A Geometric Approach to Combinatorial Fixed-Point Theorems

We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological fixed-point theorems. Our work naturally suggests several interesting open questions for future research.Comment: 10 pages; an extended abstract appeared at Eurocomb 201

A Static Optimality Transformation with Applications to Planar Point Location

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.Comment: 13 pages, 1 figure, a preliminary version appeared at SoCG 201

Some properties of n-dimensional triangulations

A number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplicial tessellation, of the convex hull of an arbitrary finite set of points in n-space are described. The principal results achieved are: (1) a set of n+2 points in n-space may be triangulated in at most 2 different ways; (2) the sphere test defined in this report selects a preferred one of these two triangulations; (3) a set of parameters is defined that permits the characterization and enumeration of all sets of n+2 points in n-space that are significantly different from the point of view of their possible triangulation; (4) the local sphere test induces a global sphere test property for a triangulation; and (5) a triangulation satisfying the global sphere property is dual to the n-dimensional Dirichlet tesselation, i.e., it is a Delaunay triangulation

Data-driven Job Search Engine Using Skills and Company Attribute Filters

According to a report online, more than 200 million unique users search for jobs online every month. This incredibly large and fast growing demand has enticed software giants such as Google and Facebook to enter this space, which was previously dominated by companies such as LinkedIn, Indeed and CareerBuilder. Recently, Google released their "AI-powered Jobs Search Engine", "Google For Jobs" while Facebook released "Facebook Jobs" within their platform. These current job search engines and platforms allow users to search for jobs based on general narrow filters such as job title, date posted, experience level, company and salary. However, they have severely limited filters relating to skill sets such as C++, Python, and Java and company related attributes such as employee size, revenue, technographics and micro-industries. These specialized filters can help applicants and companies connect at a very personalized, relevant and deeper level. In this paper we present a framework that provides an end-to-end "Data-driven Jobs Search Engine". In addition, users can also receive potential contacts of recruiters and senior positions for connection and networking opportunities. The high level implementation of the framework is described as follows: 1) Collect job postings data in the United States, 2) Extract meaningful tokens from the postings data using ETL pipelines, 3) Normalize the data set to link company names to their specific company websites, 4) Extract and ranking the skill sets, 5) Link the company names and websites to their respective company level attributes with the EVERSTRING Company API, 6) Run user-specific search queries on the database to identify relevant job postings and 7) Rank the job search results. This framework offers a highly customizable and highly targeted search experience for end users.Comment: 8 pages, 10 figures, ICDM 201

There are only two nonobtuse binary triangulations of the unit nn-cube

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into $n!$ simplices. In this paper we show that, surprisingly, for each $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than $n!({\rm e}-2)$.Comment: 17 pages, 7 figure