Greedy Embedding Research Articles

Look-Ahead VNF-FG Embedding Framework for Latency-Sensitive Network Services

Dynamic and zero-touch management is expected to be the key feature of next-generation 6G networks. Network Function Virtualization (NFV) is one of the key technologies for realizing such management through software-based networks. Despite great benefits offered by NFV, deploying network services (NSs) in NFV ecosystems remains a challenge, especially for latency-sensitive NSs, as they demand stringent latency requirements and fast service provisioning. Specifically, service graphs should be embedded into an infrastructure such that these requirements are satisfied while optimizing network operator’s objectives. To cope with the scalability of optimization-based approaches, heuristic methods are known as promising alternatives to find a satisfactory solution within an acceptable execution time. However, existing VNF embedding heuristics still suffer from the so-called causality issue, which may degrade the embedding solution quality. The causality issue means that embedding decisions cannot be optimally determined before all neighboring dependencies are known. To this end, we introduce our h-horizon sequential look-ahead greedy embedding framework, which provides efficient embedding and re-embedding strategies to alleviate the impact of the causality issue. The simulation results indicate that our proposed algorithm significantly improves embedding cost, compared to the existing heuristic algorithms while being much more scalable than an optimization-based approach.

Resilient greedy routing on GPS-free surface sensor networks

We design a greedy routing scheme specifically for GPS-free large-scale wireless sensor networks deployed on surfaces of complex-connected three-dimensional settings. Compared with other greedy embedding–based surface network routing scheme, the proposed one is cut free such that no pair of nodes suffers a long detour to reach each other. The routing scheme is designed to be resilient to node or link failures especially under random node or link failure model where each node in a network has an equal and independent probability of failure during some time interval. The proposed algorithm is fully distributed and scalable to both the size and the topological complexity of a network. Each sensor node requires only limited and constant storage. Simulation results show the proposed routing scheme with a higher successful delivery ratio, a lower average stretch factor, and a lower normalized communication cost compared with other resilient routing methods.

Secure Data Sharing Framework via Hierarchical Greedy Embedding in Darknets

Geometric routing, which combines greedy embedding and greedy forwarding, is a promising approach for efficient data sharing in darknets. However, the security of data sharing using geometric routing in darknets is still an issue that has not been fully studied. In this paper, we propose a Secure Data Sharing framework (SeDS) for future darknets via hierarchical greedy embedding. SeDS adopts a hierarchical topology and uses a set of secure nodes to protect the whole topology. To support geometric routing in the hierarchical topology, a two-level bit-string prefix embedding approach (Prefix-T) is first proposed, and then a greedy forwarding strategy and a data mapping approach are combined with Prefix-T for data sharing. SeDS guarantees that the publication or request of a data item can always pass through the corresponding secure node, such that security strategies can be performed. The experimental results show that SeDS provides scalable and efficient end-to-end communication and data sharing.

Every Schnyder drawing is a greedy embedding

In this paper, we show that every Schnyder drawing is a greedy embedding. Schnyder drawings are used to represent planar (maximal) graphs. It is a way of getting coordinates in R2 given a graph G=(V,E) such that the representation is planar. The Schnyder technique leads to a family of representations and previous results show that a particular representation may be chosen such that the drawing has additional properties like being greedy or monotone. In this article, we relax the definition of greediness to a definition that does not rely on the geometry and the Euclidean distance in R2, but rather on the combinatorial graph G. The construction of greedy paths valid for all Schnyder representations shows that, provided the relaxed definition, every Schnyder drawing is a greedy embedding.

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

AbstractWe show that for every there exists C > 0 such that if then asymptotically almost surely the random graph contains the kth power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi‐polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi‐polynomial algorithm for finding a tight Hamilton cycle in the random k‐uniform hypergraph for . The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of p. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.

On Planar Greedy Drawings of 3-Connected Planar Graphs

A graph drawing is greedy if, for every ordered pair of vertices (x, y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination “greedily” forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra. In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra’s result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture.

Collaborative Dynamic Virtual Network Embedding Algorithm Based on Resource Importance Measures

Many existing virtual network embedding (VNE) algorithms adopt the greedy embedding strategies, which embed the in-progress virtual network requests (VNRs) onto the substrate nodes and links with more residual resources. However, the previous VNRs may overconsume the critical substrate resources and lead to the resource fragmentation problem in the substrate networks, thus reducing the acceptance ratio of the latter VNRs. To address this problem, we propose a novel collaborative VNE algorithm based on resource importance measures, RIM-ViNE, in which the global node importance and link importance in the initial substrate network are measured using multiple topological attributes and are used to set the embedding cost of VNRs. Then, the VNE problem is defined as a linear programming model and is solved by minimizing the total embedding cost, which could bring about the coordinative embedding between different VNRs, and thus preventing the critical resources from being over occupied by the previous VNRs with small requirements and improving the resource fragmentation problem. Moreover, we propose a dynamic reconfiguration mechanism based on critical nodes protection (CNP-Re) to improve the resource fragmentation problem further. Extensive experiments are conducted under two network scenarios, and the results show that the proposed algorithms outperform the VNE algorithms that only consider the coordination between node mapping and link mapping or only measure the node importance in the residual substrate network, and the average VNR acceptance ratio, average revenue, and average resource utilization ratio are effectively improved.

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 \({\mathbb {R}}^2\) is known for certain graph classes such as 3-connected planar graphs. We completely characterize the trees that admit a greedy embedding in \({\mathbb {R}}^2\). This answers a question by Angelini et al. (Networks 59(3):267–274, 2012) and is a further step in characterizing the graphs that admit Euclidean greedy embeddings.

Hyperbolic Embedding for Efficient Computation of Path Centralities and Adaptive Routing in Large-Scale Complex Commodity Networks

Computing the most central nodes in large-scale commodity networks is rather important for improving routing and associated applications. In this paper, we introduce a novel framework for the analysis and efficient computation of routing path-based centrality measures, focusing on betweenness and traffic load centrality. The proposed framework enables efficient approximation and in special cases accurate computation of the aforementioned measures in large-scale complex networks, as well as improving/adapting commodity (traffic) routing by identifying and alleviating key congestion points. It capitalizes on network embedding in hyperbolic space and exploits properties of greedy routing over hyperbolic coordinates. We show the computational benefits and approximation precision of our approach by comparing it with state-of-the-art path centrality computation techniques. We demonstrate its applicability on real topologies, characteristic of actual large-scale commodity networks, e.g., data, utility networks. Focusing on two graph embedding types, Rigel and greedy, we compare their impact on the performance of our framework. Then, we exemplify and statistically analyze the dynamic routing adaptation, via the variation of the minimum-depth spanning tree employed for greedy embedding in hyperbolic space. Notably, this allows for efficient routing adaptation according to a simple, distributed computation that can be applied during network operation to alleviate arising bottlenecks.

SLICE: Enabling Greedy Routing in High Genus 3-D WSNs With General Topologies

In this paper, we propose a highly efficient scheme, SLICE (a scalable and low stretch routing scheme), enabling greedy routing for wireless sensor networks (WSNs) deployed in complex-connected 3-D settings, whose topologies are often theoretically modeled as high genus 3-D WSNs. Compared to previous 3-D greedy embedding techniques, SLICE improves both the robustness and applicability. 1) It achieves a smaller distance distortion and a lower routing stretch with guaranteed delivery. While it follows the basic idea to embed the surface network to a planar topology to enable greedy routing, the embedding method proposed in SLICE is novel. We first slice the surface network to a genus-0 open surface with exactly one boundary. Then, to achieve a lower distance distortion, we purposely propose a variation of the Ricci flow algorithm, by which this open surface is flattened not to a planar annulus, but to a planar convex polygon, resulting in a lower routing stretch. 2) This is the first work, to the best of our knowledge, that enables greedy routing in high genus 3-D WSNs with general topologies. SLICE not only works for high genus 3-D surface WSNs, but also can be easily adapted to more general cases: high genus 3-D surface networks with holes, and high genus 3-D volume networks. For a high genus 3-D surface network with holes, SLICE embeds it to a planar convex polygon with circular holes, where our proposed greedy routing variation can be applied. For a high genus 3-D volume network, SLICE embeds the inner nodes to a height structure attached to the convex polygon, and a variation of greedy routing scheme with guaranteed delivery is proposed in this structure. The effectiveness of SLICE is validated by extensive simulations.

Dynamic greedy routing in overlay networks using virtual coordinates from the hyperbolic plane

AbstractGreedy routing algorithms based on virtual coordinates have attracted considerable interest in recent years. Those based on coordinates taken from the hyperbolic plane have interesting theoretical scalability properties. However, their scalability and reliability are yet to be ensured when applied to large‐scale dynamic networks. In this paper, we propose a scalable and reliable solution for creating and managing dynamic overlay networks where nodes have hyperbolic coordinates. In this context, our solution provides a greedy routing algorithm based on the hyperbolic distance. To cope with network dynamics, we have defined two methods for avoiding temporary local minima and one method for maintaining the greedy embedding over time. Through analysis, we evaluate the complexity costs of our solution. Through simulations, we assess the scalability of our solution on static networks and its reliability on dynamic networks. Results show that using our solution based on hyperbolic geometry provides scalability and reliability to both addressing and routing tasks in dynamic overlay networks. Copyright © 2015 John Wiley & Sons, Ltd.

Geometric Name Routing for ICN in dynamic world

A new paradigm of scalable routing for ICN is to combine a geometric routing scheme with a Distributed Hash Table. However, for most routing schemes in this paradigm, when a node joins or leaves, large numbers of nodes, even the whole topology, need to be re-embedded, and a great number of contents need to be re-registered. In this paper, we propose D-Griffin, a geometric routing scheme on flat names for dynamic topologies. D-Griffin provides two advantages. First, it avoids re-embedding the topology by using an online greedy embedding scheme and a void handling greedy forwarding scheme. Second, it decreases the number of re-registrations by using a name mapping scheme with a tradeoff between topology independence and load balancing. Theoretical and experimental analyses show that D-Griffin provides guaranteed content lookup, low description complexity, low path stretch, scalable routing update, and acceptable load balancing.

Experimental validation of resilient tree-based greedy geometric routing

Geometric routing is an alternative for IP routing based on longest prefix matching. Using this routing paradigm, every node in the network is assigned a coordinate and packets are forwarded towards their intended destination following a distance-decreasing policy (greedy forwarding). This approach makes the routers significantly more memory-efficient compared to the current IP routers. In this routing, greedy embeddings are used to guarantee a 100% successful delivery to every destination in the network. Most of the existing proposals lack resiliency mechanisms to react efficiently to network changes. We propose a distributed algorithm to calculate a greedy embedding based on a spanning tree of the network. In this algorithm, nodes are triggered to re-calculate their coordinates upon a change in the topology such as link or node failures. The advantage of this approach is that it recovers from topology failures within a very short period of time. We further extend the algorithm to generate backups to apply protection in distributed setups. Different trade-offs and trends of re-convergence for geometric routing have been evaluated in an emulation environment. Realistic results are achieved through emulation as no model or abstraction is involved. The proposed routing scheme is implemented in Quagga routing software and new elements are developed in Click modular router to enable greedy forwarding. For the first time, the performance of this scheme is evaluated through emulation on a large topology of 1000 nodes and the results are compared with BGP. The experimental results indicate that the proposed scheme has interesting characteristics in terms of convergence time upon a change in the network topology.

Space lower bounds for low-stretch greedy embeddings

Greedy (geometric) routing is an important paradigm for routing in communication networks. It uses an embedding of the nodes of a network into points of a space (e.g., Rd) equipped with a distance function (e.g., the Euclidean distance ℓ2) and uses as address of each node the coordinates of the corresponding point. The embedding has particular properties so that the routing decision for a packet is taken greedily based only on the addresses of the current node, its neighbors, and the destination node and the distances of the corresponding points. In this way, there is no need to keep extensive routing information (e.g., routing tables) at the nodes. Embeddings that allow for this functionality are called greedy embeddings. Even though greedy embeddings do exist for several spaces and distance functions, they usually result in paths of high stretch, i.e., significantly longer than the corresponding shortest paths.In this paper, we show that greedy embeddings in low-dimensional (Euclidean) spaces necessarily have high stretch. In particular, greedy embeddings of n-node graphs with optimal stretch require at least Ω(n) dimensions for distance ℓ2. This result disproves a conjecture by Maymounkov (2006) [14] stating that greedy embeddings of optimal stretch are possible in Euclidean spaces with polylogarithmic number of dimensions. Furthermore, we present trade-offs between the stretch and the number of dimensions of the host space. Our results imply that every greedy embedding into a space with polylogarithmic number of dimensions (and Euclidean distance) has stretch Ω(log⁡nlog⁡log⁡n). We extend this result for a distance function used by an O(log⁡n)-stretch greedy embedding presented by Flury et al. (2009) [9]. Our lower bound implies that their embedding has almost best possible stretch.

GPS-Free Greedy Routing With Delivery Guarantee and Low Stretch Factor on 2-D and 3-D Surfaces

This paper focuses on greedy routing in wireless networks deployed on 2-D and 3-D surfaces. It introduces a distributed embedding scheme based on the conformal map theory. The proposed scheme identifies the convex hull of each boundary and employs Yamabe flow to compute flat metric under convex hull boundary condition to establish virtual coordinates. Such virtual coordinates are then used for greedy routing. Since the proposed embedding algorithm maps the outer boundary to a convex shape and an interior concave void to a circle-like convex polygon, it effectively eliminates local minimum and attains guaranteed delivery. At the same time, it introduces a small distortion only and consequently achieves a low stretch factor. Our simulations show that its stretch factor is lower than any existing greedy embedding algorithms. Moreover, the proposed scheme is merely based on local connectivity and consumes a small constant storage, thus scaling to arbitrarily large networks.

A distributed routing for wireless sensor networks with mobile sink based on the greedy embedding

Mobile sinks can be a solution to solve the problem that energy consumption of sensor nodes is not balanced in Wireless Sensor Networks (WSN). Caused by the sink mobility, the paths between the sensor nodes and the sink change with time. It is necessary to find a protocol that can find efficient routings between the mobile sink and sensor nodes but do not consume too many network resources. In this paper, we propose an algorithm called virtual-node greedy embedding (VGE) to embed the topology of a WSN into hyperbolic plane without the physical geographic information. This algorithm gives each node a virtual coordinate that guarantees the greedy forwarding of the packets from sensor nodes to the sink when there is no failed node in WSN, even if some obstacles exist in the network. VGE keeps the greedy property when new nodes join the network after the initial embedding and does not change the node coordinates assigned previously. Based on the characteristic of embedding process, modified gravity-pressures (MGP) algorithm is proposed to find a routing path quickly when there are some failed nodes in WSN. Furthermore, particular solutions of the parameters in VGE are presented with concise style, and the simulations show that our algorithm has an efficient performance.

On the choice of a spanning tree for greedy embedding of network graphs

Greedy embedding is a graph embedding that makes the simple greedy geometric packet forwarding successful for every source-destination pair. It is desirable that graph embeddings also yield low hop overhead (stretch) of the greedy paths over the corresponding shortest paths. In this paper we study how topological and geometric properties of embedded graphs influence the hop stretch. Based on the obtained insights, we design embedding heuristics that yield minimal hop stretch greedy embeddings and verify their effectiveness on models of synthetic graphs. Finally, we verify the effectiveness of our heuristics on instances of several classes of large, real-world network graphs.

An optimal greedy routing algorithm for triangulated polygons

In this paper, we prove that a triangulated polygon G admits a greedy embedding into an appropriate semi-metric space such that using an appropriate distance definition, for any two vertices u and w in G, a most virtual distance decreasing path is always a minimum-edge path between u and w. Therefore, our greedy routing algorithm is optimal. The greedy embedding of G can be obtained in linear time. To the best of our knowledge, this is the first optimal greedy routing algorithm for a nontrivial subcategory of graphs.

Greedy routing via embedding graphs onto semi-metric spaces

In this paper, we generalize the greedy routing concept to use semi-metric spaces. We prove that any connected n-vertex tree T admits a greedy embedding, onto an appropriate semi-metric space such that (1) each vertex v of the tree is represented by k=deg(v) virtual coordinates (where the numbers are from 1 to 2n−2); and (2) using an appropriate distance definition, the unique simple path between any two vertices in T is always a distance decreasing path between the two vertices. Applying the result to an arbitrary connected graph G, such a greedy embedding of any of its spanning tree T onto a semi-metric space becomes a greedy embedding of G onto the same semi-metric space. In particular, for 3-connected plane graphs, we prove that, for a 3-connected plane graph G, there is a greedy embedding of G such that: (1) the greedy embedding can be obtained in linear time; (2) each vertex can be represented by at most 3 virtual coordinates from 1 to 2n−2; and (3) the distance between two vertices can be computed in constant time. To our best knowledge, this is the first greedy routing algorithm for 3-connected plane graphs, albeit with non-standard notions of greedy embedding and greedy routing, such that: (1) it runs in linear time to compute the virtual coordinates for the vertices; and (2) the virtual coordinates are represented succinctly in O(logn) bits.

Weak Greedy Routing over Graph Embedding for Wireless Sensor Networks

In this paper we classify the greedy routing in sensor networks into two categories, strong greedy routing and weak greedy routing. Most existing work mainly focuses on weak greedy routing over geographic location network or strong greedy routing over greedy embedding network. It is a difficult job and needs much cost to obtain geographic location or greedy embedding of the network. We propose a light-weight Tree-based graph embedding (TGE) for sensor networks. Over the TGE, we design a weak greedy routing protocol, TGR. TGR can archive good performance on path stretch factor and load balance factor.