Ring Tree Research Articles

The Sandpile Group of Polygon Rings and Twisted Polygon Rings

Let $$C_{k_1}, \ldots , C_{k_n}$$ be cycles with $$k_i\ge 2$$ vertices ( $$1\le i\le n$$ ). By attaching these n cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these n cycles together in a cyclic order, we obtain a graph, which is called a polygon ring if it can be embedded on the plane; and called a twisted polygon ring if it can be embedded on the Möbius band. It is known that the sandpile group of a polygon chain is always cyclic. Furthermore, there exist edge generators. In this paper, we not only show that the sandpile group of any (twisted) polygon ring can be generated by at most three edges, but also give an explicit relation matrix among these edges. So we obtain a uniform method to compute the sandpile group of arbitrary (twisted) polygon rings, as well as the number of spanning trees of (twisted) polygon rings. As an application, we compute the sandpile groups of several infinite families of polygon rings, including some that have been done before by ad hoc methods, such as, generalized wheel graphs, ladders and Möbius ladders.

A multi-objective antlion optimizer for the ring tree problem with secondary sub-depots

This article proposes a multi-objective ring tree problem with secondary sub-depots (MORTPSSD), which focusses on the problems of telecommunication and logistics networks. In this problem, we have considered a fixed node as the main depot. Other nodes are divided into primary sub-depots, secondary sub-depots, and left-out nodes referred to type 1, type 2, and type 3 customers. The first objective of the proposed model MORTPSSD is to minimize the circuits’ total routing cost through type 1 and type 2 customers added by the minimal spanning tree cost of type 3 customers. The second objective is to minimize the total number of type 3 customers, which influences the first objective. The model is solved by a discrete multi-objective antlion optimizer (DMOALO) with a ternary encoding. The proposed algorithm is also tested on some instances derived from TSP benchmark problems. Statistical analyses are performed to compare the convergence and the diversity of the proposed DMOALO against NSGAII and MOPSO, which yields a better efficiency of DMOALO for most instances.

Reliable Formation Protocol for Bluetooth Hybrid Single-hop and Multi-hop Networks

There are presently many applications where a non-uniform distribution of devices needs to be established for Bluetooth scatternets. Under the scenario of one dense zone and multiple sparse zones, the dense area has a high probability of generating a single-hop scenario since most devices are within radio range, whereas most devices are out of radio range under the multi-hop scenario in other sparse areas. Thus, both situations have to be considered in the formation of an algorithm design for most real-life situations. This work proposes a reliable formation protocol, called Dual-Ring Tree, for hybrid single-hop/ multi-hop instances. To benefit from the advantages of the hybrid scenarios, a dual-ring subnet is presented as a single-hop solution for dense areas, while a tree-shaped subnet is designed as a multi-hop solution for sparse areas. To the best of the authors' knowledge, this is the first time such an algorithm has been designed to deal with both single-hop and multi-hop scenarios. The computer simulation results suggest that the reliable Dual- Ring Tree outperforms conventional BlueHRT in terms of routing efficiency and network reliability for Bluetooth multi-hop networks.

Exact algorithms for bi-objective ring tree problems with reliability measures

We introduce bi-objective models for ring tree network design with a focus on network reliability within telecommunication applications. Our approaches generalize the capacitated ring tree problem (CRTP) which asks for a partially reliable topology that connects customers with different security requirements to a depot node by combined ring and tree graphs. While the CRTP aims at optimizing the edge installation costs, we propose four alternative, reliability-oriented objective functions. We study the case of service interruptions due to single-edge failures, and consider the overall number of tree customers and tree edges, the maximal number of subtree customers, and the maximal number of tree hops from rings as additional measures. To model the corresponding novel bi-objective problems, we develop mathematical multi-commodity flow formulations and identify relationships between the new objectives. For identifying the Pareto fronts, we apply an ϵ-constraint method based on integer programming. The computational efficiency is increased by employing local search heuristics in order to tighten upper bounds and by valid inequalities to strengthen lower bounds in the subproblems. In a computational study we report results, illustrate solution network topologies and extensively analyze the algorithm performance for instances from the literature.

A hybrid WDM ring–tree topology delivering efficient utilization of bandwidth over resilient infrastructure

This paper proposed a hybrid WDM ring---tree topology which employs spectrally efficient 60 Gbps non-return-to-zero/polarization shift keying optical orthogonal modulated signals. The reconfigurable optical add/drop multiplexers, optical cross-connects and semiconductor optical amplifiers are utilized to make hybrid ring---tree architecture. Each optical add/drop multiplexer node is connected to tree topology to further increase the number of users. The main contribution of the proposed spectrally efficient hybrid optical network is to support the maximum number of users within limited bandwidth for future communication networks. Also, the proposed resilient ring---tree architecture has advantages such as high optical node density and bandwidth/spectral efficiency as compared to conventional packet-switched optical access network.

RTCP: reliable topology construction protocol of bluetooth hybrid single‐hop and multi‐hop networks

In this work, a reliable topology construction protocol (RTCP) called Dual-Ring Tree scatternet with a self-routing algorithm is proposed for hybrid single-hop/multi-hop situations. To benefit from the advantages of the hybrid scenarios, a dual-ring subnet is presented as a single-hop solution for one dense area, while a tree-shaped subnet is designed as a multi-hop solution for other sparse areas. To the best of the author's knowledge, this is the first time that such an algorithm has been presented to deal with the hybrid scenarios. Since the dual-ring structure serves as the core network design, the resulting architecture demonstrates better fault tolerance capability than the conventional single ring networks do. In addition, a self-routing protocol is introduced to efficiently deliver packets over the scatternet. Simulation results suggest that the RTCP scheme outperforms conventional Bluetooth hybrid ring tree (BlueHRT) and Bluetree methods in terms of routing performance and network survivability for Bluetooth ad hoc networks.

Generalized local branching heuristics and the capacitated ring tree problem

In this paper we present a heuristic framework that is based on mathematical programming to solve network design problems. Our techniques combine local branching with locally exact refinements. In an iterative strategy an existing solution is refined by sequentially solving restricted mixed integer programs (MIPs) to optimality. These are obtained from the master problem MIP by limiting the number of variable flips for structured subsets of the binary edge variables which are selected based on the underlying network cost structure. We introduce generalized local branching cuts which enforce the latter using two parameters at the same time: the number of considered variables and the number of allowed variable flips.Using this concept we develop an efficient algorithm for the capacitated ring tree problem (CRTP), a recent network design model for partially reliable capacitated networks that combines cycle and tree structures. Our implementation operates on top of an efficient branch and cut algorithm for the CRTP. The sets of refinement variables are deduced from single-ball network node clusters. We provide computational results and an extensive analysis of the algorithm for a set of literature instances. We show that the approach is capable of improving existing best results for the CRTP and outperforms the pure refinement or local branching approaches.

A hierarchical P2P clustering framework for video streaming systems

In this study, a novel overlay architecture for constructing hierarchical and scalable clustering of Peer-to-Peer (P2P) networks is proposed. The proposed architecture attempts to enhance the clustering of peers by incorporating join, split, merge and cluster leader election mechanisms in a fully distributed manner. It takes delay proximity of peers into account as distance measure. By constructing hierarchical clustering of peers, the control message overhead and maintenance such as host departure/host join overhead are decreased. Theoretical comparisons on overheads of the proposed system with that of other systems from literature are studied. The control mechanism for dynamic peer behavior of the architecture is tested over PlanetLab. The performance metrics used are end-to-end delay, diameter, cluster head distance, occupancy rate, peer join latency, accuracy and correctness. The test results are compared with Hierarchical Ring Tree (HRT) and mOverlay architecture. In addition, a P2P video streaming application is run over the proposed network overlay. Streaming tests show that video streaming applications perform well in terms of received video quality if hierarchical clusters considering delay proximity are used as underlying network architecture.

A Reconfigurable Formation and Disjoint Hierarchical Routing for Rechargeable Bluetooth Networks

In this paper, a reconfigurable mesh-tree with a disjoint hierarchical routing protocol for the Bluetooth sensor network is proposed. First, a designated root constructs a tree-shaped subnet and propagates parameters k and c in its downstream direction to determine new roots. Each new root asks its upstream master to start a return connection to convert the first tree-shaped subnet into a mesh-shaped subnet. At the same time, each new root repeats the same procedure as the designated root to build its own tree-shaped subnet, until the whole scatternet is formed. As a result, the reconfigurable mesh-tree constructs a mesh-shaped topology in one densely covered area that is extended by tree-shaped topology to other sparsely covered areas. To locate the optimum k layer for various sizes of networks, a peak-search method is introduced in the designated root to determine the optimum mesh-tree configuration. In addition, the reconfigurable mesh-tree can dynamically compute the optimum layer k when the size of the network changes in the topology maintenance phase. In order to deliver packets over the mesh-tree networks, a disjoint hierarchical routing protocol is designed during the scatternet formation phase to efficiently forward packets in-between the mesh-subnet and the tree-subnet. To achieve the energy balance design, two equal disjoint paths are generated, allowing each node to alleviate network congestion, since most traffic occurs at the mesh-subnet. Simulation results show that the joint reconfigurable method and routing algorithm generate an efficient scatternet configuration by achieving better scatternet and routing performance than BlueHRT (bluetooth hybrid ring tree). Furthermore, the disjoint routing with rechargeable battery strategy effectively improves network lifetime and demonstrates better energy efficiency than conventional routing methods.

Independent Spanning Trees of 2-Chordal Rings

Independent Spanning Trees of 2-Chordal Rings

The Ring Tree Facility Location Problem

In this work we discuss a facility location variant of the capacitated ring tree problem. The new model generalizes vehicle routing problems and Steiner tree problems, yielding applicability in telecommunication and transportation networks. In this ring tree facility location problem (RTFLP) two layers of networks have to be designed to connect two different types of customers to a central depot. The first, or inner layer, consists of cycles that intersect in the depot and collect all type 2 customers, and some of the type 1 customers. An outer layer is represented by a forest that contains the remaining type 1 customers such that each tree shares exactly one vertex with the inner layer. Capacity bounds apply to the number of connected substructures emanating from the depot, the number of customers in each of these so-called ring trees, and in each tree of the forest. Additional optional Steiner vertices can be used to reduce the overall costs, which are layer-dependent edge costs and facility location costs at the vertices in which the two layers coincide. Our contribution is the introduction of the RTFLP, the development of two mathematical formulations, and preliminary computational results for the first RTFLP test set derived from instances from the literature.

Optimal capacitated ring trees

We study a new network design model combining ring and tree structures under capacity constraints. The solution topology of this capacitated ring tree problem (CRTP) is based on ring trees which are the union of trees and 1-trees. The objective is the minimization of edge costs but could also incorporate other types of measures. This overall problem generalizes prominent capacitated vehicle routing and Steiner tree problem variants. Two customer types have to be connected to a distributor ensuring single and double node connectivity, respectively, while installing optional Steiner nodes. The number of ring trees and the number of customers supplied by such a single structure are bounded. After embedding this combinatorial optimization model in existing network design concepts, we develop a mathematical formulation and introduce several valid inequalities for the CRTP that are separated in our exact algorithm. For a set of literature-derived instances we consider various reliability scenarios and present computational results.

On the Hyperbolicity of Small-World and Treelike Random Graphs

Hyperbolicity is a property of a graph that may be viewed as a “soft” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here we consider Gromov's notion of δ-hyperbolicity and establish several positive and negative results for small-world and treelike random graph models. First, we study the hyperbolicity of the class of Kleinberg small-world random graphs <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_828336_o_ilm0001.gif"></inline-graphic>, where _n_ is the number of vertices in the graph, _d_ is the dimension of the underlying base grid _B_, and γ is the small-world parameter such that each node _u_ in the graph connects to another node _v_ in the graph with probability proportional to 1/_d_B_(_u_, _v_)^γ, with _d_B_(_u_, _v_) the grid distance from _u_ to _v_ in the base grid _B_. We show that when γ=_d_, the parameter value allowing efficient decentralized routing in Kleinberg's small-world network,the hyperbolic δ is <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_828336_o_ilm0002.gif"></inline-graphic> with probability 1−_o_(1) for every ϵ>0 independent of _n_. We see that hyperbolicity is not significantly improved in relation to graph diameter even when the long-range connections greatly improve decentralized navigation. We also show that for other values of γ, the hyperbolic δ is very close to the graph diameter, indicating poor hyperbolicity in these graphs as well. Next we study a class of treelike graphs called ringed trees that have constant hyperbolicity. We show that adding random links among the leaves in a manner similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides one of the first significant analytic results on the hyperbolicity of a rich class of random graphs, which sheds light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.

Scalable group key exchange protocol with provable security

PurposeThe purpose of this paper is to introduce a scalable group key exchange (GKE) protocol with provable security, which is used to establish a confidential channel.Design/methodology/approachThe protocol in this paper uses line structure, rather than formal ring structure or tree structure, to reduce the amount of calculation and communication.FindingsIn the protocol, every participant just need send two messages. The security can be proved under the Decisional Diffie‐Hellman (DDH) problem.Originality/valueUsing the protocol provides an efficient and secure method to allow group users to share a common key.

An Optimal Algorithm for Conflict-Free Coloring for Tree of Rings

An optimal algorithm is presented about Conflict-Free Coloring for connected subgraphs of tree of rings. Suppose the number of the rings in the tree is |T| and the maximum length of rings is |R|. A presented algorithm in [1] for a Tree of rings used O(log|T|.log|R|) colors but this algorithm uses O(log|T|+log|R|) colors. The coloring earned by this algorithm has the unique-min property, that is, the unique color is also minimum.

A scatternet formation algorithm for Bluetooth networks with a non-uniform distribution of devices

We address the problem of scatternet formation for network scenarios with a non-uniform distribution of Bluetooth devices. The assumption of having a non-uniform distribution of devices in a given area of interest is motivated by examples that can be encountered in real scenarios. We propose a new scatternet formation protocol called BlueHRT (Bluetooth Hybrid Ring Tree) that results in a hybrid ring tree topology. The proposed protocol operates in multiple phases that include node discovery, identification of a dense area within the network, role assignment with ring based piconets in the dense area and tree based piconets in the surrounding lightly loaded areas, and interconnection of the piconets to each other via slave–slave and master–slave bridges. General protocol design analysis and ns-2 simulation results are presented in order to highlight the main performance characteristics.

Enhancing P2P overlay network architecture for live multimedia streaming

The number of live multimedia streaming applications is increasing, explaining the use of many overlay network topologies. Application-layer multicast (ALM) that it is a feasible alternative to multimedia stream has attracted considerable attention. However, a serious problem of ALM is that the multicast tree may be fragile, and peer failure causes tree partitions. This work presents a novel Hierarchical Ring Tree (HRT) architecture for Peer-to-Peer (P2P) live multimedia streaming. The proposed architecture combines ring-based and tree-based structures in a robust, scalable, reliable and resilient structure that can be used practically as an ALM topology. When peers enter or leave the system, the topology can be recovered rapidly such that live multimedia stream can be delivered smoothly with a low latency. The proposed HRT topology is maintained efficiently without splitting or merging trees. The performance of the proposed architecture and algorithms is evaluated experimentally. Experimental results indicate that the proposed topology can be used in a high-churn P2P network with a small delay. Simulation and experiment results reveal that the proposed architecture has a lower overhead than the ZIGZAG approach when handling peers’ joining or leaving, exhibits faster recovery, better quality-of-service during streaming, and a more robust topology, even with an extremely high number of peers joining/leaving.

Efficient algorithms for wavelength assignment on trees of rings

A fundamental problem in communication networks is wavelength assignment (WA): given a set of routing paths on a network, assign a wavelength to each path such that the paths with the same wavelength are edge-disjoint, using the minimum number of wavelengths. The WA problem is NP-hard for a tree of rings network which is well used in practice. In this paper, we give an efficient algorithm which solves the WA problem on a tree of rings with an arbitrary (node) degree using at most 3 L wavelengths and achieves an approximation ratio of 2.75 asymptotically, where L is the maximum number of paths on any link in the network. The 3 L upper bound is tight since there are instances of the WA problem that require 3 L wavelengths even on a tree of rings with degree four. We also give a 3 L and 2-approximation (resp. 2.5-approximation) algorithm for the WA problem on a tree of rings with degree at most six (resp. eight). Previous results include: 4 L (resp. 3 L ) wavelengths for trees of rings with arbitrary degrees (resp. degree at most eight), and 2-approximation (resp. 2.5-approximation) algorithm for trees of rings with degree four (resp. six).

Virtual Private Network Design: A Proof of the Tree Routing Conjecture on Ring Networks

A basic question in virtual private network (VPN) design is if the symmetric version of the problem always has an optimal solution which is a tree network. An affirmative answer would imply that the symmetric VPN problem is solvable in polynomial time. We give an affirmative answer in case the communication network, within which the VPN must be created, is a circuit. This seems to be an important step towards answering the general question. The proof relies on a dual pair of linear programs and actually implies an even stronger property of VPNs. We show that this property also holds for some other special cases of the problem, in particular when the network is a tree of rings.

Inhibiting peach-tree growth with abscisic acid, hinokitiol and tropolone applied to partially ringed bark strips

SummaryThe dwarfing effects of growth inhibitors [abscisic acid (ABA), hinokitiol and tropolone] applied to a connecting bark strip on a partially ringed trunk were evaluated using 2-year-old peach trees. A 2 cm strip of bark was removed from the circumference of the trunk, leaving a 2 mm-wide connecting bark band to which aqueous solutions of the chemicals were applied. Positive correlations were found between bark regeneration and tree growth. High concentrations of tropolone and hinokitiol retarded both bark regeneration and shoot growth. Tropolone at 500 µg ml–1 and 1,000 µg ml–1 completely inhibited bark regeneration, even resulting in tree-death similar to complete ringing. Regenerated bark at 1,000 µg ml–1 ABA was half that in the partially ringed water control, and regeneration was completely inhibited at 2,000 µg ml–1 ABA. Root weight was also reduced in accordance with the decline in bark regeneration. These results indicate that tree growth can be controlled by chemically modifying bark regeneration, or by translocation to the shoots of partially ringed trees. The cost of chemicals will be less than that of whole-tree sprays for growth inhibition.