MAP: Medial Axis Based Geometric Routing in Sensor Networks

One of the challenging tasks in the deployment of dense wireless networks (like sensor networks) is in devising a routing scheme for node to node communication. Important consideration includes scalability, routing complexity, the length of the communication paths and the load sharing of the routes. In this paper, we show that a compact and expressive abstraction of network connectivity by the medial axis enables efficient and localized routing. We propose MAP, a Medial Axis based naming and routing Protocol that does not require locations, makes routing decisions locally, and achieves good load balancing. In its preprocessing phase, MAP constructs the medial axis of the sensor field, defined as the set of nodes with at least two closest boundary nodes. The medial axis of the network captures both the complex geometry and non-trivial topology of the sensor field. It can be represented compactly by a graph whose size is comparable with the complexity of the geometric features (e.g., the number of holes). Each node is then given a name related to its position with respect to the medial axis. The routing scheme is derived through local decisions based on the names of the source and destination nodes and guarantees delivery with reasonable and natural routes. We show by both theoretical analysis and simulations that our medial axis based geometric routing scheme is scalable, produces short routes, achieves excellent load balancing, and is very robust to variations in the network model

Sandpiles, spanning trees, and plane duality

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.Comment: 13 pages, 9 figure

Toward incremental FIB aggregation with quick selections (FAQS)

Several approaches to mitigating the Forwarding Information Base (FIB) overflow problem were developed and software solutions using FIB aggregation are of particular interest. One of the greatest concerns to deploy these algorithms to real networks is their high running time and heavy computational overhead to handle thousands of FIB updates every second. In this work, we manage to use a single tree traversal to implement faster aggregation and update handling algorithm with much lower memory footprint than other existing work. We utilize 6-year realistic IPv4 and IPv6 routing tables from 2011 to 2016 to evaluate the performance of our algorithm with various metrics. To the best of our knowledge, it is the first time that IPv6 FIB aggregation has been performed. Our new solution is 2.53 and 1.75 times as fast as the-state-of-the-art FIB aggregation algorithm for IPv4 and IPv6 FIBs, respectively, while achieving a near-optimal FIB aggregation ratio

Control Plane Compression

We develop an algorithm capable of compressing large networks into a smaller ones with similar control plane behavior: For every stable routing solution in the large, original network, there exists a corresponding solution in the compressed network, and vice versa. Our compression algorithm preserves a wide variety of network properties including reachability, loop freedom, and path length. Consequently, operators may speed up network analysis, based on simulation, emulation, or verification, by analyzing only the compressed network. Our approach is based on a new theory of control plane equivalence. We implement these ideas in a tool called Bonsai and apply it to real and synthetic networks. Bonsai can shrink real networks by over a factor of 5 and speed up analysis by several orders of magnitude.Comment: Extended version of the paper appearing in ACM SIGCOMM 201

Euclidean Greedy Drawings of Trees

Greedy embedding (or drawing) is a simple and efficient strategy to route messages in wireless sensor networks. For each source-destination pair of nodes s, t in a greedy embedding there is always a neighbor u of s that is closer to t according to some distance metric. The existence of greedy embeddings in the Euclidean plane R^2 is known for certain graph classes such as 3-connected planar graphs. We completely characterize the trees that admit a greedy embedding in R^2. This answers a question by Angelini et al. (Graph Drawing 2009) and is a further step in characterizing the graphs that admit Euclidean greedy embeddings.Comment: Expanded version of a paper to appear in the 21st European Symposium on Algorithms (ESA 2013). 24 pages, 20 figure

Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

A greedily routable region (GRR) is a closed subset of $\mathbb R^2$, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.Comment: full version of a paper appearing in ISAAC 201