A simultaneous embedding with fixed edges (SEFE) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to the common vertices and edges of R and B. We show that whenever R and B admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if R and B are trees ...

The structure of a sentence can be represented as a network where vertices are words and edges indicate syntactic dependencies. Interestingly, crossing syntactic dependencies have been observed to be infrequent in human languages. This leads to the question of whether the scarcity of crossings in languages arises from an independent and specific constraint on crossings. We provide statistical e...

An important problem in signal analysis is to define a general purpose signal representation which is well adapted for developing pattern recognition algorithms. In this paper we will show that such a representation can be defined from the position of the zero-crossings and the local energy values of a dyadic wavelet decomposition. This representation is experimentally complete and admits a sim...

The problem of drawing a graph in the plane with a minimum number of edge crossings—called the crossing number of a graph—is a well-studied problem which dates back to the first half of the twentieth century, as mentioned in [11], and was formulated in full generality in [3]. It was shown that this problem is NP-Complete [4], and that it remains so even when restricted to cubic graphs [5]. Many...

We review and discuss our recent studies of subwavelength grating (SWG) structures for engineering the refractive index of silicon microphotonic waveguides. Various applications are presented, such as fiber-chip coupling structures, waveguide crossings and athermal waveguides made with a polymer-silicon hybrid core. Keywords-Silicon photonics; subwavelength gratings; fiberchip coupler; waveguid...

We provide linear-time algorithms for geometric graphs with sublinearly many edge crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in...