Small Stretch Factor Research Articles

Pseudo-Anosov homeomorphisms of punctured nonorientable surfaces with small stretch factor

Pseudo-Anosov homeomorphisms of punctured nonorientable surfaces with small stretch factor

Time-Dependent Alternative Route Planning: Theory and Practice

We consider the problem of computing a set of meaningful alternative origin-to-destination routes, in real-world road network instances whose arcs are accompanied by travel-time functions rather than fixed costs. In this time-dependent alternative route scenario, we present a novel query algorithm, called Time-Dependent Alternative Graph (TDAG), that exploits the outcome of a time-consuming preprocessing phase to create a manageable amount of travel-time metadata, in order to provide answers for arbitrary alternative-routes queries, in only a few milliseconds for continental-size instances. The resulting set of alternative routes is aggregated in the form of a time-dependent alternative graph, which is characterized by the minimum route overlap, small stretch factor, small size, and low complexity. To our knowledge, this is the first work that deals with the time-dependent setting in the framework of alternative routes. The preprocessed metadata prescribe the minimum travel-time informations between a small set of “landmark” nodes and all other nodes in the graph. The TDAG query algorithm carries out the work in two distinct phases: initially, a collection phase constructs candidate alternative routes; consequently, a pruning phase cautiously discards uninteresting or low-quality routes from the candidate set. Our experimental evaluation on real-world, time-dependent road networks demonstrates that TDAG performed much better (by one or two orders of magnitude) than the existing baseline approaches.

Pseudo-Anosov maps with small stretch factors on punctured surfaces

Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus $g$ with $n$ punctures. We determine the behaviour of this minimum number for a certain large subset of the $(g,n)$ plane, up to a multiplicative constant. In particular it has been shown that for fixed $n$, this minimum value behaves as $\frac{1}{g}$, proving what Penner speculated in 1991.

The Arnoux-Yoccoz mapping classes via Penner's construction

We give a new description of the Arnoux-Yoccoz mapping classes as a product of two Dehn twists and a finite order element. The construction is analogous to Penner's construction of mapping classes with small stretch factors.

Computing the Teichmüller polynomial

The Teichmüller polynomial of a fibered 3-manifold, introduced in [McM00], plays a useful role in the construction of mapping classes having a small stretch factor. We provide a general algorithm for computing the Teichmüller polynomial given a pseudo-Anosov mapping class obtained as a loop in a train track automaton. As a byproduct, our algorithm allows us to derive all the relevant information on the topology of various fibers that belong to a fibered face.

Fine-Grained Location-Free Planarization in Wireless Sensor Networks

Extracting planar graph from network topologies is of great importance for efficient protocol design in wireless ad hoc and sensor networks. Previous techniques of planar topology extraction are often based on ideal assumptions, such as UDG communication model and accurate node location measurements. To make these protocols work effectively in practice, we need extract a planar topology in a location-free and distributed manner with small stretch factors. The planar topologies constructed by current location-free methods often have large stretch factors. In this paper, we present a fine-grained and location-free network planarization method under ρ-quasi-UDG communication model with ρ≥1/√2. Compared with existing location-free planarization approaches, our method can extract a provably connected planar graph, called topological planar simplification (TPS), from the connectivity graph in a fine-grained manner using local connectivity information. We evaluate our design through extensive simulations and compare with the state-of-the-art approaches. The simulation results show that our method produces high-quality planar graphs with a small stretch factor in practical large-scale networks.