Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems. Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by "on" vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in O (m^ 2 3 ) amortized time, where m denotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC'02], which required fast matrix multiplication and had an update time of O(m^0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We provide similarly improved update times, O (n^ 2 3 ), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries, such as arbitrary 2D line segments, d-dimensional simplices, and d-dimensional balls.
For more difficult dynamic graph problems, the goal is typically changed from getting polylogarithmic bounds to finding better exponents in polynomial bounds; for example, see all the papers on directed reachability @cite_1 @cite_26 @cite_35 @cite_22 . Evidence suggests that dynamic subgraph connectivity fits this category. It was observed @cite_32 that finding triangles (3-cycles) or quadrilaterals (4-cycles) in directed graphs can be reduced to @math vertex updates. Thus, an update bound better than @math appears unlikely without FMM, since the best running time for finding triangles without FMM is @math , dating back to STOC'77 @cite_28 . Even with FMM, known results are only slightly better: finding triangles and quadrilaterals takes time @math @cite_3 and @math @cite_6 respectively. Thus, current knowledge prevents an update bound better than @math .
@cite_35 @cite_26 @cite_22 @cite_28 @cite_1 @cite_32 @cite_3 @cite_6
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We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Finding minimum circuits in graphs and digraphs is discussed. An almost minimum circuit is a circuit which may have only one edge more than the minimum. To find an almost minimum circuit an @math algorithm is presented. A direct algorithm for finding a minimum circuit has an @math behavior. It is refined to yield an @math average time algorithm. An alternative method is to reduce the problem of finding a minimum circuit to that of finding a triangle in an auxiliary graph. Three methods for finding a triangle in a graph are given. The first has an @math worst case bound ( @math for planar graphs); the second takes @math time on the average; the third has an @math worst case behavior. For digraphs, results of Bloniarz, Fisher and Meyer are used to obtain an algorithm with @math average behavior. In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an @math round matrix multiplication algorithm, where @math is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in @math rounds, improving upon the @math triangle detection algorithm of [DISC 2012], -- a @math -approximation of all-pairs shortest paths in @math rounds, improving upon the @math -round @math -approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in @math rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles. Purpose – The purpose of this paper is to suggest an alternative to the widely used Times Cited criterion for analysing citation networks. The approach involves taking account of the natures of the papers that cite a given paper, so as to differentiate between papers that attract the same number of citations.Design methodology approach – ArticleRank is an algorithm that has been derived from Google's PageRank algorithm to measure the influence of journal articles. ArticleRank is applied to two datasets – a citation network based on an early paper on webometrics, and a self‐citation network based on the 19 most cited papers in the Journal of Documentation – using citation data taken from the Web of Knowledge database.Findings – ArticleRank values provide a different ranking of a set of papers from that provided by the corresponding Times Cited values, and overcomes the inability of the latter to differentiate between papers with the same numbers of citations. The difference in rankings between Times Cited ...