47345540
Shape Fitting with Outliers
Given a set $\mathcal{H}$ of n hyperplanes in ${\Bbb R}^d$, we present an algorithm that $\eps$-approximates the extent between the top and bottom k levels of the arrangement of $\mathcal{H}$ in time $O(n + (k/\eps)^c)$, where c is a constant depending on d. The algorithm relies on computing a subset of $\mathcal{H}$ of size $O(k/\eps^{d-1})$, in near linear time, such that the k-level of the arrangement of the subset approximates that of the original arrangement. Using this algorithm, we propose efficient approximation algorithms for shape fitting with outliers for various shapes. These are the first algorithms to handle outliers efficiently for the shape fitting problems considered.
378133
LFM-Pro: a tool for detecting significant local structural sites in proteins
MOTIVATION
The rapidly growing protein structure repositories have opened up new opportunities for discovery and analysis of functional and evolutionary relationships among proteins. Detecting conserved structural sites that are unique to a protein family is of great value in identification of functionally important atoms and residues. Currently available methods are computationally expensive and fail to detect biologically significant local features.


RESULTS
We propose Local Feature Mining in Proteins (LFM-Pro) as a framework for automatically discovering family-specific local sites and the features associated with these sites. Our method uses the distance field to backbone atoms to detect geometrically significant structural centers of the protein. A feature vector is generated from the geometrical and biochemical environment around these centers. These features are then scored using a statistical measure, for their ability to distinguish a family of proteins from a background set of unrelated proteins, and successful features are combined into a representative set for the protein family. The utility and success of LFM-Pro are demonstrated on trypsin-like serine proteases family of proteins and on a challenging classification dataset via comparison with DALI. The results verify that our method is successful both in identifying the distinctive sites of a given family of proteins, and in classifying proteins using the extracted features.


AVAILABILITY
The software and the datasets are freely available for academic research use at http://bioinfo.ceng.metu.edu.tr/Pub/LFMPro.
2273352
Near-Linear Time Approximation Algorithms for Curve Simplification
AbstractWe consider the problem of approximating a polygonal
 curve P under a given error criterion by another
 polygonal curve P’ whose vertices are a
 subset of the vertices of P. The goal is to minimize
 the number of vertices of P’ while ensuring that the
 error between P’ and P is below a certain threshold.
 We consider two different error measures: 
 Hausdorff and Frechet. For both error criteria, we present
 near-linear time approximation algorithms that, given a
 parameter ε > 0, compute a simplified polygonal
 curve P’ whose error is less than ε and
 size at most the size of an optimal simplified
 polygonal curve with error ε/2. We consider
 monotone curves in ℝ2 in the case of the Hausdorff error measure
 under the uniform distance metric
 and arbitrary curves in any dimension for the Frechet error measure
 under Lp metrics. 
 We present
 experimental results demonstrating that our algorithms
 are simple and fast, and produce close to optimal
 simplifications in practice.
312809
Segmenting molecular surfaces
This paper presents a new method for segmentation of molecular surfaces. Topological analysis of a scalar function defined on the surface and its associated gradient field reveals the relationship between the features of interest and critical points of the scalar function. The segmentation is obtained by associating segments with local minima/maxima. Controlled simplification of the function merges segments resulting in a hierarchical segmentation of the molecular surface. This segmentation is used to identify rigid components of protein molecules and to study the role of cavities and protrusions in protein-protein interactions.
37729130
Extreme Elevation on a 2-Manifold
Given a smoothly embedded 2-manifold in ${\Bbb R}^3,$ we define the elevation of a point as the height difference to a canonically defined second point on the same manifold. Our definition is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation. We give an algorithm for finding points of locally maximum elevation, which we suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.
15443235
Computing the Writhing Number of a Polygonal Knot
AbstractThe writhing number measures the global geometry of
 a closed space curve or knot.
 We show that this measure is related to the average winding number
 of its Gauss map.
 Using this relationship, we give an algorithm for computing
 the writhing number for a polygonal knot with n edges
 in time roughly proportional to n1.6.
 We also implement a different, simple algorithm and provide
 experimental evidence for its practical efficiency.


18478207
Prefix-Free Regular-Expression Matching
We explore the regular-expression matching problem with respect to prefix-freeness of the pattern. We show that the prefix-free regular expression gives only linear number of matching substrings in the size of a given text. Based on this observation, we propose an efficient algorithm for the prefix-free regular-expression matching problem. Furthermore, we suggest an algorithm to determine whether or not a given regular language is prefix-free.
14047084
Efficient Algorithm for Embedding Hypergraphs in a Cycle
The problem of Minimum Congestion Hypergraph Embedding in a Cycle (MCHEC) is to embed the hyperedges of a hypergraph as paths in a cycle such that the maximum congestion (the maximum number of paths that use any single link in the cycle) is minimized. This problem has many applications, including minimizing communication congestions in computer networks and parallel computations. The MCHEC problem is NP-hard. We give a 1.8-approximation algorithm for the problem. This improves the previous 2-approximation results. The algorithm has the optimal O(mn) time for the hypergraph with m hyperedges and n nodes.
570803
Provable dimension detection using principal component analysis
We present simple algorithms for detecting the dimension k of a smooth manifold M ⊂ Rd from a set P of point samples, provided that P satisfies a standard sampling condition as in previous results. The best running time so far is O(d2O(k7 log k)) worst-case by Giesen and Wagner after the adaptive neighborhood graph is constructed in O(d|P|2) worst-case time. Given the adaptive neighborhood graph, for any l ≥ 1, our algorithm outputs the true dimension with probability at least 1-2-l in O(2O(k)kd(k + l log d)) expected time. Our experimental results validate the effectiveness of our approach in computing the dimension. A further advantage is that both the algorithm and its analysis can be generalized to the noisy case, in which outliers and a small perturbation of the samples are allowed.
207130271
Frobenius maps
A graph is called Frobenius if it is a connected orbital regular graph of a Frobenius group. A Frobenius map is a regular Cayley map whose underlying graph is Frobenius. In this paper, we show that almost all low-rank Frobenius graphs admit regular embeddings and enumerate non-isomorphic Frobenius maps for a given Frobenius graph. For some Frobenius groups, we classify all Frobenius maps derived from these groups. As a result, we construct some Frobenius maps with trivial exponent groups as a partial answer of a question raised by Nedela and Skoviera (Exponents of orientable maps, Proc. London Math. Soc. 75(3) (1997) 1-31).
2456151
Subdivision Schemes and Refinement Equations with Nonnegative Masks
We consider the two-scale refinement equation f(x)=@?^N"n"="0c"nf(2x-n) with @?"nc"2"n=@?"nc"2"n"+"1=1 where c"0, c"N 0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all c"n>=0. It has long been conjectured that under such an assumption the subdivision algorithm converge, and the cascade algorithm converge uniformly to a continuous function, if and only if only if 00} is 1. We prove the conjecture for a large class of refinement equations.