Bond-Propagation Algorithm for Thermodynamic Functions in General 2D Ising Models

Recently, we developed and implemented the bond propagation algorithm for calculating the partition function and correlation functions of random bond Ising models in two dimensions. The algorithm is the fastest available for calculating these quantities near the percolation threshold. In this paper, we show how to extend the bond propagation algorithm to directly calculate thermodynamic functions by applying the algorithm to derivatives of the partition function, and we derive explicit expressions for this transformation. We also discuss variations of the original bond propagation procedure within the larger context of Y-Delta-Y-reducibility and discuss the relation of this class of algorithm to other algorithms developed for Ising systems. We conclude with a discussion on the outlook for applying similar algorithms to other models.Comment: 12 pages, 10 figures; submitte

Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs

A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of \epsilon and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(n lg^2 n). However, our constructions have slower query times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our linear-space results, we can in fact ensure, for any delta > 0, that the space required is only 1 + delta times the space required just to represent the graph itself

Valiant's model: from exponential sums to exponential products

12 pagesWe study the power of big products for computing multivariate polynomials in a Valiant-like framework. More precisely, we define a new class \vpip as the set of families of polynomials that are exponential products of easily computable polynomials. We investigate the consequences of the hypothesis that these big products are themselves easily computable. For instance, this hypothesis would imply that the nonuniform versions of P and NP coincide. Our main result relates this hypothesis to Blum, Shub and Smale's algebraic version of P versus NP. Let $K$ be a field of characteristic 0. Roughly speaking, we show that in order to separate \p_K from \np_K using a problem from a fairly large class of ``simple'' problems, one should first be able to show that exponential products are not easily computable. The class of ``simple'' problems under consideration is the class of NP problems in the structure $(K,+,-,=)$, in which multiplication is not allowed

Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators

In this work we consider balanced Schnyder woods for planar graphs, which are Schnyder woods where the number of incoming edges of each color at each vertex is balanced as much as possible. We provide a simple linear-time heuristic leading to obtain well balanced Schnyder woods in practice. As test applications we consider two important algorithmic problems: the computation of Schnyder drawings and of small cycle separators. While not being able to provide theoretical guarantees, our experimental results (on a wide collection of planar graphs) suggest that the use of balanced Schnyder woods leads to an improvement of the quality of the layout of Schnyder drawings, and provides an efficient tool for computing short and balanced cycle separators.Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019

Investigation of pathogenic mechanisms in multiple colorectal adenoma patients without germline APC or MYH/MUTYH mutations

Patients with multiple (5–100) colorectal adenomas (MCRAs) often have no germline mutation in known predisposition genes, but probably have a genetic origin. We collected a set of 25 MCRA patients with no detectable germline mutation in APC, MYH/MUTYH or the mismatch repair genes. Extracolonic tumours were absent in these cases. No vertical transmission of the MCRA phenotype was found. Based on the precedent of MYH-associated polyposis (MAP), we searched for a mutational signature in 241 adenomatous polyps from our MCRA cases. Somatic mutation frequencies and spectra at APC, K-ras and BRAF were, however, similar to those in sporadic colorectal adenomas. Our data suggest that the genetic pathway of tumorigenesis in the MCRA patients' tumours is very similar to the classical pathway in sporadic adenomas. In sharp contrast to MAP tumours, we did not find evidence of a specific mutational signature in any individual patient or in the overall set of MCRA cases. These results suggest that hypermutation of APC does not cause our patients' disease and strongly suggests that MAP is not a paradigm for the remaining MCRA patients. Our MCRA patients' colons showed no evidence of microadenomas, unlike in MAP and familial adenomatous polyposis (FAP). However, nuclear β-catenin expression was significantly greater in MCRA patients' tumours than in sporadic adenomas. We suggest that, at least in some cases, the MCRA phenotype results from germline variation that acts subsequent to tumour initiation, perhaps by causing more rapid or more likely progression from microadenoma to macroadenoma

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value $\phi(S_T)$ at expiry $T$ if neither of the continuous time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time interval $[0,T]$. Using a probabilistic approach we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions $\phi$, barrier functions $b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier premium can be expressed as a sum of integrals along the barriers $b_\pm$ of the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic