A Hierarchical Spatial Network Index for Arbitrarily Distributed Spatial Objects

The range query is one of the most important query types in spatial data processing. Geographic information systems use it to find spatial objects within a user-specified range, and it supports data mining tasks, such as density-based clustering. In many applications, ranges are not computed in unrestricted Euclidean space, but on a network. While the majority of access methods cannot trivially be extended to network space, existing network index structures partition the network space without considering the data distribution. This potentially results in inefficiency due to a very skewed node distribution. To improve range query processing on networks, this paper proposes a balanced Hierarchical Network index (HN-tree) to query spatial objects on networks. The main idea is to recursively partition the data on the network such that each partition has a similar number of spatial objects. Leveraging the HN-tree, we present an efficient range query algorithm, which is empirically evaluated using three different road networks and several baselines and state-of-the-art network indices. The experimental evaluation shows that the HN-tree substantially outperforms existing methods.

Frequency-hiding order-preserving encryption with small client storage

The range query on encrypted databases is usually implemented using the order-preserving encryption (OPE) technique which preserves the order of plaintexts. Since the frequency leakage of plaintexts makes OPE vulnerable to frequency-analyzing attacks, some frequency-hiding order-preserving encryption (FH-OPE) schemes are proposed. However, existing FH-OPE schemes require either the large client storage of size O ( n ) or O (log n ) rounds of interactions for each query, where n is the total number of plaintexts. To this end, we propose a FH-OPE scheme that achieves the small client storage without additional client-server interactions. In detail, our scheme achieves O ( N ) client storage and 1 interaction per query, where N is the number of distinct plaintexts and N ≤ n . Especially, our scheme has a remarkable performance when N ≪ n . Moreover, we design a new coding tree for producing the order-preserving encoding which indicates the order of each ciphertext in the database. The coding strategy of our coding tree ensures that encodings update in the low frequency when inserting new ciphertexts. Experimental results show that the single round interaction and low-frequency encoding updates make our scheme more efficient than previous FH-OPE schemes.

On the VC-dimension of half-spaces with respect to convex sets

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.