Convex Hulls under Uncertainty

We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a \emph{null} location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of \some-hull that may be a useful representation of uncertain hulls

Synthetic, Structural, and Biochemical Studies of Organotin(IV) With Schiff Bases Having Nitrogen and Sulphur Donor Ligands

Three bidentate Schiff bases having nitrogen and sulphur donor sequences were prepared by condensing S-benzyldithiocarbazate (NH(2)NHCS(2)CH(2)C(6)H(5)) with heterocyclic aldehydes. The reaction of diphenyltin dichloride with Schiff bases leads to the formation of a new series of organotin(IV) complexes. An attempt has been made to prove their structures on the basis of elemental analyses, conductance measurements, molecular weights determinations, UV, infrared, and multinuclear magnetic resonance ((1)H, (13)C, and (119)Sn) spectral studies. Organotin(IV) complexes were five- and six-coordinate. Schiff bases and their corresponding organotin complexes have also been screened for their antibacterial and antifungal activities and found to be quite active in this respect

Computing Optimal Kernels in Two Dimensions

Let $P$ be a set of $n$ points in $\mathbb{R}^2$. A subset $C\subseteq P$ is an $\varepsilon$-kernel of $P$ if the projection of the convex hull of $C$ approximates that of $P$ within $(1-\varepsilon)$-factor in every direction. The set $C$ is a weak $\varepsilon$-kernel if its directional width approximates that of $P$ in every direction. We present fast algorithms for computing a minimum-size $\varepsilon$-kernel as well as a weak $\varepsilon$-kernel. We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of $\varepsilon$-core, a convex polygon lying inside $CH(P)$, prove that it is a good approximation of the optimal $\varepsilon$-kernel, present an efficient algorithm for computing it, and use it to compute an $\varepsilon$-kernel of small size

Computing the Similarity Between Moving Curves

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality

The Maximum-Level Vertex in an Arrangement of Lines

Let $L$ be a set of $n$ lines in the plane, not necessarily in general position. We present an efficient algorithm for finding all the vertices of the arrangement $A(L)$ of maximum level, where the level of a vertex $v$ is the number of lines of $L$ that pass strictly below $v$. The problem, posed in Exercise~8.13 in de Berg etal [BCKO08], appears to be much harder than it seems, as this vertex might not be on the upper envelope of the lines. We first assume that all the lines of $L$ are distinct, and distinguish between two cases, depending on whether or not the upper envelope of $L$ contains a bounded edge. In the former case, we show that the number of lines of $L$ that pass above any maximum level vertex $v_0$ is only $O(\log n)$. In the latter case, we establish a similar property that holds after we remove some of the lines that are incident to the single vertex of the upper envelope. We present algorithms that run, in both cases, in optimal $O(n\log n)$ time. We then consider the case where the lines of $L$ are not necessarily distinct. This setup is more challenging, and the best we have is an algorithm that computes all the maximum-level vertices in time $O(n^{4/3}\log^{3}n)$. Finally, we consider a related combinatorial question for degenerate arrangements, where many lines may intersect in a single point, but all the lines are distinct: We bound the complexity of the weighted $k$-level in such an arrangement, where the weight of a vertex is the number of lines that pass through the vertex. We show that the bound in this case is $O(n^{4/3})$, which matches the corresponding bound for non-degenerate arrangements, and we use this bound in the analysis of one of our algorithms

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

Synthetic and Spectroscopic Characterization of Organotin(IV) Complexes of Biologically Active Schiff Bases Derived from Sulpha Drugs

A number of diorganotin(IV) complexes with Schiffbase have been synthesized and characterized by elemental analysis, conductance measurements, molecular weight determinations, infrared, electronic and multinuclear magnetic resonance ((1)H, (13)C and (119)Sn NMR) spectral data. The molar conductivity data shows non-electrolytic nature of complexes. The bidentate nature of the ligands is inferred from IR and NMR spectral studies. The antimicrobial activities of the ligands and their tin complexes have been screened in vitro against the organism Escherichia coli; Staphylococus aureus, Prouteus mirabilis, Bacillus thurengiensis, Penicillium co.,sogenum, Aspergillus niger and Fusarium oxysporum