Sam Greenberg

Generation of simple quadrangulations of the sphere

A simple quadrangulation of the sphere is a finite simple graph embedded on the sphere such that every face is bounded by a walk of 4 edges. We consider the following classes of simple quadrangulations: arbitrary, minimum degree 3, 3-connected, and 3-connected without non-facial 4-cycles. In each case, we show how the class can be generated by starting with some basic graphs in the class and applying a sequence of local modifications. The duals of our algorithms generate classes of quartic (4-regular) planar graphs. In the case of minimum degree 3, our result is a strengthening of a theorem of Nakamoto and almost implicit in Nakamotos proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batageljs proof contained a serious error which cannot easily be corrected. We also give a theoretical enumeration of rooted planar quadrangulations of minimum degree 3, and some counts obtained by a program of Brinkmann and McKay that implements our algorithm.

Improved analysis of D

D* is a planning method that always routes a robot in initially unknown terrain from its current location to a given goal location along a shortest presumed unblocked path. The robot moves along the path until it discovers new obstacles and then repeats the procedure. D* has been used on a large number of robots. It is therefore important to analyze the resulting travel distance. Previously, there has been only one analysis of D*, and it has two shortcomings. First, to prove the lower bound, it uses a physically unrealistic example graph which has distances that do not correspond to distances on a real map. We show that the lower bound is not smaller for grids, the kind of map-based graph on which D* is usually used. Second, there is a large gap between the upper and lower bounds on the travel distance. We considerably reduce this gap by decreasing the upper bound on arbitrary graphs, including grids. To summarize, we provide new, substantially tighter bounds on the travel distance of D* on grids, thus providing a realistic analysis for the way D* is actually used.

Convergence rates of Markov chains for some self-assembly and non-saturated Ising models

Algorithms based on Markov chains are ubiquitous across scientific disciplines as they provide a method for extracting statistical information about large, complicated systems. For some self-assembly models, Markov chains can be used to predict both equilibrium and non-equilibrium dynamics. In fact, the efficiency of these self-assembly algorithms can be related to the rate at which simple chains converge to their stationary distribution. We give an overview of the theory of Markov chains and show how many natural chains, including some relevant in the context of self-assembly, undergo a phase transition as a parameter representing temperature is varied in the model. We illustrate this behavior for the non-saturated Ising model in which there are two types of tiles that prefer to be next to other tiles of the same type. Unlike the standard Ising model, we also allow empty spaces that are not occupied by either type of tile. We prove that for a local Markov chain that allows tiles to attach and detach from the lattice, the rate of convergence is fast at high temperature and slow at low temperature.

Bounds on the Travel Cost of a Mars Rover Prototype Search Heuristic

D* is a greedy heuristic planning method that is widely used in robotics, including several Nomad class robots and the Mars rover prototype, to reach a destination in unknown terrain. We obtain nearly sharp lower and upper bounds of

The effect of boundary conditions on mixing rates of markov chains

\Omega(n\log n/\log\log n)

Slow Mixing of Markov Chains Using Fault Lines and Fat Contours

and O(n log n), respectively, on the worst-case total distance traveled by the robot, for the grid graphs on n vertices typically used in robotics applications. For arbitrary graphs we prove an O(n log2 n) upper bound.

Sampling biased lattice configurations using exponential metrics

Many natural Markov chains undergo a phase transition as a temperature parameter is varied; a chain can be rapidly mixing at high temperature and slowly mixing at low temperature. Moreover, it is believed that even at low temperature, the rate of convergence is strongly dependent on the environment in which the underlying system is placed. It is believed that the boundary conditions of a spin configuration can determine whether a local Markov chain mixes quickly or slowly, but this has only been verified previously for models defined on trees. We demonstrate that the mixing time of Broders Markov chain for sampling perfect and near-perfect matchings does have such a dependence on the environment when the underlying graph is the square-octagon lattice. We show the same effect occurs for a related chain on the space of Ising and “near-Ising” configurations on the two-dimensional Cartesian lattice.

Sampling stable marriages: why spouse-swapping won't work

We show that local dynamics require exponential time for two sampling problems: independent sets on the triangular lattice (the hard-core lattice gas model) and weighted even orientations of the Cartesian lattice (the 8-vertex model). For each problem, there is a parameter i¾?known as the fugacity such that local Markov chains are expected to be fast when i¾?is small and slow when i¾?is large. However, establishing slow mixing for these models has been a challenge because standard contour arguments typically used to show that a chain has small conductance do not seem sufficient. We modify this approach by introducing the notion of fat contoursthat can have nontrivial d-dimensional volume and use these to establish slow mixing of local chains defined for these models.