We introduce and evaluate a very simple landmark-based network partition technique called Hierarchical Bipartition Routing (HBR) to support routing with delivery guarantee in wireless ad hoc sensor networks. It is a simple routing protocol that can easily be combined with any other greedy routing algorithm to obtain delivery guarantee. The efficiency of HBR increases if the network is sparse and contains obstacles. The space necessary to store the additional routing information at a node u is on average not larger than the size necessary to store the IDs of the neighbors of u. The amount of work to setup the complete data structure is on average proportional to flooding the entire network log(n) times, where n is the total number of sensor nodes. We evaluate the performance of HBR in combination with two simple energy-aware geographic greedy routing algorithms based on physical coordinates and virtual coordinates, respectively. Our simulations show that the difference between using HBR and a weighted shortest path to escape a dead-end is only a few percent in typical cases.
There are several attempts to obtain delivery guarantee for greedy routing algorithms. The authors of @cite_25 propose face routing , which guarantees delivery in two-dimensional unit disk graphs (UDG). Face routing is applied to a planar sub-network obtained by considering the Gabriel Graph @cite_25 @cite_16 , the Relative Neighborhood Graph @cite_30 , or the Morelia Graph @cite_0 . In @cite_19 a greedy-face-greedy (GFG) approach is considered, where greedy routing is based on COP as in @cite_20 and face routing is similar to the one in @cite_25 . Energy-aware routing is also proposed in LEARN @cite_17 , SPFSP @cite_21 , End-to-End (EtE) @cite_11 , and EEGR @cite_8 .
@cite_30 @cite_11 @cite_8 @cite_21 @cite_0 @cite_19 @cite_16 @cite_25 @cite_20 @cite_17
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It was recently reported that all known face and combined greedy-face routing variants cannot guarantee message delivery in arbitrary undirected planar graphs. The purpose of this article is to clarify that this is not the truth in general. We show that specifically in relative neighborhood and Gabriel graphs recovery from a greedy routing failure is always possible without changing between any adjacent faces. Guaranteed delivery then follows from guaranteed recovery while traversing the very first face. In arbitrary graphs, however, a proper face selection mechanism is of importance since recovery from a greedy routing failure may require visiting a sequence of faces before greedy routing can be restarted again. A prominent approach is to visit a sequence of faces which are intersected by the line connecting the source and destination node. Whenever encountering an edge which is intersecting with this line, the critical part is to decide if face traversal has to change to the next adjacent one or not. Failures may occur from incorporating face routing procedures that force to change the traversed face at each intersection. Recently observed routing failures which were produced by the GPSR protocol in arbitrary planar graphs result from incorporating such a face routing variant. They cannot be constructed by the well known GFG algorithm which does not force changing the face anytime. Beside methods which visit the faces intersected by the source destination line, we discuss face routing variants which simply restart face routing whenever the next face has to be explored. We give the first complete and formal proofs that several proposed face routing, and combined greedyface routing schemes do guarantee delivery in specific graph classes or even any arbitrary planar graphs. We also discuss the reasons why other methods may fail to deliver a message or even end up in a loop. We propose a novel localized routing protocol for wireless sensor networks (WSN) that is energy-efficient and guarantees delivery. We prove that it is constant factor of the optimum for dense networks. To forward a packet, a node s in graph G computes the cost of the energy weighted shortest path (SP) between s and each of its neighbors which are closer to the destination than itself. It then selects node x which minimizes the ratio of the cost of the SP to the progress towards the destination. It then sends the message to the first node on the SP from s to x: say node x'. Node x' restarts the same greedy routing process until the destination is reached or the routing fails. To recover from failure, our algorithm invokes face routing that guarantees delivery. This work is the first to optimize energy consumption of face routing. First, we build a connected dominating set from graph G, second we compute its Gabriel graph to obtain the planar graph G'. Face routing is applied on G' only to decide which edges to follow in the recovery process. On each edge, greedy routing is used. This two-phase (greedy-Face) end-to-end routing process (EtE) reiterates until the final destination is reached. Simulation results show that EtE outperforms several existing geographical routing on energy consumption metric. We conjecture that any planar 3-connected graph can be embedded in the plane in such a way that for any nodes s and t, there is a path from s to t such that the Euclidean distance to t decreases monotonically along the path. A consequence of this conjecture would be that in any ad hoc network containing such a graph as a spanning subgraph, two-dimensional virtual coordinates for the nodes can be found for which the method of purely greedy geographic routing is guaranteed to work. We discuss this conjecture and its equivalent forms show that its hypothesis is as weak as possible, and show a result delimiting the applicability of our approach: any 3-connected K3,3-free graph has a planar 3-connected spanning subgraph. We also present two alternative versions of greedy routing on virtual coordinates that provably work. Using Steinitz's theorem we show that any 3-connected planar graph can be embedded in three dimensions so that greedy routing works, albeit with a modified notion of distance; we present experimental evidence that this scheme can be implemented effectively in practice. We also present a simple but provably robust version of greedy routing that works for any graph with a 3-connected planar spanning subgraph. We propose an embedding and routing scheme for arbitrary network connectivity graphs, based on greedy routing and utilizing virtual node coordinates. In dynamic multihop packet-switching communication networks, routing elements can join or leave during network operation or exhibit intermittent failures. We present an algorithm for online greedy graph embedding in the hyperbolic plane that enables incremental embedding of network nodes as they join the network, without disturbing the global embedding. Even a single link or node removal may invalidate the greedy routing success guarantees in network embeddings based on an embedded spanning tree subgraph. As an alternative to frequent reembedding of temporally dynamic network graphs in order to retain the greedy embedding property, we propose a simple but robust generalization of greedy distance routing called Gravity-Pressure (GP) routing. Our routing method always succeeds in finding a route to the destination provided that a path exists, even if a significant fraction of links or nodes is removed subsequent to the embedding. GP routing does not require precomputation or maintenance of special spanning subgraphs and, as demonstrated by our numerical evaluation, is particularly suitable for operation in tandem with our proposed algorithm for online graph embedding. Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomial-time algorithm that embeds combinatorial unit disk graphs (CUDGs - a CUDG is a UDG without any geometric information) into O(log 2 n)- dimensional space, permitting greedy routing with constant stretch. To the best of our knowledge, this is the first greedy embedding with stretch guarantees for this class of networks. Our main technical contribution involves extracting, in polynomial time, a constant number of isometric and balanced tree separators from a given CUDG. We do this by extending the celebrated Lipton-Tarjan separator theorem for planar graphs to CUDGs. Our techniques extend to other classes of graphs; for example, for general graphs, we obtain an O(log n)-stretch greedy embedding into O(log 2 n)-dimensional space. The greedy embeddings constructed by our algorithm can also be viewed as a constant-stretch compact routing scheme in which each node is assigned an O(log 3 n)-bit label. To the best of our knowledge, this result yields the best known stretch-space trade-off for compact routing on CUDGs. Extensive simulations on random wireless networks indicate that the average routing overhead is about 10 ; only few routes have a stretch above 1.5. We propose a scalable and reliable point-to-point routing algorithm for ad hoc wireless networks and sensor-nets. Our algorithm assigns to each node of the network a virtual coordinate in the hyperbolic plane, and performs greedy geographic routing with respect to these virtual coordinates. Unlike other proposed greedy routing algorithms based on virtual coordinates, our embedding guarantees that the greedy algorithm is always successful in finding a route to the destination, if such a route exists. We describe a distributed algorithm for computing each node's virtual coordinates in the hyperbolic plane, and for greedily routing packets to a destination point in the hyperbolic plane. (This destination may be the address of another node of the network, or it may be an address associated to a piece of content in a Distributed Hash Table. In the latter case we prove that the greedy routing strategy makes a consistent choice of the node responsible for the address, irrespective of the source address of the request.) We evaluate the resulting algorithm in terms of both path stretch and node congestion. An important technique for discovering routes between two nodes in an ad-hoc network involves applying the face routing algorithm on a planar spanner of the network. Face routing guarantees message delivery in networks that contains large holes, where greedy algorithms fail. Existing techniques for constructing a suitable planar subgraph involve local tests that eliminate crossings between existing links by deleting some links. They do not test whether the deleted links actually create some crossings and some of the links are deleted needlessly. As a result, some of the routes found in face routing will have an unnecessarily large number of hops from source to destination. We consider a new local test for preprocessing a wireless network that produces a planar subgraph. The test is relatively simple, requires low overhead and does not eliminate existing links unless it is needed to eliminate a crossing, thus reducing overhead associated with multiple hops. With the development of ad hoc networks, some researchers proposed several geometric routing protocols which depend on the planarization of the network connectivity graph to guarantee the delivery of the packet between any pair of nodes in the network. In this paper, we proposed a new online routing algorithm GLNFR (greedy and local neighbor face routing) for finding paths between the nodes of the ad hoc networks by storing a small amount of local face information at each node. The localized Delaunay triangulation was used to be the backbone of wireless network on which the GLNFR routing algorithm runs. It has the better scalability and adaptability for the change of ad hoc networks. Experiment on NS have been conducted. The results show that the delivery rate of packets and routing protocol message cost under such novel routing protocols performs better than others proposed before. We show that greedy geometric routing schemes exist for the Euclidean metric in R 2, for 3-connected planar graphs, with coordinates that can be represented succinctly, that is, with O(logn) bits, where n is the number of vertices in the graph. Greedy geographic routing is attractive for large multi-hop wireless networks because of its simple and distributed operation. However, it may easily result in dead ends or hotspots when routing in a network with obstacles (regions without sufficient connectivity to forward messages). In this paper, we propose a distributed routing algorithm that combines greedy geographic routing with two non-Euclidian distance metrics, chosen so as to provide load balanced routing around obstacles and hotspots. The first metric, Local Shortest Path, is used to achieve high probability of progress, while the second metric, Weighted Distance Gain, is used to select a desirable node among those that provide progress. The proposed Load Balanced Local Shortest Path (LBLSP) routing algorithm provides loop freedom, guarantees delivery when a path exists, is able to efficiently route around obstacles, and provides good load balancing. This paper considers the problem of designing power efficient routing with guaranteed delivery for sensor networks with known distances between neighbors but unknown geographic locations. We propose Hector, a hybrid energy efficient tree-based optimized routing protocol, based on two sets of virtual coordinates. One set is based on rooted tree coordinates, and the other is based on hop distances toward several landmarks. In our algorithm, the node currently holding the packet forwards it to its neighbor that optimizes ratio of power cost over distance progress with landmark coordinates, among nodes that reduce landmark coordinates and do not increase tree coordinates. If such a node does not exist then forwarding is made to the neighbor that reduces tree based distance and optimizes power cost over tree distance progress ratio. Our simulations show the superiority of our algorithm over existing alternatives while guaranteeing delivery, and only up to 30 additional power compared to centralized shortest weighted path algorithm. We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 Several peer-to-peer networks are based upon randomized graph topologies that permit efficient greedy routing, e. g., randomized hypercubes, randomized Chord, skip-graphs and constructions based upon small-world percolation networks. In each of these networks, a node has out-degree Θ(log n), where n denotes the total number of nodes, and greedy routing is known to take O(log n) hops on average. We establish lower-bounds for greedy routing for these networks, and analyze Neighbor-of-Neighbor (NoN)- greedy routing. The idea behind NoN, as the name suggests, is to take a neighbor's neighbors into account for making better routing decisions.The following picture emerges: Deterministic routing networks like hypercubes and Chord have diameter Θ(log n) and greedy routing is optimal. Randomized routing networks like randomized hypercubes, randomized Chord, and constructions based on small-world percolation networks, have diameter Θ(log n log log n) with high probability. The expected diameter of Skip graphs is also Θ(log n log log n). In all of these networks, greedy routing fails to find short routes, requiring Ω(log n) hops with high probability. Surprisingly, the NoN- greedy routing algorithm is able to diminish route-lengths to Θ(log n log log n) hops, which is asymptotically optimal. This paper considers max-min fair rate allocation and routing in energy harvesting networks where fairness is required among both the nodes and the time slots. Unlike most previous work on fairness, we focus on multihop topologies and consider different routing methods. We assume a predictable energy profile and focus on the design of efficient and optimal algorithms that can serve as benchmarks for distributed and approximate algorithms. We first develop an algorithm that obtains a max-min fair rate assignment for any given (time-variable or time-invariable) unsplittable routing or a routing tree. For time-invariable unsplittable routing, we also develop an algorithm that finds routes that maximize the minimum rate assigned to any node in any slot. For fractional routing, we study the joint routing and rate assignment problem. We develop an algorithm for the time-invariable case with constant rates. We show that the time-variable case is at least as hard as the 2-commodity feasible flow problem and design an FPTAS to combat the high running time. Finally, we show that finding a max-min fair unsplittable routing or a routing tree is NP-hard, even for a time horizon of a single slot. Our analysis provides insights into the problem structure and can be applied to other related fairness problems. We investigate the construction of greedy embeddings in polylogarithmic dimensional Euclidian spaces in order to achieve scalable routing through geographic routing. We propose a practical algorithm which uses random projection to achieve greedy forwarding on a space of dimension O(log(n)) where nodes have coordinates of size O(log(n)), thus achieving greedy forwarding using a route table at each node of polylogarithmic size with respect to the number of nodes. We further improve this algorithm by using a quasi-greedy algorithm which ensures greedy forwarding works along a path-wise construction, allowing us to further reduce the dimension of the embedding. The proposed algorithm, denoted GLoVE-U, is fully distributed and practical to implement. We evaluate the performance using extensive simulations and show that our greedy forwarding algorithm delivers low path stretch and scales properly. Constructing a route connecting a source node to a destination node is one of the central problems in sensor networks and mobile computing. In this paper we consider the application of a data structure called ExtDCEL that can be very effective in implementing several location-based routing algorithms that include face routing, greedy most forward routing, and hybrid routing. The ExtDCEL data structure is very convenient for representing unit disk graphs as well as planar components. Many network properties can be computed locally when the graph is represented in ExtDCEL. In particular, we show how the popular hybrid greedy face routing algorithm can be implemented efficiently by using ExtDCEL. We adapt the compact routing scheme by Thorup and Zwick to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the power-law graph model for a compact routing scheme. In particular, we prove that, for stretch 3, instead of routing tables with O(n1 2) bits as in the general scheme by Thorup and Zwick, expected sizes of O(nγ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n1 + γ log n), with γ = τ - 2 2τ - 3 + Ɛ, where τ ∈ (2, 3) is the power-law exponent and Ɛ > 0. Both bounds also hold with probability at least 1 - 1 n (independent of Ɛ). The routing scheme is a labeled scheme, requiring a stretch-5 handshaking step and using addresses and message headers with O(log n log log n) bits, with probability at least 1 - o(1). We further demonstrate the effectiveness of our scheme by simulations on real-world graphs as well as synthetic power-law graphs. With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick for stretch 3 and obtain a new upper bound of expected O(n1+γ) for space and preprocessing. Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones. We consider the following network design problem. We are given an undirected graph G=(V,E) with edges costs c(e) and a set of terminal nodes W. A hose demand matrix for W is any symmetric matrix [Dij] such that for each i, ∑ j ≠ i Dij ≤ 1. We must compute the minimum cost edge capacities that are able to support the oblivious routing of every hose matrix in the network. An oblivious routing template, in this context, is a simple path Pij for each pair i,j ∈ W. Given such a template, if we are to route a demand matrix D, then for each i,j we send Dij units of flow along each Pij. and obtained a 2-approximation for this problem, using a solution template in the form of a tree. It has been widely asked and subsequently conjectured [Italiano 2006] that this solution actually results in the optimal capacity for the single path VPN design problem; this has become known as the VPN conjecture. The conjecture has previously been proven for some restricted classes of graphs [Hurkens 2005, Grandoni 2007, Fiorini 2007]. Our main theorem establishes that this conjecture is true in general graphs. This also gives the first polynomial time algorithm for the single path VPN problem. We also show that the multipath version of the conjecture is false. Compact routing addresses the tradeoff between table sizes and stretch, which is the worst-case ratio between the length of the path a packet is routed through by the scheme and the length of an actual shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwick [2001] to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by [2000]. Our result is the first analytical bound coupled to the parameter of the power-law graph model for a compact routing scheme. Let n denote the number of nodes in the network. We provide a labeled routing scheme that, after a stretch--5 handshaking step (similar to DNS lookup in TCP IP), routes messages along stretch--3 paths. We prove that, instead of routing tables with O(n1 2) bits (O suppresses factors logarithmic in n) as in the general scheme by Thorup and Zwick, expected sizes of O(nγ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n1+γ log n), with γ = τ-22 τ-3 + e, where τ∈(2,3) is the power-law exponent and e 0 (which implies e With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick [2001, 2005] for stretch-3 and we obtain a new upper bound of expected O(n1+γ) for space and preprocessing for random power-law graphs. Our distance oracle is the first one optimized for power-law graphs. Furthermore, we provide a linear-space data structure that can answer 5--approximate distance queries in time at most O(n1 4+e) (similar to γ, the exponent actually depends on τ and lies between e and 1 4 + e).