Sam Greenberg

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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(More)
We study the behavior of random walks along the edges of the stable marriage lattice for various restricted families of allowable preference sets. In the "<i>k</i>-attribute model," each man is valued in each of <i>k</i> attributes, and each woman's ranking of the men is determined by a linear function, representing her relative ranking of those attributes;(More)
Monotonic surfaces spanning finite regions of Zd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z2, and for bias λ > d2 in Zd,(More)
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(More)
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(More)
We show that local dynamics require exponential time for two sampling problems motivated by statistical physics: independent sets on the triangular lattice (the hard-core lattice gas model) and weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model). For each problem, there is a parameter λ known as the fugacity, such that(More)
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 Ω(n logn/ log logn) and O(n logn), respectively, on the worst-case total distance traveled by the robot, for the grid graphs(More)
Monotonic surfaces spanning finite regions of Z arise in many contexts, including DNAbased self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that iteratively adds or removes a single cube below the surface during a step. We consider a biased version of the chain, where we(More)