Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with $|E(G)| = O(n)$, for which we show a lowerbound of $\frac{n}{2^{O(\sqrt{\log n})}}$ for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size $O(\frac{n}{RS(n)^{c}})$ for some $0 < c < 1$, where $RS(n)$ is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to $\frac{n}{2^{(\log n)^{o(1)}}}$ would require a breakthrough in the study of lower bounds on $RS(n)$, which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of $\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$, where $SumIndex(n)$ is the communication complexity of the Sum-Index problem over $Z_n$. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be $\Theta(\frac{n}{2^{(\log n)^c}})$ for some $0<c < 1$

Finding 3-edge-connected components in parallel

A parallel algorithm for finding 3-edge-connected components of an undirected graph on a CRCW PRAM is presented. The time and work complexity of this algorithm is O(logn) and O((m+n)loglogn), respectively, where n is the number of vertices and m is the number of edges in the input graph. The algorithm is based on ear decomposition and reduction of 3-edge-connectivity to 1-vertex-connectivity. This is the first 3-edge-connected component algorithm of a parallel model

Dielectrophoresis of sub-micrometre particles

The aim of this PhD project was to develop the technology of dielectrophoresis on the sub-micrometre scale and to use DEP to manipulate sub-micrometre particles and measure their dielectric properties. Of particular interest was the application of DEP to viruses, the largest of which is approximately 250 nm in diameter. A system for virus characterisation, identification and separation based on DEP would be a major milestone in this field of research, as well as having beneficial medical and biotechnological uses. Particles with a diameter between 1nm and 1m are referred to as Colloidal particles and the dynamics of their movement are complicated by the effects of thermal energy and Brownian motion. High electric fields are required to dominate these effects but signals with high potentials and high frequencies are difficult to generate. Semiconductor manufacturing techniques can be used to fabricate micro-electrode structures which can produce high electric fields from relatively low potentials. Lithography based manufacturing techniques were developed to produce suitable electrodes for dielectrophoresis on a scale small enough to manipulate sub-micrometre particles. Detailed electric field patterns were numerically calculated for these electrodes, so that the dielectrophoretic force could be simulated, predicted and compared with experimental measurements of particle movement. The dielectric properties of latex spheres with diameters from 93 nm to 557 nm were determined through observation and measurement of the DEP movement; new theories were postulated to account for the results which did not conform to accepted theories. A rod shaped plant virus, Tobacco Mosaic Virus (TMV) was also studied and its dielectric properties determined from the experimental results. TMV is 300 nm long with a cylindrical radius of 9nm, a shape of particle which is very different from a sphere and one which has not been studied by this method previously. An expression for the frequency dependent dielectrophoretic force on such a particle was derived and values of the dielectrophoretic force on the particle were measured and compared with the theoretical model

An overview on polynomial approximation of NP-hard problems

The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled

Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered

Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph $\mathcal{P}(n,q)$ which results from taking the union of two paths of length $n\geq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter $0<q(n)\leq 1$. This random graph model, the tangled path, goes through an evolution: if $q$ is close to $0$ the graph bears resemblance to a path and as $q$ tends to $1$ it becomes an expander. In an effort to understand the evolution of $\mathcal{P}(n,q)$ we determine the treewidth and cutwidth of $\mathcal{P}(n,q)$ up to log factors for all $q$. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of $q$.Comment: 40 pages, 7 figure

Construction and Characterization of a Single Stage Dual Diaphragm Gas Gun

In the interest of studying the propagation of shock waves, this work sets out to design, construct, and characterize a pneumatic accelerator that performs high-velocity flyer plate impact tests. A single stage gas gun with a dual diaphragm breach allows for a non-volatile, reliable experimental testing platform for shock phenomena. This remotely operated gas gun utilizes compressed nitrogen to launch projectiles down a 14 foot long, 2 inch diameter bore barrel, which subsequently impacts a target material of interest. A dual diaphragm firing mechanism allows the 4.5 liter breech to reach a total pressure differential of 10ksi before accelerating projectiles to velocities as high as 1,000 m/s (1570-2240 mph). The projectile’s velocity is measured using a series of break pin circuits. The target response can be measured with Photon Doppler Velocimetry (PDV) and/or stress gauge system. A vacuum system eliminates the need for pressure relief in front of the projectile, while additionally allowing the system to remain closed over the entire firing cycle. Characterization of the system will allow for projectile speed to be estimated prior to launching based on initial breach pressure

A laboratory study of localized boundary mixing in a rotating stratified fluid

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2003Oceanic observations indicate that abyssal mixing is localized in regions of rough topography. How locally mixed fluid interacts with the ambient fluid is an open question. Laboratory experiments explore the interaction of mechanically induced boundary mixing and an interior body of linearly stratified rotating fluid. A single oscillating bar produces a small region of turbulence along the wall at middepth. Mixed fluid quickly reaches a steady state height set by a turbulent-buoyant balance, independent of rotation. Initially, the bar is exposed on three sides. Mixed fluid intrudes directly into the interior rather than forming a boundary current. The circulation patterns suggest a model of unmixed fluid being laterally entrained into the turbulent zone. In accord with the model, observed outflux is constant, independent of stratification and restricted by rotation. Later the bar is laterally confines between two walls, which form a channel opening into the basin. A small percentage of mixed fluid enters a boundary current, which exits the channel. The bulk forms a cyclonic circulation in front of the bar, which blocks the channel and restricts horizontal entrainment. In the confined case, the volume flux of mixed fluid decays with time.This work was supported by the Ocean Ventures Fund, the Westcott Fund and the WHOI Education Office. Financial support was also provided by the National Science Foundation through grant OCE-9616949