[Paper Review] Space Filling Curves for 3D Sensor Networks with Complex Topology

Mayank Goswami, Siming Li+4 more|arXiv (Cornell University)|Jul 10, 2015

Computational Geometry and Mesh Generation25 references3 citations

TL;DR

This paper proposes a distributed, conformal geometry-based method to generate space-filling curves on 3D sensor networks deployed on complex 2D surfaces with non-trivial topology. By leveraging Hodge decomposition and holomorphic differentials to map high-genus Riemann surfaces to flat tori with slits, it constructs a uniformly dense, progressively filling path that ensures high coverage and low average distance to unvisited nodes, outperforming Eulerian cycles and random walks in simulation.

ABSTRACT

Several aspects of managing a sensor network (e.g., motion planning for data mules, serial data fusion and inference) benefit once the network is linearized to a path. The linearization is often achieved by constructing a space filling curve in the domain. However, existing methods cannot handle networks distributed on surfaces of complex topology. This paper presents a novel method for generating space filling curves for 3D sensor networks that are distributed densely on some two-dimensional geometric surface. Our algorithm is completely distributed and constructs a path which gets uniformly, progressively denser as it becomes longer. We analyze the algorithm mathematically and prove that the curve we obtain is dense. Our method is based on the Hodge decomposition theorem and uses holomorphic differentials on Riemann surfaces. The underlying high genus surface is conformally mapped to a union of flat tori and then a proportionally-dense space filling curve on this union is constructed. The pullback of this curve to the original network gives us the desired curve.

Motivation & Objective

• To address the challenge of linearizing 3D sensor networks distributed on surfaces of complex topology for efficient data mule routing.
• To develop a distributed algorithm that generates a space-filling curve with uniformly increasing density and bounded path density.
• To extend prior work on space-filling curves in simple domains (e.g., tori) to high-genus Riemann surfaces with arbitrary topology.
• To ensure fair and efficient coverage of sensor nodes through progressive, geometrically informed path planning.
• To enable scalable, low-overlap deployment of multiple data mules using independent dense curves on the same surface.

Proposed method

• The method uses the Hodge decomposition theorem to decompose the surface into harmonic and exact forms, enabling conformal mapping of the high-genus surface to a union of flat tori with slits.
• Holomorphic differentials on the Riemann surface are used to construct a canonical representation of the surface as a pair of tori connected by slits.
• A straight-line trajectory with irrational slope is defined on the torus union, which is then mapped back to the original surface via pullback to generate a dense curve.
• The discrete path is generated by expanding the dense curve into a belt of width δ and greedily traversing to the nearest neighbor within the belt at each step.
• Multiple data mules are supported by assigning distinct slopes and starting points to independent dense curves, minimizing path overlap.
• Performance is evaluated using coverage rate and average shortest distance to unvisited nodes, comparing against Eulerian cycles and random walks.

Experimental results

Research questions

• RQ1 Can a space-filling curve be constructed on a 3D sensor network distributed on a high-genus surface with complex topology?
• RQ2 Does the proposed method ensure that the curve becomes uniformly dense and progressively covers the network as path length increases?
• RQ3 How does the coverage performance of the proposed method compare to Eulerian cycles and random walks in terms of coverage rate and distance to unvisited nodes?
• RQ4 Can multiple data mules be deployed using independent dense curves with minimal path overlap?
• RQ5 Can the method be extended to truly volumetric 3D sensor networks?

Key findings

• The proposed method achieves 90% network coverage with a path length of approximately 1.25 times the network size, significantly faster than Eulerian cycles and random walks.
• After covering 5,000 nodes, the average shortest distance to unvisited nodes is 3.5 hops under the proposed method, compared to 16.6 and 22.4 under Eulerian and random walk methods, respectively.
• With only 1,000 nodes visited, the average distance to unvisited nodes is 6.5 hops under two independent dense curves, compared to 20 under a single curve.
• The method ensures minimal overlap between multiple data mule paths, with only 1,865 overlapping nodes out of the first 10,000 covered nodes when using two curves.
• The curve is mathematically proven to be dense, meaning every point on the surface is eventually approached arbitrarily closely as path length increases.
• The method outperforms both Eulerian cycles and random walks in both coverage rate and average distance metrics across all path lengths.