Discrete Mathematics - Topics on domination
GPSR: greedy perimeter stateless routing for wireless networks
MobiCom '00 Proceedings of the 6th annual international conference on Mobile computing and networking
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
Robust position-based routing in wireless Ad Hoc networks with unstable transmission ranges
DIALM '01 Proceedings of the 5th international workshop on Discrete algorithms and methods for mobile computing and communications
Routing with guaranteed delivery in ad hoc wireless networks
Wireless Networks
Asymptotically optimal geometric mobile ad-hoc routing
DIALM '02 Proceedings of the 6th international workshop on Discrete algorithms and methods for mobile computing and communications
Algorithmic aspects of constrained unit disk graphs
Algorithmic aspects of constrained unit disk graphs
Ad-hoc networks beyond unit disk graphs
DIALM-POMC '03 Proceedings of the 2003 joint workshop on Foundations of mobile computing
Coverage and connectivity in three-dimensional networks
Proceedings of the 12th annual international conference on Mobile computing and networking
POSANT: A Position Based Ant Colony Routing Algorithm for Mobile Ad-hoc Networks
ICWMC '07 Proceedings of the Third International Conference on Wireless and Mobile Communications
Ad Hoc Networking
On routing with guaranteed delivery in three-dimensional ad hoc wireless networks
ICDCN'08 Proceedings of the 9th international conference on Distributed computing and networking
Position-based routing in ad hoc networks
IEEE Communications Magazine
Practical 3D geographic routing for wireless sensor networks
Proceedings of the 8th ACM Conference on Embedded Networked Sensor Systems
Movement assisted-topology control and geographic routing protocol for underwater sensor networks
Proceedings of the 16th ACM international conference on Modeling, analysis & simulation of wireless and mobile systems
Lexicographic ranking and unranking of derangements in cycle notation
Discrete Applied Mathematics
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We study the problem of routing in three-dimensional ad hoc networks. We are interested in routing algorithms that guarantee delivery and are k-local, i.e., each intermediate node v's routing decision only depends on knowledge of the labels of the source and destination nodes, of the subgraph induced by nodes within distance k of v, and of the neighbour of v from which the message was received. We model a three-dimensional ad hoc network by a unit ball graph, where nodes are points in three-dimensional space, and for each node v, there is an edge between v and every node u contained in the unit-radius ball centred at v. The question of whether there is a simple local routing algorithm that guarantees delivery in unit ball graphs has been open for some time. In this paper, we answer this question in the negative: we show that for any fixed k, there can be no k-local routing algorithm that guarantees delivery on all unit ball graphs. This result is in contrast with the two-dimensional case, where 1-local routing algorithms that guarantee delivery are known. Specifically, we show that guaranteed delivery is possible if the nodes of the unit ball graph are contained in a slab of thickness $$1/\sqrt{2}.$$ However, there is no k-local routing algorithm that guarantees delivery for the class of unit ball graphs contained in thicker slabs, i.e., slabs of thickness $$1/\sqrt{2} + \epsilon$$ for some $$ \epsilon 0.$$ The algorithm for routing in thin slabs derives from a transformation of unit ball graphs contained in thin slabs into quasi unit disc graphs, which yields a 2-local routing algorithm. We also show several results that further elaborate on the relationship between these two classes of graphs.