g where the domain of g is relatively larger than that of f , we seek to find a translated copy f ′ of f such that the domain of f ′ is a sub-domain of g and the L∞ or the L1 distance of f ′ and g restricted to the domain of f ′ is minimized. In this paper, we show a tight bound on the number of different combinatorial structures that f and g can have under translation in their projections on the xy-plane. We give a deterministic algorithm and a randomized algorithm that compute an optimal translation of f with respect to g under L∞ metric. We also give a deterministic algorithm that computes an optimal translation of f with respect to g under L1 metric.