Disk Graph Research Articles

Approximation scheme for burst scheduling with minimum overhead in time slicing mobile TV

In this paper, we consider the problem of minimizing switching overhead of burst scheduling in time slicing mobile TV broadcast systems. This problem was formulated by Hsu and Hefeeda, and was given an elegant approximation scheme with an approximation ratio of at least two. Our proposed scheme significantly improves the approximation ratio of the previous scheme. More concretely, it generates a burst schedule for mobile TV systems whose switching overhead is at most $$(1/\ln 2)+\epsilon \simeq 1.4427$$ ( 1 / ln 2 ) + ∈ ? 1.4427 times of optimum, where $$\ln $$ ln is the natural logarithm and $$\epsilon $$ ∈ is arbitrary positive constant. Our scheme is a combination of a binary partion of burst cycles and a shifting method which is commonly used for the analysis of approximation algorithms designed for unit disk graphs.

Geometry of the augmented disk graph

AbstractFor a handlebodyH, we define two graphs, the augmented disk graph$\mathcal{ADG}(H)$and the truncated augmented disk graph$\mathcal{TADG}(H)$, and we show they are hyperbolic in the sense of Gromov. In the process, we show they are quasi-isometric to two other disk graphs defined by U. Hamenstädt, the super conducting disk graph$\mathcal{SDG}(H)$and the electrified disk graph$\mathcal{EDG}(H)$respectively. So we reprove two theorems of Hamenstädt [12].Our approach uses techniques from Masur–Schleimer's study on the hyperbolicity of the disk graph$\mathcal{DG}(H)$[21].

Broadcast Scheduling with Latency and Redundancy Analysis for Cognitive Radio Networks

Cognitive radio networks (CRNs) introduce a new communication paradigm that enables unlicensed users to opportunistically access spectrum bands assigned to licensed users. Interestingly, the broadcast problem, which is one of the most fundamental operations in CRNs, has not been well studied. Existing works for the broadcast issue in CRNs are heuristic solutions either without a performance guarantee or with performance far from the optimal solution. In this paper, we study the minimum-latency broadcast scheduling (MLBS) issue for CRNs. Our contributions are threefold. First, we propose a mixed broadcast scheduling (MBS) algorithm under the unit disk graph (UDG) model, denoted by MBS-UDG. MBS-UDG finishes a broadcast task by employing mixed unicast and broadcast communication modes in two phases. We show that the latency performance of MBS-UDG is $O(\hbar+\Delta_{T})$ when $\Delta_{T}\leq \hbox{1}/p$ or $O(\hbar+\log_{1-p}(\hbox{1}/p\Delta_{T}))$ when $\Delta_{T}>\hbox{1}/p$ , where $\hbar$ and $\Delta_{T}$ are the height and the maximum number of leaf nodes connected by a second user (SU)of the broadcasting tree, respectively, and $p$ is the spectrum opportunity for a secondary communication. Furthermore, the redundancy performance of MBS-UDG is analyzed. Second, we extend MBS-UDG to a more general MBS algorithm under the protocol interference model and analyze its latency and redundancy performance. Finally, simulations are conducted to validate MBS, which indicate that MBS significantly improves existing algorithms with respect to both latency and redundancy.

Colouring stability two unit disk graphs

We prove that every stability two unit disk graph has chromatic number at most 3/2 times its clique number.

A new bound on maximum independent set and minimum connected dominating set in unit disk graphs

It is a conjecture that in any unit disk graph $$G$$G, $$\alpha (G) \le 3 \cdot \gamma _c(G) + 3$$?(G)≤3·?c(G)+3 where $$\alpha (G)$$?(G) is the size of the maximum independent set in $$G$$G and $$\gamma _c(G)$$?c(G) is the size of minimum connected dominating set in $$G$$G. In this paper, we show that in any unit disk graph $$G$$G, $$\alpha (G) \le 3.399 \cdot \gamma _c(G) + 4.874$$?(G)≤3.399·?c(G)+4.874. Currently, this is the best-known bound.

Optimizing Communication Overhead while Reducing Path Length in Beaconless Georouting with Guaranteed Delivery for Wireless Sensor Networks

Beaconless or contention-based geographic routing algorithms forward packets toward a geographical destination reactively without the knowledge of the neighborhood. Such algorithms allow greedy forwarding, where only the next hop neighbor responds after a timer-based contention using only three messages (RTS, CTS, and DATA) per forwarding step. However, existing contention-based schemes for recovery from greedy failures do not have this property. In this paper, we show that recovery is possible within this 3-message scheme: the Rotational Sweep (RS) algorithm directly identifies the next hop after timer-based contention and constructs a traversal path that ensures progress after a greedy failure. It uses a traversal scheme (called Sweep Circle) that forwards a message along the $(\alpha)$-shape of the network and provides recovery paths shorter or equal to the prominent face routing with Gabriel graph planarization. An alternative traversal scheme (called Twisting Triangle) provides even shorter routes on average, as shown by simulations. They both also reduce per-node overall network contention delays. We prove that both traversal schemes guarantee delivery in unit disk graphs. Our traversal schemes avoid planarization and are easy to implement, based merely on the evaluation of a function of a neighbor node's relative position. They can also be used for boundary detection and improve path-length in conventional beacon-based routing.

The Euclidean Bottleneck Steiner Path Problem and Other Applications of (α,β)-Pair Decomposition

We consider a geometric optimization problem that arises in network design. Given a set P of n points in the plane, source and destination points s, tźP, and an integer k>0, one has to locate k Steiner points, such that the length of the longest edge of a bottleneck path between s and t is minimized. In this paper, we present an O(nlog2n)-time algorithm that computes an optimal solution, for any constant k. This problem was previously studied by Hou et al. (in Wireless Networks 16, 1033---1043, 2010), who gave an O(n2logn)-time algorithm. We also study the dual version of the problem, where a value ź>0 is given (instead of k), and the goal is to locate as few Steiner points as possible, so that the length of the longest edge of a bottleneck path between s and t is at most ź. Our algorithms are based on two new geometric structures that we develop--an (ź,β)-pair decomposition of P and a floor (1+ź)-spanner of P. For real numbers β>ź>0, an (ź,β)-pair decomposition of P is a collection $\mathcal{W}=\{(A_{1},B_{1}),\ldots,(A_{m},B_{m})\}$ of pairs of subsets of P, satisfying the following: (i) For each pair $(A_{i},B_{i}) \in\mathcal {W}$, both minimum enclosing circles of Ai and Bi have a radius at most ź, and (ii) for any p, qźP, such that |pq|≤β, there exists a single pair $(A_{i},B_{i}) \in\mathcal{W}$, such that pźAi and qźBi, or vice versa. We construct (a compact representation of) an (ź,β)-pair decomposition of P in time O((β/ź)3nlogn). In some applications, a simpler (though weaker) grid-based version of an (ź,β)-pair decomposition of P is sufficient. We call this version a weak (ź,β)-pair decomposition of P. For ź>0, a floor (1+ź)-spanner of P is a (1+ź)-spanner of the complete graph over P with weight function w(p,q)=ź|pq|ź. We construct such a spanner with O(n/ź2) edges in time O((1/ź2)nlog2n), even though w is not a metric. Finally, we present two additional applications of an (ź,β)-pair decomposition of P. In the first, we construct a strong spanner of the unit disk graph of P, with the additional property that the spanning paths also approximate the number of substantial hops, i.e., hops of length greater than a given threshold. In the second application, we present an O((1/ź2)nlogn)-time algorithm for computing a one-sided approximation for distance selection (i.e., given k, $1 \le k \le{n \choose2}$, find the k'th smallest Euclidean distance induced by P), significantly improving the running time of the algorithm of Bespamyatnikh and Segal.

Approximating 2-cliques in unit disk graphs

This paper studies distance-based clique relaxations in unit disk graphs arising in wireless networking applications. Namely, a 2-clique is a subset of nodes with pairwise distance at most two in the graph, and a 2-club is a subset of nodes inducing a subgraph of diameter two. It is shown that in a unit disk graph any 2-clique is 4-dominated and any 2-club is 3-dominated. The former observation is used to develop a 12-approximation algorithm for the maximum 2-clique problem in unit disk graphs. Moreover, this also implies polynomial solvability of the minimum dominating set problem in unit disk graphs of diameter two, whereas the same problem is shown to be hard in general diameter-two graphs. The paper also poses several related open questions of interest.

On Construction of Quality Fault-Tolerant Virtual Backbone in Wireless Networks

In this paper, we study the problem of computing quality fault-tolerant virtual backbone in homogeneous wireless network, which is defined as the k-connected m-dominating set problem in a unit disk graph. This problem is NP-hard, and thus many efforts have been made to find a constant factor approximation algorithm for it, but never succeeded so far with arbitrary k ≥ 3 and m ≥ 1 pair. We propose a new strategy for computing a smaller-size 3-connected m-dominating set in a unit disk graph with any m ≥ 1. We show the approximation ratio of our algorithm is constant and its running time is polynomial. We also conduct a simulation to examine the average performance of our algorithm. Our result implies that while there exists a constant factor approximation algorithm for the k-connected m-dominating set problem with arbitrary k ≤ 3 and m ≥ 1 pair, the k-connected m-dominating set problem is still open with k > 3.

SPANNING PROPERTIES OF GRAPHS INDUCED BY DIRECTIONAL ANTENNAS

Let S be a set of points in the plane, such that the unit disk graph with vertex set S is connected. We address the problem of finding orientations and a minimum radius for directional antennas of a fixed cone angle placed at the points of S, such that the induced communication graph G[S] is a hop t-spanner of the unit disk graph for S (meaning that G[S] is strongly connected, and contains a directed path with at most t edges between any pair of points within unit distance). We consider problem instances in which antenna angles are bounded below by 120° and 90°. We show that, in the case of 120° angles, a radius of 5 suffices to establish a hop 4-spanner; and in the case of 90° angles, a radius of 7 suffices to establish a hop 5-spanner.

Efficient and practical resource block allocation for LTE-based D2D network via graph coloring

In this paper, we construct a practical framework for efficiently allocating long term evolution (LTE) resource blocks (RB) among the users in a device-to-device (D2D) network. For such network that presumably operates under the LTE cellular network, our aim is to improve the overall throughput of D2D connections using opportunistic or fairness-based approach. Taking the practical considerations into account, our proposed framework allows a number of connections to share a single RB whenever possible, thus utilizing the radio resources. To do so, our solution first identifies a superior set of the interference-free D2D reuse groups via graph modeling and graph coloring approach. In particular, we model a D2D network with a two-overlapping disk graph for which a suitable coloring algorithm is proposed and its performance bound is calculated. Once the reuse groups are known, our solution optimizes the RB allocation among these groups based on their reported channel condition as well as the scheduling criterion, whether it is fairness-based or opportunistic. Through numerical experiments, we show that our solution can significantly improve the throughput performance of a D2D network.

The number of disk graphs

A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of unit radius disks in the plane. We give upper and lower bounds on the number of labeled unit disk and disk graphs on n vertices. We show that the number of unit disk graphs on n vertices is n2n⋅α(n)n and the number of disk graphs on n vertices is n3n⋅β(n)n, where α(n) and β(n) are Θ(1). We conjecture that there exist constants α,β such that the number of unit disk graphs is n2n⋅(α+o(1))n and the number of disk graphs is n3n⋅(β+o(1))n.

WRSR: wormhole-resistant secure routing for wireless mesh networks

Wormhole attack is one of the most severe security threats in wireless mesh network that can disrupt majority of routing communications, when strategically placed. At the same time, most of the existing wormhole defence mechanisms are not secure against wormhole attacks that are launched in participation mode. In this paper, we propose WRSR, a wormhole-resistant secure routing algorithm that detects the presence of wormhole during route discovery process and quarantines it. Unlike other existing schemes that initiate wormhole detection process after observing packet loss, WRSR identifies route requests traversing a wormhole and prevents such routes from being established. WRSR uses unit disk graph model to determine the necessary and sufficient condition for identifying a wormhole-free path. The most attractive features of the WRSR include its ability to defend against all forms of wormhole (hidden and Byzantine) attacks without relying on any extra hardware like global positioning system, synchronized clocks or timing information, and computational intensive traditional cryptographic mechanisms.

Hypocomb: Bounded-Degree Localized Geometric Planar Graphs for Wireless Ad Hoc Networks

We propose a radically new family of geometric graphs, i.e., Hypocomb (HC), Reduced Hypocomb (RHC), and Local Hypocomb (LHC). HC and RHC are extracted from a complete graph; LHC is extracted from a Unit Disk Graph (UDG). We analytically study their properties including connectivity, planarity, and degree bound. All these graphs are connected (provided that the original graph is connected) planar. Hypocomb has unbounded degree while Reduced Hypocomb and Local Hypocomb have maximum degree 6 and 8, respectively. To our knowledge, Local Hypocomb is the first strictly localized, degree-bounded planar graph computed using merely 1-hop neighbor position information. We present a construction algorithm for these graphs and analyze its time complexity. Hypocomb family graphs are promising for wireless ad hoc networking. We report our numerical results on their average degree and their impact on FACE routing. We discuss their potential applications and pinpoint some interesting open problems for future research.

Bridging the Gap between Protocol and Physical Models for Wireless Networks

This paper tries to reconcile the tension between the physical model and the protocol model that have been used to characterize interference relationship in a multihop wireless network. The physical model (a.k.a. signal-to-interference-and-noise ratio model) is widely considered as a reference model for physical layer behavior but its application in multihop wireless networks is limited by its complexity. On the other hand, the protocol model (a.k.a. disk graph model) is simple but there have been doubts on its validity. This paper explores the following fundamental question: How to correctly use the protocol interference model? We show that, in general, solutions obtained under the protocol model may be infeasible and, thus, results based on blind use of protocol model can be misleading. We propose a new concept called "reality check” and present a method of using a protocol model with reality check for wireless networks. Subsequently, we show that by appropriate setting of the interference range in the protocol model, it is possible to narrow the solution gap between the two models. Our simulation results confirm that this gap is indeed small (or even negligible). Thus, our methodology of joint reality check and interference range setting retains the protocol model as a viable approach to analyze multihop wireless networks.

Symmetric connectivity with directional antennas

Let P be a set of points in the plane, representing transceivers equipped with a directional antenna of angle α and range r. The coverage area of the antenna at point p is a circular sector of angle α and radius r, whose orientation can be adjusted. For a given assignment of orientations, the induced symmetric communication graph (SCG) of P is the undirected graph, in which two vertices (i.e., points) u and v are connected by an edge if and only if v lies in uʼs sector and vice versa. In this paper we ask what is the smallest angle α for which there exists an integer n=n(α), such that for any set P of n antennas of angle α and unbounded range, one can orient the antennas so that (i) the induced SCG is connected, and (ii) the union of the corresponding wedges is the entire plane. We show (by construction) that the answer to this question is α=π/2, for which n=4. Moreover, we prove that if Q1 and Q2 are two quadruplets of antennas of angle π/2 and unbounded range, separated by a line, to which one applies the above construction, independently, then the induced SCG of Q1∪Q2 is connected. This latter result enables us to apply the construction locally, and to solve the following two further problems.In the first problem (replacing omni-directional antennas with directional antennas), we are given a connected unit disk graph, corresponding to a set P of omni-directional antennas of range 1, and the goal is to replace the omni-directional antennas by directional antennas of angle π/2 and range r=O(1) and to orient them, such that the induced SCG is connected, and, moreover, is an O(1)-spanner of the unit disk graph, w.r.t. hop distance. In our solution r=142 and the spanning ratio is 8. In the second problem (orientation and power assignment), we are given a set P of directional antennas of angle π/2 and adjustable range. The goal is to assign to each antenna p, an orientation and a range rp, such that the resulting SCG is (i) connected, and (ii) ∑p∈Prpβ is minimized, where β⩾1 is a constant. For this problem, we devise an O(1)-approximation algorithm.

On the Discovery of Critical Links and Nodes for Assessing Network Vulnerability

The assessment of network vulnerability is of great importance in the presence of unexpected disruptive events or adversarial attacks targeting on critical network links and nodes. In this paper, we study Critical Link Disruptor (CLD) and Critical Node Disruptor (CND) optimization problems to identify critical links and nodes in a network whose removals maximally destroy the network's functions. We provide a comprehensive complexity analysis of CLD and CND on general graphs and show that they still remain NP-complete even on unit disk graphs and power-law graphs. Furthermore, the CND problem is shown NP-hard to be approximated within Ω([( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> )/( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ε</sup> )] ) on general graphs with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> vertices and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> critical nodes. Despite the intractability of these problems, we propose HILPR, a novel LP-based rounding algorithm, for efficiently solving CLD and CND problems in a timely manner. The effectiveness of our solutions is validated on various synthetic and real-world networks.

Bounded-degree minimum-radius spanning trees in wireless sensor networks

In this paper, we study the problem of computing a spanning tree of a given undirected disk graph such that the radius of the tree is minimized subject to a given degree constraint Δ⁎. We first introduce an (8,4)-bicriteria approximation algorithm for unit disk graphs (which is a special case of disk graphs) that computes a spanning tree such that the degree of any nodes in the tree is at most Δ⁎+8 and its radius is at most 4⋅OPT, where OPT is the minimum possible radius of any spanning tree with degree bound Δ⁎. We also introduce an (α,2)-bicriteria approximation algorithm for disk graphs that computes a spanning tree whose maximum node degree is at most Δ⁎+α and whose radius is bounded by 2⋅OPT, where α is a non-constant value that depends on M and k with M being the number of distinct disk radii and k being the ratio of the largest and the smallest disk radius.

CDS-Based Virtual Backbone Construction with Guaranteed Routing Cost in Wireless Sensor Networks

Inspired by the backbone concept in wired networks, virtual backbone is expected to bring substantial benefits to routing in wireless sensor networks (WSNs). Virtual backbone construction based on Connected Dominating Set (CDS) is a competitive approach among the existing methods used to establish virtual backbone in WSNs. Traditionally, CDS size was the only factor considered in the CDS-based approach. The motivation was that smaller CDS leads to simplified network maintenance. However, routing cost in terms of routing path length is also an important factor for virtual backbone construction. In our research, both of these two factors are taken into account. Specifically, we attempt to devise a polynomial-time constant-approximation algorithm that leads to a CDS with bounded CDS size and guaranteed routing cost. We prove that, under general graph model, there is no polynomial-time constant-approximation algorithm unless P = NP. Under Unit Disk Graph (UDG) model, we propose an innovative polynomial-time constant-approximation algorithm, GOC-MCDS-C, that produces a CDS D whose size I D is within a constant factor from that of the minimum CDS. In addition, for each node pair u and v, there exists a routing path with all intermediate nodes in D and path length at most 7 · d(u, v), where d(u, v) is the length of the shortest path between u and v. Our theoretical analysis and simulation results show that the distributed version of the proposed algorithm, GOC-MCDS-D, outperforms the existing approaches.

Distributed minimum dominating set approximations in restricted families of graphs

A dominating set is a subset of the nodes of a graph such that all nodes are in the set or adjacent to a node in the set. A minimum dominating set approximation is a dominating set that is not much larger than a dominating set with the fewest possible number of nodes. This article summarizes the state-of-the-art with respect to finding minimum dominating set approximations in distributed systems, where each node locally executes a protocol on its own, communicating with its neighbors in order to achieve a solution with good global properties. Moreover, we present a number of recent results for specific families of graphs in detail. A unit disk graph is given by an embedding of the nodes in the Euclidean plane, where two nodes are joined by an edge exactly if they are in distance at most one. For this family of graphs, we prove an asymptotically tight lower bound on the trade-off between time complexity and approximation ratio of deterministic algorithms. Next, we consider graphs of small arboricity, whose edge sets can be decomposed into a small number of forests. We give two algorithms, a randomized one excelling in its approximation ratio and a uniform deterministic one which is faster and simpler. Finally, we show that in planar graphs, which can be drawn in the Euclidean plane without intersecting edges, a constant approximation factor can be ensured within a constant number of communication rounds.