Local Routing on a Convex Polytope in R^3

TL;DR

Addresses local routing on the boundary $\partial P$ of a convex polytope in $\mathbb{R}^3$ by combining a patch-based geometric approximation with a two-level routing structure. The authors partition $\partial P$ into $\delta$-patches, construct a surrogate polytope $P'$ and representative vertices, build geodesic spanners $G_i'$ and a global $G'$, and apply Thorup–Zwick routing to populate compact routing tables on $P$; the resulting scheme achieves an amortized per-vertex space of $O(\min(n, 1/\epsilon^{3/2}))$, a label size of $O\bigl((\log(\min(n, 1/\epsilon)))^2\bigr)$, and a worst-case stretch of $\frac{8+\epsilon}{\sin{\theta_m}}(D+d(s,t))$, with preprocessing time $O\bigl(n+\min(n^3, 1/\epsilon^7)\bigr)$. The work introduces an explicit, local-routing protocol for polytopal surfaces, enabling efficient routing between any pair of vertices and opening directions for extending to non-convex domains or obstacle-rich 3D spaces. It blends geometric sketching, spanner-based connectivity, and graph-routing techniques to achieve scalable routing on 3D polyhedra.

Abstract

Given a convex polytope $P$ defined with $n$ vertices in $\mathbb{R}^3$, this paper presents an algorithm to preprocess $P$ to compute routing tables at every vertex of $P$ so that a data packet can be routed on $P$ from any vertex of $P$ to any other vertex of $P$. At every vertex $v$ of $P$ along the routing path, until the packet reaches its destination, the next hop is determined using the routing tables at $v$ and the information stored in the packet header. In $O(n+\min(n^3, \frac{1}{ε^7}))$ time, the preprocessing algorithm computes a routing table at every vertex of $P$ of amortized size $O(\min(n, \frac{1}{ε^{3/2}}))$ bits. If the optimal shortest distance between $s$ and $t$ on $P$ is $d(s, t)$, then the routing path produced by this algorithm has length at most $\frac{8+ε}{\sin{θ_m}}(D+d(s,t))$. Here, $ε\in (0, 1)$ is an input parameter, $D$ is the maximum length of the diagonal of any cell when $\partial P$ is partitioned into $\frac{1}{ε^3}$ geodesic cells of equal size, and $θ_m$ is half the minimum angle between any two neighbouring edges of $P$ on $\partial P$.

Figures (4)

• Figure 1: Illustrates patch $\delta_i$ consisting of a contiguous section of faces from $P$, a representative face $f$ of $\delta_i$, and the corresponding representative face $\delta_i'$ containing $f$. Also illustrated the point of projection $v'$ on $\delta_i'$ for a vertex $v$ of $\delta_i$.
• Figure 2: Illustrates selecting one representative (red) per cell of a grid set up on $\delta_i'$. Also illustrated is a plane $\eta$ orthogonal to $\delta_i'$ containing the representative (red) and a non-representative (blue) belonging to the same grid cell.
• Figure 3: Illustrates how the next hop for the packet is determined at vertex $p_1$ when the packet is at $p_1$. In the case shown, since the plane $\eta$ intersects edge $p_2p_4$ of $f_2$, the packet at $p_1$ is forwarded to $p_2$.
• Figure 4: Illustrates how the next hop for the packet is determined at $p_1$ when the packet is at vertex $p_1$. Among all the faces incident to $p_1$, in the sequence of faces intersected by $\eta$, faces $f_1$ and $f_2$ are the first and last ones. Given the packet was at $p$, since $\eta$ intersects $p_2p_4$ of $f_3$ in the case depicted, the packet is forwarded to $p_2$.

Theorems & Definitions (12)

• Theorem
• lemma thmcounterlemma
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• lemma thmcounterlemma
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