Abstract

In this work we show that given a set S of n points with coordinates on an n × n grid, we can construct data structures for (i) reporting and (ii) counting the maximal points in an axes-parallel query rectangle in sub-logarithmic time. We assume our model of computation to be the word RAM with size of each word being log n bits.

Abstract

Let S be a set of n points in the plane. We present a method where, using O(n log2 n) time and space, S can be pre-processed into a data structure such that given an axis-parallel query rectangle q, we can report the radius of the smallest enclosing disk of the points lying in S ∩ q in O(log6 n) time per query

Abstract

We revisit the problem of anonymous communication where nodes communicate without revealing their identity. We propose its use in the choosing of a node as an anonymous leader and subsequent communication to and from this leader such that its identity is not revealed. We give indistinguishability definitions for anonymous leader and prove it to be secure in an arbitrary reliable network.

Abstract

We present output sensitive techniques for the generalized reporting versions of the planar range maxima problem and the planar range convex hull problem. Our solutions are in the pointer machine model, for orthogonal range queries on a static point set. We solve the planar range maxima problem for two-sided, three-sided and four-sided queries. We achieve a query time of O(logn + c) using O(n) space for the two-sided case, where n denotes the number of stored points and c the number of colors reported. For the three-sided case, we achieve query time O(log2 n + clogn) using O(n) space while for four-sided queries we answer queries in O(log3 n + clog2 n) using O(nlogn) space. For the planar range convex hull problem, we provide a solution that answers queries in O(log2 n + clogn) time, using O(nlog2 n) space.

Abstract

In this work, we consider the problem of finding the closest pair (in L 1 metric) of points in an orthogonal query rectangle. Given a set of n static points on a U× U grid, we preprocess these points into a data structure of size O (m f (m) log 2 m) that can be queried in time O ((g (m)+ log log m) log 3 m), for m= O (n log U) and (i) f (m)= O (1) and g (m)= O (log ε m);(ii) f (m)= O (log log m) and g (m)= O (log log m);(iii) f (m)= O (log ε m) and g (m)= O (1). Here ε: 0< ε< 1 is a small but arbitrary constant.

Abstract

Reliable message transmission is a fundamental problem in distributed communication networks. Of late, several interesting results have been obtained by modelling the network as a directed graph. An important result among them was due to Bhavani et al. [Unconditionally reliable message transmission in directed networks, 18 in: SODA’08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, USA, 2008, pp. 1048–1055], where it was shown that if a negligible probability of error is allowed, the connectivity required in a synchronous network for unconditionally reliable message transmission (URMT) is significantly reduced. We refer to the variant of URMT studied by them as Monte Carlo URMT; here the receiver is allowed to output an incorrect message with small probability. Another interesting variant which has received little attention so far in the literature is the Las Vegas variant, where the receiver is allowed to abort with a small probability, but never to output an incorrect message. We show that the minimum connectivity requirements for the existence of Las Vegas URMT protocols over synchronous networks are the same as that of Monte Carlo URMT protocols over asynchronous networks—a surprising equivalence between two very different models. Furthermore, we show that the higher connectivity requirements of Las Vegas URMT over asynchronous networks match exactly with that of zero-error (perfect) variant over (a)synchronous networks. We also show that there exists a family of graphs where in the number of critical edges for the ‘easier’ randomized variants are asymptotically higher than that for the perfect variants. Hence, our results demonstrate an interesting interplay between (im)perfectness, synchrony and connectivity for the case of reliable message transmission.

Abstract

In this paper, we show that certain phrases although not present in a given question/query, play a very important role in answering the question. Exploring the role of such phrases in answering questions not only reduces the dependency on matching question phrases for extracting answers, but also improves the quality of the extracted answers. Here matching question phrases means phrases which co-occur in given question and candidate answers. To achieve the above discussed goal, we introduce a bigram-based word graph model populated with semantic and topical relatedness of terms in the given document. Next, we apply an improved version of ranking with a prior-based approach, which ranks all words in the candidate document with respect to a set of root words (ie non-stopwords present in the question and in the candidate document). As a result, terms logically related to the root words are scored higher than terms that are not related to the root words. Experimental results show that our devised system performs better than state-of-the-art for the task of answering Why-questions.

Abstract

Summarization mainly provides the major topics or theme of document in limited number of words. However, in extract summary we depend upon extracted sentences, while in abstract summary, each summary sentence may contain concise information from multiple sentences. The major facts which affect the quality of summary are: (1) the way of handling noisy or less important terms in document, (2) utilizing information content of terms in document (as, each term may have different levels of importance in document) and (3) finally, the way to identify the appropriate thematic facts in the form of summary. To reduce the effect of noisy terms and to utilize the information content of terms in the document, we introduce the graph theoretical model populated with semantic and statistical importance of terms. Next, we introduce the concept of weighted minimum vertex cover which helps us in identifying the most representative and thematic facts in the document. Additionally, to generate abstract summary, we introduce the use of vertex constrained shortest path based technique, which uses minimum vertex cover related information as valuable resource. Our experimental results on DUC-2001 and DUC-2002 dataset show that our devised system performs better than baseline systems.

Abstract

In this work, we propose a solution with sub-logarithmic query time for counting the number of maximal points in an axis parallel query rectangle. The problem has been previously studied in [3]and [5]. To the best of our knowledge, this is the first sub-logarithmic query time solution for the problem. Our model of computation is the word RAM with word size of Θ(logn) bits.

Abstract

We consider the problem of reporting convex hull points in an orthogonal range query in two dimensions. Formally, let $ P $ be a set of $ n $ points in $\mathbb {R}^{2} $. A point lies on the convex hull of a point set $ S $ if it lies on the boundary of the minimum convex polygon formed by $ S $. In this paper, we are interested in finding the points that lie on the boundary of the convex hull of the points in $ P $ that also fall with in an orthogonal range $[x_ {lt}, x_ {rt}]\times {}[y_b, y_t] $. We propose a $ O (n\log^{2} n) $ space data structure that can support reporting points on a convex hull inside an orthogonal range query, in time $ O (\log^{3} n+ h) $. Here $ h $ is the size of the output. This work improves the result of (Brass et al. 2013)\cite {brass} that builds a data structure that uses $ O (n\log^{2} n) $ space and has a $ O (\log^{5} n+ h) $ query time. Additionally, we show that our data structure can be modified slightly to solve other related problems. For instance, for counting the number of points on the convex hull in an orthogonal query rectangle, we propose an $ O (n\log^{2} n) $ space data structure that can be queried upon in $ O (\log^{3} n) $ time. We also propose a $ O (n\log^{2} n) $ space data structure that can compute the $ area $ and $ perimeter $ of the convex hull inside an orthogonal range query in $ O (\log^{3} n $) time.