Deploying robots with two sensors in K1,6K_{1,6}-free graphs

Let $G$ be a graph of minimum degree at least two with no induced subgraph isomorphic to $K_{1,6}$. We prove that if $G$ is not isomorphic to one of eight exceptional graphs, then it is possible to assign two-element subsets of $\{1,2,3,4,5\}$ to the vertices of $G$ in such a way that for every $i\in\{1,2,3,4,5\}$ and every vertex $v\in V(G)$ the label $i$ is assigned to $v$ or one of its neighbors. It follows that $G$ has fractional domatic number at least $5/2$. This is motivated by a problem in robotics and generalizes a result of Fujita, Yamashita and Kameda who proved that the same conclusion holds for all $3$-regular graphs

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))

Efficient embedding of virtual hypercubes in irregular WDM optical networks

This thesis addresses one of the important issues in designing future WDM optical networks. Such networks are expected to employ an all-optical control plane for dissemination of network state information. It has recently been suggested that an efficient control plane will require non-blocking communication infrastructure and routing techniques. However, the irregular nature of most WDM networks does not lend itself to efficient non-blocking communications. It has been recently shown that hypercubes offer some very efficient non-blocking solutions for, all-to-all broadcast operations, which would be very attractive for control plane implementation. Such results can be utilized by embedding virtual structures in the physical network and doing the routing using properties of a virtual architecture. We will emphasize the hypercube due to its proven usefulness. In this thesis we propose three efficient heuristic methods for embedding a virtual hypercube in an irregular host network such that each node in the host network is either a hypercube node or a neighbor of a hypercube node. The latter will be called a “satellite” or “secondary” node. These schemes follow a step-by-step procedure for the embedding and for finding the physical path implementation of the virtual links while attempting to optimize certain metrics such as the number of wavelengths on each link and the average length of virtual link mappings. We have designed software that takes the adjacency list of an irregular topology as input and provides the adjacency list of a hypercube embedded in the original network. We executed this software on a number of irregular networks with different connectivities and compared the behavior of each of the three algorithms. The algorithms are compared with respect to their performance in trying to optimize several metrics. We also compare our algorithms to an already existing algorithm in the literature

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

Design, Analysis and Computation in Wireless and Optical Networks

abstract: In the realm of network science, many topics can be abstracted as graph problems, such as routing, connectivity enhancement, resource/frequency allocation and so on. Though most of them are NP-hard to solve, heuristics as well as approximation algorithms are proposed to achieve reasonably good results. Accordingly, this dissertation studies graph related problems encountered in real applications. Two problems studied in this dissertation are derived from wireless network, two more problems studied are under scenarios of FIWI and optical network, one more problem is in Radio- Frequency Identification (RFID) domain and the last problem is inspired by satellite deployment. The objective of most of relay nodes placement problems, is to place the fewest number of relay nodes in the deployment area so that the network, formed by the sensors and the relay nodes, is connected. Under the fixed budget scenario, the expense involved in procuring the minimum number of relay nodes to make the network connected, may exceed the budget. In this dissertation, we study a family of problems whose goal is to design a network with “maximal connectedness” or “minimal disconnectedness”, subject to a fixed budget constraint. Apart from “connectivity”, we also study relay node problem in which degree constraint is considered. The balance of reducing the degree of the network while maximizing communication forms the basis of our d-degree minimum arrangement(d-MA) problem. In this dissertation, we look at several approaches to solving the generalized d-MA problem where we embed a graph onto a subgraph of a given degree. In recent years, considerable research has been conducted on optical and FIWI networks. Utilizing a recently proposed concept “candidate trees” in optical network, this dissertation studies counting problem on complete graphs. Closed form expressions are given for certain cases and a polynomial counting algorithm for general cases is also presented. Routing plays a major role in FiWi networks. Accordingly to a novel path length metric which emphasizes on “heaviest edge”, this dissertation proposes a polynomial algorithm on single path computation. NP-completeness proof as well as approximation algorithm are presented for multi-path routing. Radio-frequency identification (RFID) technology is extensively used at present for identification and tracking of a multitude of objects. In many configurations, simultaneous activation of two readers may cause a “reader collision” when tags are present in the intersection of the sensing ranges of both readers. This dissertation ad- dresses slotted time access for Readers and tries to provide a collision-free scheduling scheme while minimizing total reading time. Finally, this dissertation studies a monitoring problem on the surface of the earth for significant environmental, social/political and extreme events using satellites as sensors. It is assumed that the impact of a significant event spills into neighboring regions and there will be corresponding indicators. Careful deployment of sensors, utilizing “Identifying Codes”, can ensure that even though the number of deployed sensors is fewer than the number of regions, it may be possible to uniquely identify the region where the event has taken place.Dissertation/ThesisDoctoral Dissertation Computer Science 201