Traversals of Infinite Graphs with Random Local Orientations

We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks. We define analogues in the setting of random basic walks of the notions of recurrence and transience in the theory of simple random walks, and we study the question of which graphs have a cycling random basic walk and which a transient random basic walk. We prove that cycles of arbitrary length are possible in any regular graph, but that they are unlikely. We give upper bounds on the expected number of vertices a random basic walk will visit on the infinite graphs studied and on their finite analogues of sufficiently large size. We then study random basic walks on complete graphs, and prove that the class of complete graphs has random basic walks asymptotically visit a constant fraction of the nodes. We end with numerous conjectures and problems for future study, as well as ideas for how to approach these problems.Comment: This is my masters thesis from Wesleyan University. Currently my advisor and I are selecting a journal where we will submit a shorter version. We plan to split this work into two papers: one for the case of infinite graphs and one for the finite case (which is not fully treated here

The looping rate and sandpile density of planar graphs

We give a simple formula for the looping rate of loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well-suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagome lattice, the dice lattice, and the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental)

Contains and Inside relationships within combinatorial Pyramids

Irregular pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids are used within the segmentation framework to encode a hierarchy of partitions. The different graph models used within the irregular pyramid framework encode different types of relationships between regions. This paper compares different graph models used within the irregular pyramid framework according to a set of relationships between regions. We also define a new algorithm based on a pyramid of combinatorial maps which allows to determine if one region contains the other using only local calculus.Comment: 35 page

Random sampling of lattice configurations using local Markov chains

Algorithms based on Markov chains are ubiquitous across scientific disciplines, as they provide a method for extracting statistical information about large, complicated systems. Although these algorithms may be applied to arbitrary graphs, many physical applications are more naturally studied under the restriction to regular lattices. We study several local Markov chains on lattices, exploring how small changes to some parameters can greatly influence efficiency of the algorithms. We begin by examining a natural Markov Chain that arises in the context of "monotonic surfaces", where some point on a surface is sightly raised or lowered each step, but with a greater rate of raising than lowering. We show that this chain is rapidly mixing (converges quickly to the equilibrium) using a coupling argument; the novelty of our proof is that it requires defining an exponentially increasing distance function on pairs of surfaces, allowing us to derive near optimal results in many settings. Next, we present new methods for lower bounding the time local chains may take to converge to equilibrium. For many models that we study, there seems to be a phase transition as a parameter is changed, so that the chain is rapidly mixing above a critical point and slow mixing below it. Unfortunately, it is not always possible to make this intuition rigorous. We present the first proofs of slow mixing for three sampling problems motivated by statistical physics and nanotechnology: independent sets on the triangular lattice (the hard-core lattice gas model), weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model), and non-saturated Ising (tile-based self-assembly). Previous proofs of slow mixing for other models have been based on contour arguments that allow us prove that a bottleneck in the state space constricts the mixing. The standard contour arguments do not seem to apply to these problems, so we modify this approach by introducing the notion of "fat contours" that can have nontrivial area. We use these to prove that the local chains defined for these models are slow mixing. Finally, we study another important issue that arises in the context of phase transitions in physical systems, namely how the boundary of a lattice can affect the efficiency of the Markov chain. We examine a local chain on the perfect and near-perfect matchings of the square-octagon lattice, and show for one boundary condition the chain will mix in polynomial time, while for another it will mix exponentially slowly. Strikingly, the two boundary conditions only differ at four vertices. These are the first rigorous proofs of such a phenomenon on lattice graphs.Ph.D.Committee Chair: Randall, Dana; Committee Member: Heitsch, Christine; Committee Member: Mihail, Milena; Committee Member: Trotter, Tom; Committee Member: Vigoda, Eri

Exploring Topological Environments

Simultaneous localization and mapping (SLAM) addresses the task of incrementally building a map of the environment with a robot while simultaneously localizing the robot relative to that map. SLAM is generally regarded as one of the most important problems in the pursuit of building truly autonomous mobile robots. This thesis considers the SLAM problem within a topological framework, in which the world and its representation are modelled as a graph. A topological framework provides a useful model within which to explore fundamental limits to exploration and mapping. Given a topological world, it is not, in general, possible to map the world deterministically without resorting to some type of marking aids. Early work demonstrated that a single movable marker was sufficient but is this necessary? This thesis shows that deterministic mapping is possible if both explicit place and back-link information exist in one vertex. Such 'directional lighthouse' information can be established in a number of ways including through the addition of a simple directional immovable marker to the environment. This thesis also explores non-deterministic approaches that map the world with less marking information. The algorithms are evaluated through performance analysis and experimental validation. Furthermore, the basic sensing and locomotion assumptions that underlie these algorithms are evaluated using a differential drive robot and an autonomous visual sensor

Exploring Topological Environments

Simultaneous localization and mapping (SLAM) addresses the task of incrementally building a map of the environment with a robot while simultaneously localizing the robot relative to that map. SLAM is generally regarded as one of the most important problems in the pursuit of building truly autonomous mobile robots. This thesis considers the SLAM problem within a topological framework, in which the world and its representation are modelled as a graph. A topological framework provides a useful model within which to explore fundamental limits to exploration and mapping. Given a topological world, it is not, in general, possible to map the world deterministically without resorting to some type of marking aids. Early work demonstrated that a single movable marker was sufficient but is this necessary? This thesis shows that deterministic mapping is possible if both explicit place and back-link information exist in one vertex. Such 'directional lighthouse' information can be established in a number of ways including through the addition of a simple directional immovable marker to the environment. This thesis also explores non-deterministic approaches that map the world with less marking information. The algorithms are evaluated through performance analysis and experimental validation. Furthermore, the basic sensing and locomotion assumptions that underlie these algorithms are evaluated using a differential drive robot and an autonomous visual sensor

Perception-driven sparse graphs for optimal motion planning

Most existing motion planning algorithms assume that a map (of some quality) is fully determined prior to generating a motion plan. In many emerging applications of robotics, e.g., fast-moving agile aerial robots with constrained embedded computational platforms and visual sensors, dense maps of the world are not immediately available, and they are computationally expensive to construct. We propose a new algorithm for generating plan graphs which couples the perception and motion planning processes for computational efficiency. In a nutshell, the proposed algorithm iteratively switches between the planning sub-problem and the mapping sub-problem, each updating based on the other until a valid trajectory is found. The resulting trajectory retains a provable property of providing an optimal trajectory with respect to the full (unmapped) environment, while utilizing only a fraction of the sensing data in computational experiments.Comment: 2018 IEEE/RSJ International Conference on Intelligent Robots and System

Parallel methods for short read assembly

This work is on the parallel de novo assembly of genomic sequences from short sequence reads. With short reads eliminating the reliability of read overlaps in predicting genomic co-location, a revival of graph-based methods has underpinned the development of short-read assemblers. While these methods predate short read technology, their reach has not extended significantly beyond bacterial genomes due to the memory resources required in their use. These memory limitations are exacerbated by the high coverage needed to compensate for shorter read lengths. As a result, prior to our work, short-read de novo assembly had been demonstrated on relatively small genome sizes with a few million bases. In our work, we advance the field of short sequence assembly in a number of ways. First, we extend models and ideas proposed and tested with small genomes on serial machines to large-scale distributed memory parallel machines. Second, we present ideas for assembly that are especially suited to the reconstruction of very large genomes on these machines. Additionally, we present the first assembler that specifically takes advantage a variable number of fragment sizes or insert lengths concurrently when making assembly decisions, while still working well for data with one insertion length