Effectiveness of landmark analysis for establishing locality in p2p networks

Locality to other nodes on a peer-to-peer overlay network can be established by means of a set of landmarks shared among the participating nodes. Each node independently collects a set of latency measures to landmark nodes, which are used as a multi-dimensional feature vector. Each peer node uses the feature vector to generate a unique scalar index which is correlated to its topological locality. A popular dimensionality reduction technique is the space filling Hilbert’s curve, as it possesses good locality preserving properties. However, there exists little comparison between Hilbert’s curve and other techniques for dimensionality reduction. This work carries out a quantitative analysis of their properties. Linear and non-linear techniques for scaling the landmark vectors to a single dimension are investigated. Hilbert’s curve, Sammon’s mapping and Principal Component Analysis have been used to generate a 1d space with locality preserving properties. This work provides empirical evidence to support the use of Hilbert’s curve in the context of locality preservation when generating peer identifiers by means of landmark vector analysis. A comparative analysis is carried out with an artificial 2d network model and with a realistic network topology model with a typical power-law distribution of node connectivity in the Internet. Nearest neighbour analysis confirms Hilbert’s curve to be very effective in both artificial and realistic network topologies. Nevertheless, the results in the realistic network model show that there is scope for improvements and better techniques to preserve locality information are required

Hamiltonian triangular refinements and space-filling curves

We have introduced here the concept of Hamiltonian triangular refinement. For any Hamiltonian triangulation it is shown that there is a refinement which is also a Hamiltonian triangulation and the corresponding Hamiltonian path preserves the nesting condition of the corresponding space-filling curve. We have proved that the number of such Hamiltonian triangular refinements is bounded from below and from above. The relation between Hamiltonian triangular refinements and space-filling curves is also explored and explained

Onion Curve: A Space Filling Curve with Near-Optimal Clustering

Space filling curves (SFCs) are widely used in the design of indexes for spatial and temporal data. Clustering is a key metric for an SFC, that measures how well the curve preserves locality in moving from higher dimensions to a single dimension. We present the {\em onion curve}, an SFC whose clustering performance is provably close to optimal for the cube and near-cube shaped query sets, irrespective of the side length of the query. We show that in contrast, the clustering performance of the widely used Hilbert curve can be far from optimal, even for cube-shaped queries. Since the clustering performance of an SFC is critical to the efficiency of multi-dimensional indexes based on the SFC, the onion curve can deliver improved performance for data structures involving multi-dimensional data.Comment: The short version is published in ICDE 1

Modification of Hilbert's Space-Filling Curve to Avoid Obstacles: A Robotic Path-Planning Strategy

This paper addresses the problem of exploring a region using the Hilbert's space-filling curve in the presence of obstacles. No prior knowledge of the region being explored is assumed. An online algorithm is proposed which can implement evasive strategies to avoid obstacles comprising a single or two blocked unit squares placed side by side and successfully explore the entire region. The strategies are specified by the change in the waypoint array which robot going to follow. The fractal nature of the Hilbert's space-filling curve has been exploited in proving the validity of the solution