GRIP: Greedy Routing through dIstributed Parametrization for guaranteed delivery in WSNs

Abstract

Although stateless greedy routing is well investigated in 2D wireless sensor networks (WSNs), it is widely believed to be impossible in 3D. In this paper, we aim at overcoming the impossibility through a distributed parametrization that equips a WSN with virtual coordinates favoring greedy routing. We propose a fundamentally new parametrization to embed the network domain, the resulting embedding domain allows greedy routing to have guaranteed delivery. We also present localized algorithms to realize this map in WSNs. To combat the load concentration caused by greedy routing that applies the distance greedy principle, we further propose tunable greedy routing, which relies on tuning a parameter in the greedy objective to naturally balance routing load. These two proposals form our Greedy Routing through dIstributed Parametrization (GRIP). We prove the correctness and efficiency of GRIP and use simulations to evaluate its performance in terms of complexity, load balancing, and energy efficiency.

Appendices

Appendix 1: Proof of Proposition 1

Note $f$ is smooth function, its gradient vector fields are curl-free. Thus, no integral curve can form a loop inside $M$. Furthermore, the function $f$ is harmonic and there is no critical points (where the gradient vanishes) inside $M$. Thus, the function value is strictly monotonic along the integral curve. Note that all points on the same boundary curve have the same function value, so the ending points of each integral curve must be on different boundary curve.

Then we show that two integral curves do not intersect. Assume two integral curves ${\mathbf{x}}_1\in M$ and ${\mathbf{x}}_2\in M$ intersect at a point $p$. Then $p$ is a critical point and the gradient $\nabla f$ vanishes at $p$. Since $f$ is harmonic, the maximum and minimum must be on the boundaries. Therefore the Hessian matrix at $p$ has negative eigenvalue values. Suppose $f(p)=s$, then according to Morse theory, the homotopy types of the level sets $f^{-1}(s-\epsilon )$ and $f^{-1}(s+\epsilon )$ will be different, where $\epsilon$ is a small positive value. At all the interior critical points, the Hessian matrices have negative eigenvalues, the homotopy type of the level sets will be changed. Therefore, the homotopy type of $\gamma _0$ is different from that of $\gamma _1$. This contradicts the given condition that $M$ is 2-connected. Therefore ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ have no intersection points anywhere.

Appendix 2: Correctness of Algorithm 4

Due to the page limit, we only sketch the proofs.

2D WSN. Consider a 2D multiply connected network domain $\mathcal {N}\in \mathbb {R}^2$ with boundaries $\partial \mathcal {N}=\gamma _0-\gamma _1-\cdots -\gamma _l$, where $\gamma _0$ is the outer boundary and $\gamma _i$, $1\le i\le l$, are the hole boundaries.

Each iteration of the parametrization algorithm contains $l$ steps: in the first step, we conformally map $\gamma _0$ to the unit circle and $\gamma _1$ to a concentric circle. Let $\phi _1:\mathcal {N}\rightarrow \mathbb {D}$ be the conformal map of the first step, where $\mathbb {D}$ is the unit disk. Then in the $i$-th step, $i>1$, we conformally map $\phi _{i-1}\circ \cdots \circ \phi _{1}(S)$ to the unit disk $\mathbb {D}$ such that $\gamma _0$ is mapped to the unit circle and $\gamma _i$ to a concentric circle, i.e., $\phi _i:\phi _{i-1}\circ \cdots \circ \phi _{1}(\mathcal {N})\rightarrow \mathbb {D}$. Note that $\phi _1(\gamma _1)$ is a circle. After the mapping, $\phi _2\circ \phi _1(\gamma _1)$ is still close to a circle. Intuitively, all the boundaries are getting rounder and rounder in the iterations, and eventually become circles. As each map $\phi _i$ is conformal, Henrici’s theorem (see [7], pp.502-505) on complex analysis guarantees the convergence of our method.

3D WSN. The proof for volumetric case is similar to the proof of planar case. Each time we map an inner void to the center spherical void, the mapping can be extended to the volume with its reflection with respect to the boundary surface of the inner void. Eventually, the complement of the union of reflected volumes is a Cantor set, with zero measure. The harmonic map can be extended to the whole volume with all voids filled. By using the Poisson integral formula for harmonic maps [20], it can be shown that the images of the inner boundary surfaces are spherical.