Distributed Dominating Sets on Grids

This paper presents a distributed algorithm for finding near optimal dominating sets on grids. The basis for this algorithm is an existing centralized algorithm that constructs dominating sets on grids. The size of the dominating set provided by this centralized algorithm is upper-bounded by $\lceil\frac{(m+2)(n+2)}{5}\rceil$ for $m\times n$ grids and its difference from the optimal domination number of the grid is upper-bounded by five. Both the centralized and distributed algorithms are generalized for the $k$-distance dominating set problem, where all grid vertices are within distance $k$ of the vertices in the dominating set.Comment: 10 pages, 9 figures, accepted in ACC 201

A LOAD-BASED APPROACH TO FORMING A CONNECTED DOMINATING SET FOR AN AD HOC NETWORK

Efficient routing in mobile ad hoc networks (MANETs) is highly desired and connected dominating sets (CDS) have been gaining significant popularity in this regard. The CDS based approach reduces the search for a minimum cost path between a pair of source and destination terminals to the set of terminals forming the backbone network. Researchers over the years have developed numerous distributed and localized algorithms for constructing CDSs which minimize the number of terminals forming the backbone or which provide multiple node-disjoint paths between each pair of terminals. However none of this research focuses on minimizing the load at the bottleneck terminal of the backbone network constructed by the CDS algorithms. A terminal becomes a bottleneck if the offered traffic load is greater than its effective transmission rate. In this thesis we analyze the load-based performance of a popular CDS algorithm which has been employed in MANET routing and a k-connected k-dominating set (k-CDS) algorithm and compare it with our new centralized algorithm which has been designed to minimize the load at the bottleneck terminal of the backbone network. We verify the effectiveness of our algorithm by simulating over a large number of random test networks

On Two Combinatorial Optimization Problems in Graphs: Grid Domination and Robustness

In this thesis, we study two problems in combinatorial optimization, the dominating set problem and the robustness problem. In the first half of the thesis, we focus on the dominating set problem in grid graphs and present a distributed algorithm for finding near optimal dominating sets on grids. The dominating set problem is a well-studied mathematical problem in which the goal is to find a minimum size subset of vertices of a graph such that all vertices that are not in that set have a neighbor inside that set. We first provide a simpler proof for an existing centralized algorithm that constructs dominating sets on grids so that the size of the provided dominating set is upper-bounded by the ceiling of (m+2)(n+2)/5 for m by n grids and its difference from the optimal domination number of the grid is upper-bounded by five. We then design a distributed grid domination algorithm to locate mobile agents on a grid such that they constitute a dominating set for it. The basis for this algorithm is the centralized grid domination algorithm. We also generalize the centralized and distributed algorithms for the k-distance dominating set problem, where all grid vertices are within distance k of the vertices in the dominating set. In the second half of the thesis, we study the computational complexity of checking a graph property known as robustness. This property plays a key role in diffusion of information in networks. A graph G=(V,E) is r-robust if for all pairs of nonempty and disjoint subsets of its vertices A,B, at least one of the subsets has a vertex that has at least r neighbors outside its containing set. In the robustness problem, the goal is to find the largest value of r such that a graph G is r-robust. We show that this problem is coNP-complete. En route to showing this, we define some new problems, including the decision version of the robustness problem and its relaxed version in which B=V \ A. We show these two problems are coNP-hard by showing that their complement problems are NP-hard

Lower bounds for local approximation

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4 − 1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α < 1 and any r, k, and g.Peer reviewe

Architecture and sparse placement of limited-wavelength converters for optical networks

Equipping all nodes of a large optical network with full conversion capability is prohibitively costly. To improve performance at reduced cost, sparse converter placement algorithms are used to select a subset of nodes for full-conversion deployment. Further cost reduction can be obtained by deploying only limited conversion capability in the selected nodes. We present a limited wavelength converter placement algorithm based on the k-minimum dominating set (k-MDS) concept. We propose three different cost-effective optical switch designs using the technologically feasible nontunable optical multiplexers. These three switch designs are flexible node sharing, strict node sharing, and static mapping. Compared to the full search heuristic of O(N-3) complexity based on ranking nodes by blocking percentages, our algorithm not only has a better time complexity O(RN2), where R is the number of disjoint sets provided by k-MIDS, but also avoids the local minimum problem. The performance benefit of our algorithm is demonstrated by network simulation with the U.S Long Haul topology having 28 nodes (91 is 5) and the National Science Foundation (NSF) network having 16 nodes (91 is 4). Our simulation considers the case when the traffic is not uniformly distributed between node pairs in the network using a weighted placement approach, referred to as k-WMDS. From the optical network management point of view, our results also show that the limited conversion capability can achieve performance very close to that of the full conversion capability, while not only decreasing the optical switch cost but also enhancing its fault tolerance

Message and time efficient multi-broadcast schemes

We consider message and time efficient broadcasting and multi-broadcasting in wireless ad-hoc networks, where a subset of nodes, each with a unique rumor, wish to broadcast their rumors to all destinations while minimizing the total number of transmissions and total time until all rumors arrive to their destination. Under centralized settings, we introduce a novel approximation algorithm that provides almost optimal results with respect to the number of transmissions and total time, separately. Later on, we show how to efficiently implement this algorithm under distributed settings, where the nodes have only local information about their surroundings. In addition, we show multiple approximation techniques based on the network collision detection capabilities and explain how to calibrate the algorithms' parameters to produce optimal results for time and messages.Comment: In Proceedings FOMC 2013, arXiv:1310.459

Local Approximation Schemes for Ad Hoc and Sensor Networks

We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs