Bounding right-arm rotation distances

Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets.

Polynomial functors and trees

We explore the relationship between polynomial functors and trees. In the first part we characterise trees as certain polynomial functors and obtain a completely formal but at the same time conceptual and explicit construction of two categories of rooted trees, whose main properties we describe in terms of some factorisation systems. The second category is the category Ω of Moerdijk and Weiss. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. Included in Part 1 is also an explicit construction of the free monad on a polynomial endofunctor, given in terms of trees. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.

Hierarchical conditional random fields for parts based models matching

A parts based model is a parametrization of an object class using a collection of landmarks following the object structure. The matching of parts based models is one of the problems where pairwise Conditional Random Fields have been successfully applied. The main reason of their effectiveness is tractable inference and learning due to the simplicity of involved graphs, usually trees. However, these models do not consider possible patterns of statistics among sets of landmarks, and thus they sufffer from using too myopic information. To overcome this limitation, we propoese a novel structure based on a hierarchical Conditional Random Fields, which we explain in the first part of this memory. We build a hierarchy of combinations of landmarks, where matching is performed taking into account the whole hierarchy. To preserve tractable inference we effectively sample the label set. We test our method on facial feature selection and human pose estimation on two challenging datasets: Buffy and MultiPIE. In the second part of this memory, we present a novel approach to multiple kernel combination that relies on stacked classification. This method can be used to evaluate the landmarks of the parts-based model approach. Our method is based on combining responses of a set of independent classifiers for each individual kernel. Unlike earlier approaches that linearly combine kernel responses, our approach uses them as inputs to another set of classifiers. We will show that we outperform state-of-the-art methods on most of the standard benchmark datasets.

Stratification of the severity of critically ill patients with classification trees

Background: Development of three classification trees (CT) based on the CART (Classification and Regression Trees), CHAID (Chi-Square Automatic Interaction Detection) and C4.5 methodologies for the calculation of probability of hospital mortality; the comparison of the results with the APACHE II, SAPS II and MPM II-24 scores, and with a model based on multiple logistic regression (LR). Methods: Retrospective study of 2864 patients. Random partition (70:30) into a Development Set (DS) n = 1808 and Validation Set (VS) n = 808. Their properties of discrimination are compared with the ROC curve (AUC CI 95%), Percent of correct classification (PCC CI 95%); and the calibration with the Calibration Curve and the Standardized Mortality Ratio (SMR CI 95%). Results: CTs are produced with a different selection of variables and decision rules: CART (5 variables and 8 decision rules), CHAID (7 variables and 15 rules) and C4.5 (6 variables and 10 rules). The common variables were: inotropic therapy, Glasgow, age, (A-a)O2 gradient and antecedent of chronic illness. In VS: all the models achieved acceptable discrimination with AUC above 0.7. CT: CART (0.75(0.71-0.81)), CHAID (0.76(0.72-0.79)) and C4.5 (0.76(0.73-0.80)). PCC: CART (72(69- 75)), CHAID (72(69-75)) and C4.5 (76(73-79)). Calibration (SMR) better in the CT: CART (1.04(0.95-1.31)), CHAID (1.06(0.97-1.15) and C4.5 (1.08(0.98-1.16)). Conclusion: With different methodologies of CTs, trees are generated with different selection of variables and decision rules. The CTs are easy to interpret, and they stratify the risk of hospital mortality. The CTs should be taken into account for the classification of the prognosis of critically ill patients.

The complexity of random ordered structures

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Restricted walks in regular trees

Let T be the Cayley graph of a finitely generated free group F. Given two vertices in T consider all the walks of a given length between these vertices that at a certain time must follow a number of predetermined steps. We give formulas for the number of such walks by expressing the problem in terms of equations in F and solving the corresponding equations.

Optimal monitoring to implement clean technologies when pollution is random

We analyze a model where firms chose a production technology which, together with some random event, determines the final emission level. We consider the coexistence of two alternative technologies: a "clean" technology, and a "dirty" technology. The environmental regulation is based on taxes over reported emissions, and on penalties over unreported emissions. We show that the optimal inspection policy is a cut-off strategy, for several scenarios concerning the observability of the adoption of the clean technology and the cost of adopting it. We also show that the optimal inspection policy induces the firm to adopt the clean technology if the adoption cost is not too high, but the cost levels for which the firm adopts it depend on the scenario.

Generating trees for permutations avoiding generalized patterns

We construct generating trees with with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-Mélou [2]. We obtain refinements of known enumerative results and find new ones.

Random graphs on surfaces

Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set {1, . . . , n}, then with probability tending to 1 as n → ∞, this random graph either is connected or consists of one giant component together with a few nodes in small planar components.

The equation xpyq=zr and groups that act freely on Λ-trees

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A class of stochastic games with infinitely many interacting agents related to Glauber dynamics on random graphs

We introduce and study a class of infinite-horizon nonzero-sum non-cooperative stochastic games with infinitely many interacting agents using ideas of statistical mechanics. First we show, in the general case of asymmetric interactions, the existence of a strategy that allows any player to eliminate losses after a finite random time. In the special case of symmetric interactions, we also prove that, as time goes to infinity, the game converges to a Nash equilibrium. Moreover, assuming that all agents adopt the same strategy, using arguments related to those leading to perfect simulation algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity allows us to prove “fixation”, i.e. that players will adopt a constant strategy after a finite time. The resulting dynamics is related to zerotemperature Glauber dynamics on random graphs of possibly infinite volume.

The analysis of approximate quickselect and related problems

Approximate Quickselect, a simple modification of the well known Quickselect algorithm for selection, can be used to efficiently find an element with rank k in a given range [i..j], out of n given elements. We study basic cost measures of Approximate Quickselect by computing exact and asymptotic results for the expected number of passes, comparisons and data moves during the execution of this algorithm. The key element appearing in the analysis of Approximate Quickselect is a trivariate recurrence that we solve in full generality. The general solution of the recurrence proves to be very useful, as it allows us to tackle several related problems, besides the analysis that originally motivated us. In particular, we have been able to carry out a precise analysis of the expected number of moves of the ith element when selecting the jth smallest element with standard Quickselect, where we are able to give both exact and asymptotic results. Moreover, we can apply our general results to obtain exact and asymptotic results for several parameters in binary search trees, namely the expected number of common ancestors of the nodes with rank i and j, the expected size of the subtree rooted at the least common ancestor of the nodes with rank i and j, and the expected distance between the nodes of ranks i and j.

Spectral analysis of random sparse matrices

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Level-wise node size distribution of randomly generated regular trees

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Propagation connectivity of random hypergraphs

We study the concept of propagation connectivity on random 3-uniform hypergraphs. This concept is inspired by a simple linear time algorithm for solving instances of certain constraint satisfaction problems. We derive upper and lower bounds for the propagation connectivity threshold, and point out some algorithmic implications.