Sharp Bounds on Davenport-Schinzel Sequences of Every Order

One of the longest-standing open problems in computational geometry is to bound the lower envelope of $n$ univariate functions, each pair of which crosses at most $s$ times, for some fixed $s$. This problem is known to be equivalent to bounding the length of an order-$s$ Davenport-Schinzel sequence, namely a sequence over an $n$-letter alphabet that avoids alternating subsequences of the form $a \cdots b \cdots a \cdots b \cdots$ with length $s+2$. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since been applied to bounding the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements. Let $\lambda_s(n)$ be the maximum length of an order-$s$ DS sequence over $n$ letters. What is $\lambda_s$ asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when $s$ is even or $s\le 3$. However, since the work of Agarwal, Sharir, and Shor in the mid-1980s there has been a persistent gap in our understanding of the odd orders. In this work we effectively close the problem by establishing sharp bounds on Davenport-Schinzel sequences of every order $s$. Our results reveal that, contrary to one's intuition, $\lambda_s(n)$ behaves essentially like $\lambda_{s-1}(n)$ when $s$ is odd. This refutes conjectures due to Alon et al. (2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the Symposium on Computational Geometry, 201

Reka bentuk laman media sosial berteraskan dimensi kebolehlihatan untuk usahawan mikro

Social media has become an important marketing phenomenon in the business environment. However, existing guidelines on using social media for business do not focus on the visibility elements. Visibility is a key factor especially for micro entrepreneurs to strengthen their marketing department. Therefore, the main objective of this study is to propose a design guideline for building a marketing social media site based on visibility elements. In order to achieve that goal, research methodology in design science was adopted. The study involved the search for key processes in a guideline development methodology, identification of the visibility dimensions, the measurement of micro entrepreneurs’ perception towards the guideline and performance monitoring. Five main visibility elements were identified and included in the guideline (i) Developing business site, (ii) Shaping community, (iii) Content information management, (iv) Analysing customer behaviour, and (v) Optimizing market opportunity. These five elements are based by media richness, social influence, and new communication paradigm theory. The perception measures of the developed guideline were carried out through 7 different seminar series which were attended by 114 entrepreneurs. The measurement of the guideline was based on three dimensions, which are quality, content and format, and usability. The results show that the developed guideline has quality, good content and format, and high usability level. The hypotheses results show no significant difference between the experienced and inexperienced entrepreneurs when using the guideline. Generally, this guideline is well received by the experts and entrepreneurs that were involved in this study. Apart from that, the monitoring of micro entrepreneur performance show significant increase in postal feedback and sales volume. This directly confirms the guideline, usage module and visibility elements produced are the main contributions of the study

Visibility maps of segments and triangles in 3D

AbstractLet T be a set of n triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T. We consider the subdivision of T based on (in)visibility from s; this is the visibility map of the segment s with respect to T. The visibility map of the triangle t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The trivial Ω(n2) lower bound for the combinatorial complexity of the strong visibility map of both s and t is almost tight: we prove an O(n2α(n)) upper bound for both structures, where α(n) is the extremely slowly increasing inverse Ackermann function. Furthermore, we prove that the weak visibility map of s has complexity Θ(n5), and the weak visibility map of t has complexity Θ(n7). If T is a polyhedral terrain, the complexity of the weak visibility map is Ω(n4) and O(n5), both for a segment and a triangle. We also present efficient algorithms to compute all discussed structures

www.cs.uu.nl Visibility Maps of Segments and Triangles in 3D ∗

Let T be a set of n triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T. We consider the visibility map of s with respect to T, i.e., the portions of T that are visible from s. The visibility map of t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The trivial Ω(n 2) lower bound for the combinatorial complexity of the strong visibility map of both s and t is almost tight: we prove an O(n 2 log n) upper bound for both structures. Furthermore, we prove that the weak visibility map of s has; complexity Θ(n 5), and the weak visibility map of t has complexity Θ(n 7). If T is a polyhedral terrain, the complexity of the weak visibility map is Ω(n 4) and O(n 5), both for a segment and a triangle. We also present efficient algorithms to compute all discussed structures, i.e., in most cases computation is only an O(log n) factor worse than the worst-case complexity.