Approximating Nearest Neighbor Distances

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we consider the most natural metric with this property, which we call the nearest neighbor metric. Given a point set P and a path $\gamma$, our metric charges each point of $\gamma$ with its distance to P. The total charge along $\gamma$ determines its nearest neighbor length, which is formally defined as the integral of the distance to the input points along the curve. We describe a $(3+\varepsilon)$-approximation algorithm and a $(1+\varepsilon)$-approximation algorithm to compute the nearest neighbor metric. Both approximation algorithms work in near-linear time. The former uses shortest paths on a sparse graph using only the input points. The latter uses a sparse sample of the ambient space, to find good approximate geodesic paths.Comment: corrected author nam

Fast Locality-Sensitive Hashing Frameworks for Approximate Near Neighbor Search

The Indyk-Motwani Locality-Sensitive Hashing (LSH) framework (STOC 1998) is a general technique for constructing a data structure to answer approximate near neighbor queries by using a distribution $\mathcal{H}$ over locality-sensitive hash functions that partition space. For a collection of $n$ points, after preprocessing, the query time is dominated by $O(n^{\rho} \log n)$ evaluations of hash functions from $\mathcal{H}$ and $O(n^{\rho})$ hash table lookups and distance computations where $\rho \in (0,1)$ is determined by the locality-sensitivity properties of $\mathcal{H}$. It follows from a recent result by Dahlgaard et al. (FOCS 2017) that the number of locality-sensitive hash functions can be reduced to $O(\log^2 n)$, leaving the query time to be dominated by $O(n^{\rho})$ distance computations and $O(n^{\rho} \log n)$ additional word-RAM operations. We state this result as a general framework and provide a simpler analysis showing that the number of lookups and distance computations closely match the Indyk-Motwani framework, making it a viable replacement in practice. Using ideas from another locality-sensitive hashing framework by Andoni and Indyk (SODA 2006) we are able to reduce the number of additional word-RAM operations to $O(n^\rho)$.Comment: 15 pages, 3 figure

Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces

Range Quantile Queries: Another Virtue of Wavelet Trees

We show how to use a balanced wavelet tree as a data structure that stores a list of numbers and supports efficient {\em range quantile queries}. A range quantile query takes a rank and the endpoints of a sublist and returns the number with that rank in that sublist. For example, if the rank is half the sublist's length, then the query returns the sublist's median. We also show how these queries can be used to support space-efficient {\em coloured range reporting} and {\em document listing}.Comment: Added note about generalization to any constant number of dimensions

Targeting androgen receptor signaling with MicroRNAs and Curcumin: a promising therapeutic approach for Prostate Cancer Prevention and intervention

Prostate cancer (PC) is a multifactorial disease characterized by the abrogation of androgen receptor signaling. Advancement in microbiology techniques has highlighted the significant role of microRNAs (miRNAs) in the progression of PC cells from an androgen-dependent to an androgen-independent state. At that stage, prostate tumors also fail to respond to currently practiced hormone therapies. So, studies in recent decades are focused on investigating the anti-tumor effects of natural compounds in PC. Curcumin is widely recognized and now of huge prestige for its anti-proliferative abilities in different types of cancer. However, its limited solubility, compatibility, and instability in the aqueous phase are major hurdles when administering. Nanoformulations have proven to be an excellent drug delivery system for various drugs and can be used as potential delivery platforms for curcumin in PC. In this review, a shed light is given on the miRNAs-mediated regulation of androgen receptor (AR) signaling and miRNA-curcumin interplay in PC, as well as on curcumin-based nanoformulations that can be used as possible therapeutic solutions for PC

Endoscopic sinus surgery for maxillary sinus mucoceles

BACKGROUND: Maxillary sinus mucoceles are relatively rare among all paranasal sinus mucoceles. With the introduction of endoscopic sinus surgical techniques, rhinologic surgeons prefer transnasal endoscopic management of sinus mucoceles. The aim of this study is to describe the clinical presentation of maxillary sinus mucoceles and to establish the efficacy of endoscopic management of sinus mucoceles. METHODS: Between 2003 and 2005, 14 patients underwent endoscopic sinus surgery for maxillary sinus mucocele. The presenting sign and symptoms, radiological findings, surgical management and need for revision surgery were reviewed. RESULTS: There were eight males and six females with an age range of 14 to 65. Ten patients complained of nasal obstruction, five of nasal drainage, five of cheek pressure or pain and one of proptosis of the eye and cheek swelling. The maxillary sinus and ipsilateral ethmoid sinus involvement on computed tomographic studies was seen in 4 patients. Four patients had history of endoscopic ethmoidectomy surgery for ethmoid sinusitis and one had Caldwell-Luc operation in the past. Ethmoidectomy with middle meatal antrostomy and marsupialization of the mucocele was performed in all patients. Postoperative follow-up ranged between 8 to 48 months. All patients had a patent middle meatal antrostomy and healthy maxillary sinus mucosa. No patients need revision surgery. CONCLUSION: The most common causes of mucoceles are chronic infection, allergic sinonasal disease, trauma and previous surgery. In 64% of the patients of our study cause remains uncertain. Endoscopic sinus surgery is an effective treatment for maxillary sinus mucoceles with a favorable long-term outcome

Maximum gradient embeddings and monotone clustering

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica