In this note, we introduce a family of bipartite graphs called path restricted ordered bipartite graphs and present it as an abstract generalization of some well known geometric graphs like unit distance graphs on convex point sets. In the framework of convex point sets, we also focus on a generalized version of Gabriel graphs known as locally Gabriel graphs or LGGs. LGGs can also be seen as th...

We continue the investigation of computational aspects of restricted-orientation convexity (O-convexity) in two dimensions. We introduce one notion of an O-halfplane, for a set O of orientations, and we investigate O-connected convexity. The O-connected convex hull of a nite set X can be computed in time O(jXj log jXj + jOj). The O-connected hull is a basis for determining the O-convex hull of ...

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex K with n vertices, one can con...

Given a set P of n points in the plane, we wish to find a set Q ⊂ P of k points for which the convex hull conv(Q) has the minimum area. We solve this, and the related problem of finding a minimum area convex k-gon, in time O(n log n) and space O(n log n) for fixed k, almost matching known bounds for the minimum area triangle problem. Our algorithm is based on finding a certain number of nearest...

In this paper, we utilize stochastic optimization to reduce the space complexity of convex composite optimization with a nuclear norm regularizer, where the variable is a matrix of size m × n. By constructing a low-rank estimate of the gradient, we propose an iterative algorithm based on stochastic proximal gradient descent (SPGD), and take the last iterate of SPGD as the final solution. The ma...

Let P be a simple planar polygon with n vertices. We would like to find a triangulation MDT(P) of P that minimizes the diameter of the dual tree. We show that MDT(P) can be constructed in O(n log n) time. If P is convex, we show that the dual of any MDT has diameter 2 · dlog2(n/3)e or 2 · dlog2(n/3)e−1, depending on the value of n. We also investigate the relation between MDT(P) and the number ...

We give a randomized algorithm using O(n7 log’ n ) separation calls to approximate the volume of a convex body with a fixed relative error. The bound is O(n6 log4 n ) for centrally symmetric bodies and for polytopes with a polynomial number of facets, and O(n5 log4 n ) for centrally symmetric polytopes with a polynomial number of facets. We also give an O(n6 log n ) algorithm to sample a point ...

We use known data structures for ray shooting and linear programming queries to derive new output-sensitive results on convex hulls, extreme points, and related problems. We show that the f-face convex hull of an n-point set P in a xed dimension d 2 can be constructed in O(n logf + (nf) 1?1=(bd=2c+1) log O(1) n) time; this is optimal if f = O(n 1=bd=2c = log K n) for some suuciently large const...

Log-linear parsing models are often trained by optimizing likelihood, but we would prefer to optimise for a task-specific metric like Fmeasure. Softmax-margin is a convex objective for such models that minimises a bound on expected risk for a given loss function, but its naı̈ve application requires the loss to decompose over the predicted structure, which is not true of F-measure. We use softmax...

E e as balls centered at the origin, provided m ∈ [0, m0) for some (potentially computable) m0 > 0; this affirmatively answers conjecture [RCBM, Conjecture 3.12] for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurca...