A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Routing with guaranteed delivery in ad hoc wireless networks
Wireless Networks
SIGGRAPH '77 Proceedings of the 4th annual conference on Computer graphics and interactive techniques
Worst-Case optimal and average-case efficient geometric ad-hoc routing
Proceedings of the 4th ACM international symposium on Mobile ad hoc networking & computing
Geometric ad-hoc routing: of theory and practice
Proceedings of the twenty-second annual symposium on Principles of distributed computing
Local construction of planar spanners in unit disk graphs with irregular transmission ranges
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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We address the problem of online route discovery for a class of graphs that can be embedded either in two or in three-dimensional space. In two dimensions we propose the class of quasi-planar graphs and in three dimensions the class of quasi-polyhedral graphs. In the former case such graphs are geometrically embedded in R^2 and have an underlying backbone that is planar with convex faces; however within each face arbitrary edges (with arbitrary crossings) are allowed. In the latter case, these graphs are geometrically embedded in R^3 and consist of a backbone of convex polyhedra and arbitrary edges within each polyhedron. In both cases we provide a routing algorithm that guarantees delivery. Our algorithms need only ''remember'' the source and destination nodes and one (respectively, two) reference nodes used to store information about the underlying face (respectively, polyhedron) currently being traversed. The existence of the backbone is used only in proofs of correctness of the routing algorithm; the particular choice is irrelevant and does not affect the behaviour of the algorithm.