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 Nakamoto's proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batagelj's 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. © 2005 Elsevier B.V. All rights reserved

Monotonic surfaces spanning finite regions of Z d 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 Z 2 , and for bias λ > d 2 in Z d , when d > 2. Moreover, our bounds on the mixing time are optimal on d-dimensional hyper-cubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large.

Monotonic surfaces spanning finite regions of ℤ d arise in many contexts, including DNA-based 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 are more likely to add a cube than to remove it, thereby favouring surfaces that are ‘higher’ or have more cubes below it. We prove that the chain is rapidly mixing for any uniform bias in ℤ 2 and for bias λ > d in ℤ d when d > 2. In ℤ 2 we match the optimal mixing time achieved by Benjamini, Berger, Hoffman and Mossel in the context of biased card shuffling [2], but using much simpler arguments. The proofs use a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile. We show that the chain continues to be rapidly mixing if the biases are close to uniform, but that the chain can converge exponentially slowly in the general setting.

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 local Markov chains are expected to be fast when λ is small and slow when λ is large. Unfortunately, establishing slow mixing for these models has been a challenge, as standard contour arguments typically used to show that a chain has small conductance do not seem to apply. We modify this approach by introducing the notion of fat contours that can have nontrivial area, and use these to establish slow mixing of local chains defined for these models.

Abstract. 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 log n/ log log n) 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 log 2 n) upper bound.