On Sensor Network Localization Using SDP Relaxation

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them [11]. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is twofold: We present the first non-asymptotic bound on the connectivity or radio range requirement of the sensors in order to ensure the network is uniquely localizable. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. We also analyze a number objective functions for the SDP relaxation to solve the localization problem for a general graph.Comment: 20 pages, 4 figures, submitted to the Fields Institute Communications Series on Discrete Geometry and Optimizatio

Optimal Permutation Routing for Low-dimensional Hypercubes

We consider the offline problem of routing a permutation of tokens on the nodes of a d-dimensional hypercube, under a queueless MIMD communication model (under the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a single token between communication steps). For a d-dimensional hypercube, it is easy to see that d communication steps are necessary. We develop a theory of “separability ” which enables an analytical proof that d steps suffice for the case d = 3, and facilitates an experimental verification that d steps suffice for d = 4. This result improves the upper bound for the number of communication steps required to route an arbitrary permutation on arbitrarily large hypercubes to 2d − 4. We also find an interesting side-result, that the number of possible communication steps in a d-dimensional hypercube is the same as the number of perfect matchings in a (d + 1)-dimensional hypercube, a combinatorial quantity for which there is no closed-form expression. Finally we present some experimental observations which may lead to a proof of a more general result for arbitrarily large dimension d. 2

Transposition of banded matrices in hypercubes : a "nearly isotropic" task

Includes bibliographical references (p. 19).Supported by NSF. NSF-DDM-8903385 Supported by the ARO. DAAL03-92-G-0115by Emmanouel A. Varvarigos, Dimitri P. Bertsekas

Almost optimal asynchronous rendezvous in infinite multidimensional grids

Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ> 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(d δ polylog d). This bound for the case of 2d-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(d δ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(d δ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O( ( d)), where r = min(r1, r2) and for r ≥ 1. r)δpolylog ( d r

On strong Menger-connectivity of star graphs

AbstractMotivated by parallel routing in networks with faults, we study the following graph theoretical problem. Let G be a graph of minimum vertex degree d. We say that G is strongly Menger-connected if for any copy Gf of G with at most d−2 nodes removed, every pair of nodes u and v in Gf are connected by min{degf(u),degf(v)} node-disjoint paths in Gf, where degf(u) and degf(v) are the degrees of the nodes u and v in Gf, respectively. We show that the star graphs, which are a recently proposed attractive alternative to the widely used hypercubes as network models, are strongly Menger-connected. An algorithm of optimal running time is developed that constructs the maximum number of node-disjoint paths of nearly optimal length in star graphs with faults

I/O embedding and broadcasting in star interconnection networks

The issues of communication between a host or central controller and processors, in large interconnection networks are very important and have been studied in the past by several researchers. There is a plethora of problems that arise when processors are asked to exchange information on parallel computers on which processors are interconnected according to a specific topology. In robust networks, it is desirable at times to send (receive) data/control information to (from) all the processors in minimal time. This type of communication is commonly referred to as broadcasting. To speed up broadcasting in a given network without modifying its topology, certain processors called stations can be specified to act as relay agents. In this thesis, broadcasting issues in a star-based interconnection network are studied. The model adopted assumes all-port communication and wormhole switching mechanism. Initially, the problem treated is one of finding the minimum number of stations required to cover all the nodes in the star graph with i-adjacency. We consider 1-, 2-, and 3-adjacencies and determine the upper bound on the number of stations required to cover the nodes for each case. After deriving the number of stations, two algorithms are designed to broadcast the messages first from the host to stations, and then from stations to remaining nodes; In addition, a Binary-based Algorithm is designed to allow routing in the network by directly working on the binary labels assigned to the star graph. No look-up table is consulted during routing and minimum number of bits are used to represent a node label. At the end, the thesis sheds light on another algorithm for routing using parallel paths in the star network

Multiphase complete exchange on Paragon, SP2 and CS-2

The overhead of interprocessor communication is a major factor in limiting the performance of parallel computer systems. The complete exchange is the severest communication pattern in that it requires each processor to send a distinct message to every other processor. This pattern is at the heart of many important parallel applications. On hypercubes, multiphase complete exchange has been developed and shown to provide optimal performance over varying message sizes. Most commercial multicomputer systems do not have a hypercube interconnect. However, they use special purpose hardware and dedicated communication processors to achieve very high performance communication and can be made to emulate the hypercube quite well. Multiphase complete exchange has been implemented on three contemporary parallel architectures: the Intel Paragon, IBM SP2 and Meiko CS-2. The essential features of these machines are described and their basic interprocessor communication overheads are discussed. The performance of multiphase complete exchange is evaluated on each machine. It is shown that the theoretical ideas developed for hypercubes are also applicable in practice to these machines and that multiphase complete exchange can lead to major savings in execution time over traditional solutions