A Tutorial on Computational Cluster Analysis with Applications to Pattern Discovery in Microarray Data

Microarrays offer unprecedented possibilities for the so-called omic, e.g., genomic and proteomic, research. However, they are also quite challenging data to analyze. The aim of this paper is to provide a short tutorial on the most common approaches used for pattern discovery and cluster analysis as they are currently used for microarrays, in the hope to bring the attention of the Algorithmic Community on novel aspects of classification and data analysis that deserve attention and have potential for high reward. R. Giancarlo is partially supported by Italian MIUR grants PRIN “Metodi Combinatori ed Algoritmici per la Scoperta di Patterns in Biosequenze” and FIRB “Bioinformatica per la Genomica e la Proteomica” and Italy-Israel FIRB Project “Pattern Discovery Algorithms in Discrete Structures, with Applications to Bioinformatics”. D. Scaturro is supported by a MIUR Fellowship in the Italy-Israel FIRB Project “Pattern Discovery Algorithms in Discrete Structures, with Applications to Bioinformatics”.     

A New Approach to Pattern Matching in Degenerate DNA/RNA Sequences and Distributed Pattern Matching

In this paper, we consider the pattern matching problem in DNA and RNA sequences where either the pattern or the text can be degenerate, i.e., contain sets of characters. We present an asymptotically faster algorithm for the above problem that works in O(n log m) time, where n and m is the length of the text and the pattern respectively. We also suggest an efficient implementation of our algorithm, which works in linear time when the pattern size is small. Finally, we also describe how our approach can be used to solve the distributed pattern matching problem. The preliminary version of this paper appeared in [26].     

A phase transition for the metric distortion of percolation on the hypercube

Let H _n be the hypercube {0, 1} ⁿ , and denote by H _n,p Bernoulli bond percolation on H _n , with parameter p = n ^−α . It is shown that at α = 1/2 there is a phase transition for the metric distortion between H _n and H _n,p . For α < 1/2, the giant component of H _n,p is likely to be quasi-isometric to H _n with constant distortion (depending only on α). For 1/2 < α < 1 the minimal distortion tends to infinity as a power of n. We argue that the phase 1/2 < α < 1 is an analogue of the non-uniqueness phase appearing in percolation on non-amenable graphs.     

A General Tractable Density Concept for Graphs

In many applications it is an important algorithmic task to find a densest subgraph in an input graph. The complexity of this task depends on how density is defined. If density means the ratio of the number of edges and the number of vertices in the subgraph, then the algorithmic problem has long been known efficiently solvable. On the other hand, the task becomes NP-hard with closely related but somewhat modified concepts of density. To capture many possible tractable density concepts of interest in a common model, we define and analyze a general concept of density, called F-density. Here F is a family of graphs and we are looking for a subgraph of the input graph, such that this subgraph is the densest in terms of containing the highest number of graphs from F relative to the size of the subgraph. We show that for any fixed finite family F, a subgraph of maximum F-density can be found in polynomial time. As our main tool we develop an algorithm, that may be of independent interest, which can find an independent set of maximum independence ratio in a certain class of weighted graphs. The independence ratio is the weight of the independent set divided by the weight of its neighborhood. This work was supported in part by NSF grants ANI-0220001 and CCF-0634848.     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

Finiteness of the number of solutions of Plateau’s problem for polygonal boundary curves II

In this article, the author proves that a simple closed polygon can bound only finitely many immersed minimal surfaces of disc-type if it meets the following two requirements: firstly it has to bound only minimal surfaces without boundary branch points, and secondly its total curvature, i.e. the sum of the exterior angles at its N + 3 vertices, has to be smaller than 6π.      

A Cayley theorem for distributive lattices

A Cayley-like representation theorem for distributive lattices is proved. Support of the research of the first author by the Czech Government Research Project MSM 6198959214 is gratefully acknowledged.     

On the Fourier spectrum of symmetric Boolean functions

We study the following questionWhat is the smallest t such that every symmetric boolean function on κ variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t?We exclude the constant functions for which there is no such t and the parity functions for which t has to be κ. Let τ (κ) be the smallest such t. Our main result is that for large κ, τ (κ)≤4κ/logκ.The motivation for our work is to understand the complexity of learning symmetric juntas. A κ-junta is a boolean function of n variables that depends only on an unknown subset of κ variables. A symmetric κ-junta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric κ-juntas, in the uniform PAC learning model, in time n ^o(κ) . This improves on a result of Mossel, O’Donnell and Servedio in [16], who show that symmetric κ-juntas can be learned in time n ^2κ/3.     

A rigorous construction of maximum recoverable work

A simple and rigorous derivation of the maximum recoverable work is presented. In contrast to previous derivations it is based on simple and rigorous projectional methods. The principle holds if the stress has a non-trivial Newtonian component.      

Three-Color Ramsey Numbers For Paths

We prove—for sufficiently large n—the following conjecture of Faudree and Schelp:

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkähler varieties).     

Lempel–Ziv Factorization Using Less Time & Space

For 30 years the Lempel–Ziv factorization LZ _x of a string x = x[1..n] has been a fundamental data structure of string processing, especially valuable for string compression and for computing all the repetitions (runs) in x. Traditionally the standard method for computing LZ _x was based on Θ(n)-time (or, depending on the measure used, O(n log n)-time) processing of the suffix tree ST _x of x. Recently Abouelhoda et al. proposed an efficient Lempel–Ziv factorization algorithm based on an “enhanced” suffix array – that is, a suffix array SA _x together with supporting data structures, principally an “interval tree”. In this paper we introduce a collection of fast space-efficient algorithms for LZ factorization, also based on suffix arrays, that in theory as well as in many practical circumstances are superior to those previously proposed; one family out of this collection achieves true Θ(n)-time alphabet-independent processing in the worst case by avoiding tree structures altogether. The work of the first and third authors was supported in part by grants from the Natural Sciences & Engineering Research Council of Canada.     

Infinite highly connected planar graphs of large girth

We construct infinite planar graphs of arbitrarily large connectivity and girth, and study their separation properties. These graphs have no thick end but continuum many thin ones. Every finite cycle separates them, but they corroborate Diestel’s conjecture that everyk-connected locally finite graph contains a possibly infinite cycle — see [3] — whose deletion leaves it (k — 3)-connected.     

The Extremal Function For Noncomplete Minors

  We investigate the maximum number of edges that a graph G can have if it does not contain a given graph H as a minor (subcontraction). Let

A class of simple proper Bol loops

The existence of finite simple non-Moufang Bol loops has long been considered to be one of the main open problems in the theory of loops and quasigroups. In this paper, we present a class of simple proper Bol loops. This class contains finite and new infinite simple proper Bol loops. This paper was written during the author’s Marie Curie Fellowship MEIF-CT-2006-041105 at the University of Würzburg (Germany).     

Blossom V: a new implementation of a minimum cost perfect matching algorithm

We describe a new implementation of the Edmonds’s algorithm for computing a perfect matching of minimum cost, to which we refer as Blossom V. A key feature of our implementation is a combination of two ideas that were shown to be effective for this problem: the “variable dual updates” approach of Cook and Rohe (INFORMS J Comput 11(2):138–148, 1999) and the use of priority queues. We achieve this by maintaining an auxiliary graph whose nodes correspond to alternating trees in the Edmonds’s algorithm. While our use of priority queues does not improve the worst-case complexity, it appears to lead to an efficient technique. In the majority of our tests Blossom V outperformed previous implementations of Cook and Rohe (INFORMS J Comput 11(2):138–148, 1999) and Mehlhorn and Schäfer (J Algorithmics Exp (JEA) 7:4, 2002), sometimes by an order of magnitude. We also show that for large VLSI instances it is beneficial to update duals by solving a linear program, contrary to a conjecture by Cook and Rohe.     

From Individuals to Populations: A Symbolic Process Algebra Approach to Epidemiology

Is it possible to symbolically express and analyse an individual-based model of disease spread, including realistic population dynamics? This problem is addressed through the use of process algebra and a novel method for transforming process algebra into Mean Field Equations. A number of stochastic models of population growth are presented, exploring different representations based on alternative views of individual behaviour. The overall population dynamics in terms of mean field equations are derived using a formal and rigorous rewriting based method. These equations are easily compared with the traditionally used deterministic Ordinary Differential Equation models and allow evaluation of those ODE models, challenging their assumptions about system dynamics. The utility of our approach for epidemiology is confirmed by constructing a model combining population growth with disease spread and fitting it to data on HIV in the UK population. This work was supported by EPSRC through a Doctoral Training Grant (CM, from 2004–2007), and through System Dynamics from Individual Interactions: A process algebra approach to epidemiology (EP/E006280/1, all authors, 2007–2010).     

A unified theory for modifications of<Emphasis Type="Italic">g</Emphasis>-closed sets

We introduce a new set calledmg-closed which is defined on a family of sets satisfying some minimal conditions. This set enables us to unify certain kind of modifications of generalized closed sets due to Levine [17].     

Complete partitions of graphs

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [19], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/√lgn). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C·cp(G)/√lgn classes, then NP ⊆ RTime(n ^{O(lg lg n)}). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lgn) ^c ) for some constant c strictly between 0 and 1. The work reported here is a merger of the results reported in [30] and [21].