A Note on the Area Requirement of Euclidean Greedy Embeddings of Christmas Cactus Graphs

An Euclidean greedy embedding of a graph is a straight-line embedding in the plane, such that for every pair of vertices $s$ and $t$, the vertex $s$ has a neighbor $v$ with smaller distance to $t$ than $s$. This drawing style is motivated by greedy geometric routing in wireless sensor networks. A Christmas cactus is a connected graph in which every two simple cycles have at most one vertex in common and in which every cutvertex is part of at most two biconnected blocks. It has been proved that Christmas cactus graphs have an Euclidean greedy embedding. This fact has played a crucial role in proving that every 3-connected planar graph has an Euclidean greedy embedding. The proofs construct greedy embeddings of Christmas cactuses of exponential size, and it has been an open question whether exponential area is necessary in the worst case for greedy embeddings of Christmas cactuses. We prove that this is indeed the case.Comment: This problem has been stated by Ankur Moitra in his presentation at the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS'08): http://people.csail.mit.edu/moitra/docs/ftl.pd

Stateless and Delivery Guaranteed Geometric Routing on Virtual Coordinate System

Stateless geographic routing provides relatively good performance at a fixed overhead, which is typically much lower than conventional routing protocols such as AODV. However, the performance of geographic routing is impacted by physical voids, and localization errors. Accordingly, virtual coordinate systems (VCS) were proposed as an alternative approach that is resilient to localization errors and that naturally routes around physical voids. However, VCS also faces virtual anomalies, causing their performance to trail geographic routing. In existing VCS routing protocols, there is a lack of an effective stateless and delivery guaranteed complementary routing algorithm that can be used to traverse voids. Most proposed solutions use variants of flooding or blind searching when a void is encountered. In this paper, we propose a spanning-path virtual coordinate system which can be used as a complete routing algorithm or as the complementary algorithm to greedy forwarding that is invoked when voids are encountered. With this approach, and for the first time, we demonstrate a stateless and delivery guaranteed geometric routing algorithm on VCS. When used in conjunction with our previously proposed aligned virtual coordinate system (AVCS), it out-performs not only all geometric routing protocols on VCS, but also geographic routing with accurate location information