Local Routing on a Convex Polytope in R^3

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 $O(n+\min(n^3, \frac{1}{\epsilon^7}))$ time, the preprocessing algorithm computes a routing table at every vertex of $P$ of amortized size $O(\min(n, \frac{1}{\epsilon^{3/2}}))$ 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 $\frac{8+\epsilon}{\sin{\theta_m}}(D+d(s,t))$. Here, $\epsilon \in (0, 1)$ is an input parameter, $D$ is the maximum length of the diagonal of any cell when $\partial P$ is partitioned into $\frac{1}{\epsilon^3}$ geodesic cells of equal size, and $\theta_m$ is half the minimum angle between any two neighbouring edges of $P$ on $\partial P$.