Partial delaunay triangulation-based asynchronous planarization of quasi unit disk graphs

In providing new telecommunication services, the requisition for quality of network is more and more popular and sophisticated with high bandwidth, small value of delay time or packet loss etc. To assure the quality of network, the localized routing schemes have recently been researched as a promising alternative to the currently deployed global QoS routing schemes. Different from the traditional QoS routing algorithms which use global state information, the localized routing algorithms use local information collected from source node to make routing decisions. These localized routing algorithms can be solutions to meet the demand of telecom market in the near future. In this paper, we study and propose one model of localized QoS routing algorithm, which uses distributed routing type to route flows. We perform it with experiments, compare and realize the more considerable performance of this algorithm than other algorithms, which use the same conditions of simulation. From that, we imply that localized QoS routing algorithm using distributed routing works more effectively than other types of routing algorithm.

In a localized routing algorithm, node A currently holding the message forwards it based on the location of itself, its neighboring nodes and destination. We propose to use depth first search (DFS) method for routing decisions. Each node A, upon receiving the message for the first time, sorts all its neighbors according to a criteria such as their distance to destination and uses that order in DFS algorithm. It is the first localized algorithm that guarantees delivery for (connected) wireless networks modeled by arbitrary graphs, including inaccurate location information. We then propose the first localized QoS routing algorithm for wireless networks. It performs DFS routing algorithm after edges with insufficient bandwidth or insufficient connection time are deleted from the graph, and attempts to minimize hop count. This is also the first paper to apply GPS in QoS routing decisions, and to consider the connection time (estimated lifetime of a link) as a QoS criterion. The average length of measured QoS path in our experiments, obtained by DFS method, was between 1 and 1.34 times longer than the length of QoS path obtained by shortest path algorithm. The overhead is considerably reduced by applying the concept of internal nodes.

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose length is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per vertex coordinate via Schnyder's embedding scheme [W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st Annual ACM--SIAM Symposium on Discrete Algorithms (SODA 1990), ACM, New York, SIAM, Philadelphia, 1990, pp. 138--148], our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of $15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the half-$\theta_6$-graph.

The scheme of Quality of Service (QoS) routing algorithms based on local state information has recently been proposed as an alternative approach to the traditional QoS routing algorithms. By implementing this localized QoS routing algorithm, each source node predetermines and maintains a set of candidate paths for each destination. These sets of paths will help the source node to decide the most appropriate path for a connection request. Hence, it helps to avoid the problems associated with the maintenance of the global network state information. In this paper, we propose a new and effective localized QoS routing algorithm, compare its performance with those of other localized algorithms and a traditional QoS routing algorithm under the same type of network topology, QoS requirements and traffic patterns. The simulations results show that our proposed algorithm can perform better than other routing algorithms. DOI: 10.32913/rd-ict.vol3.no14.258

If a set K of nodes in a wireless network want to set up a routing structure that allows them to communicate with each other, one possible approach is to use a Steiner tree that spans all the nodes in K. However, a tree can be disconnected by the failure of a single link, and so it is desirable to employ other routing structures that are fault-tolerant. Furthermore, many real-world wireless networks are heterogeneous, meaning that the suitability of nodes for inclusion in the routing structure varies significantly. Therefore, it is meaningful to assign weights to the nodes and aim to compute a fault-tolerant routing structure of minimum total weight. In this paper, we model this problem as the problem of computing a minimum-weight 2-edge-connected Steiner subgraph spanning a given set of terminals, and we propose a constant-factor approximation algorithm for this problem in wireless networks that are modelled as unit disk graphs or quasi unit disk graphs.

In this paper, we study an algorithmic model for wireless ad hoc and sensor networks that aims to be sufficiently close to reality as to represent practical real-world networks while at the same time being concise enough to promote strong theoretical results. The quasi unit disk graph model contains all edges shorter than a parameter d between 0 and 1 and no edges longer than 1. We show that--in comparison to the cost known for unit disk graphs--the complexity results of geographic routing in this model contain the additional factor 1/d2. We prove that in quasi unit disk graphs flooding is an asymptotically message-optimal routing technique, we provide a geographic routing algorithm being most efficient in dense networks, and we show that classic geographic routing is possible with the same asymptotic performance guarantees as for unit disk graphs if d ≥ 1/√2.

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. When the input graph is planar, we present a simple and elegant streaming algorithm that, with high probability, estimates the size of a maximum matching within a constant factor using Õ( n 2/3 ) space, where n is the number of vertices. The approach generalizes to the family of graphs that have bounded arboricity, which include graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model) in o ( n ) space. We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We further reduce the required memory size to Õ(√ n ) for three restricted settings: (i) when the input graph is a forest; (ii) when we have 2-passes and the input graph has bounded arboricity; and (iii) when the edges arrive in random order and the input graph has bounded arboricity. Finally, we design a reduction from the Boolean Hidden Matching Problem to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o ( n 1/2 ) bits of space. Using the same reduction, we show that there is no deterministic algorithm that computes this kind of estimate in o ( n ) bits of space. The lower bounds hold even for graphs that are collections of paths of constant length.

We study the following all-or-nothing multicommodity flow problem in planar graphs. Input: A graph G with n vertices and k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Find: A largest subset W of {1, ...., k such that for every i in W, we can send one unit of flow between si and ti. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri et al. (STOC'04) give a poly-logarithmic factor approximation algorithm and show that a natural LP-relaxation has a poly-logarithmic integrality gap. This result is in contrast with the integrality gap Ω(√n) for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an O(1)-approximation algorithm and show that the integrality gap is O(1). In particular, in polynomial time, we can find an index set W with |W| = Ω(OPT) and eight si-ti paths for each i in W such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the recent result by S'eguin-Charbonneau and Shepherd (FOCS'11) who give an O(1)-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

Little is known about geographic properties of large-scale social networks. In this paper, we examine the geospatial attributes of a planetary-scale social network of 240 million people and 1.3 billion edges. We study the interplay among topological, geographical, and algorithmically generated paths connecting pairs of nodes in a social network. Starting in the realm of cyberspace, we find that topologically shortest paths of average length of 6.6 exist between pairs of nodes in the network and that the average degree of separation among nodes is robust to removal of hub nodes. Moving to the realm of locations and distances in geographic space, we find that topologically shortest paths in the social graph grow with increasing geographic distance between path's endpoint nodes. We discover that shortest topological paths are geographically inefficient, but that geography provides an important cue for local algorithmic policies for navigating between source and target nodes. Local algorithmic strategies for navigating the larger network structure in the absence of global navigation procedures have varying success. At the early stages of the navigation, navigating to a hub node helps, while in the middle stage, geography provides the most important clue. While local algorithms for navigating have trouble reaching the target node, they are successful in reaching nodes that are geographically close to the target. Taken together, our results demonstrate a complex interplay between topological and geographical properties of social networks and explain the success of local strategies for navigating such networks.

This chapter proposed a collaborative peer network application called 6Search (6S) to address the scalability limitations of centralized search engines. Each peer crawls the Web in a focused way, guided by its user’s information context. Through this approach, better (distributed) coverage can be achieved. Each peer also acts as a search “servent” (server + client) by submitting and responding to queries to/from its neighbors. This search process has no centralized bottleneck. Peers depend on a local adaptive routing algorithm to dynamically change the topology of the peer network and search for the best neighbors to answer their queries. We present and evaluate learning techniques to improve local query routing. We validate prototypes of the 6S network via simulations with model users based on actual Web crawls. We find that the network topology rapidly converges from a random network to a small world network, with clusters emerging from user communities with shared interests. We finally compare the quality of the results with those obtained by centralized search engines such as Google.

A vertex-subset graph problem \(\mathcal {Q}\) defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions \(S_s\) and \(S_t\) of size k, whether it is possible to transform \(S_s\) into \(S_t\) by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by \(\textsc {ISR}\) and the latter by \(\textsc {DSR}\). Both \(\textsc {ISR}\) and \(\textsc {DSR}\) are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that \(\textsc {ISR}\) is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that \(\textsc {ISR}\) is fixed-parameter tractable on planar graphs and graphs of bounded degree. For \(\textsc {DSR}\), we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes degenerate and nowhere dense graphs.

A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store. To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge $(u,v) \in E$ belongs to the output spanner. Such LCAs give the user the `illusion' that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present the following results: -For general $n$-vertex graphs and $r \in \{2,3\}$, there exists an LCA for $(2r-1)$-spanners with $\widetilde{O}(n^{1+1/r})$ edges and sublinear probe complexity of $\widetilde{O}(n^{1-1/2r})$. These size/stretch tradeoffs are best possible (up to polylogarithmic factors). -For every $k \geq 1$ and $n$-vertex graph with maximum degree $Δ$, there exists an LCA for $O(k^2)$ spanners with $\widetilde{O}(n^{1+1/k})$ edges, probe complexity of $\widetilde{O}(Δ^4 n^{2/3})$, and random seed of size $\mathrm{polylog}(n)$. This improves upon, and extends the work of [Lenzen-Levi, 2018]. We also complement our results by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with $o(m)$ edges.

In this paper, we investigate how to design energy-efficient localized routing in a large-scale three-dimensional (3D) wireless network. Several 3D localized routing protocols were proposed to seek either energy efficiency or delivery guarantee in 3D wireless networks. However, recent results [1, 2] showed that there is no deterministic localized routing algorithm that guarantees either delivery of packets or energy efficiency of its routes in 3D networks. In this paper, we focus on design of a simple localized routing method which can provide energy efficiency with high probability in a randomly deployed 3D network. In particular, we extend our previous routing method designed for 2D networks [3] to 3D networks. The proposed 3D routing method is a simple variation of 3D greedy routing and can guarantee energy efficiency of its paths with high probability in random 3D networks. We also study its asymptotic critical transmission radius to ensure the packet delivery with high probability in random 3D networks. Simulation results confirm our theoretical results.KeywordsDelivery RatioForwarding NodeRestricted RegionTransmission RadiusStretch FactorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The network time protocol (NTP), which is designed to distribute time information in a large, diverse system, is described. It uses a symmetric architecture in which a distributed subnet of time servers operating in a self-organizing, hierarchical configuration synchronizes local clocks within the subnet and to national time standards via wire, radio, or calibrated atomic clock. The servers can also redistribute time information within a network via local routing algorithms and time daemons. The NTP synchronization system, which has been in regular operation in the Internet for the last several years, is described, along with performance data which show that timekeeping accuracy throughout most portions of the Internet can be ordinarily maintained to within a few milliseconds, even in cases of failure or disruption of clocks, time servers, or networks. >

Energy preservation for effective WSN applications is one of the major challenges, as the small sensor nodes have very limited power, storage, communication and computing resources. The sensors remain unattended, and after deployment, rechargeable batteries are almost difficult. The algorithm proposed to pick a nearest hop node based on the token-ring algorithm, compared with the other current S-MAC, T-MAC and HEED algorithms, which pick the next hop node based on parameters such as shortest-path, maximum residual power or probability. More specifically, it is easier to reduce the total power consumption on a multihop route when driving by carefully choosing suitable intermediate nodes. The combination of cluster-based routing and containerized WSN routing algorithms is proposed to be used as an Advanced Energy Efficient Sensor-based MAC (AES-MAC) protocol. The routing process involves the maintenance of routing and routing. The illustrations show a simple, distributed, localized routing algorithm, which can be easily implemented for various practical applications.