It was open for more than eight years to obtain an algorithm for computing the convex hull of a set of n sorted points in sub-logarithmic time on a reconfigurable mesh of size p n pn. Our main contribution is to provide the first breakthrough: we propose an almost optimal algorithm running in O((log logn)2) time on a reconfigurable mesh of size p n pn. With slight modifications this algorithm c...

Most online optimization algorithms focus on one of two things: performing well in adversarial settings by adapting to unknown data parameters (such as Lipschitz constants), typically achieving O( √ T ) regret, or performing well in stochastic settings where they can leverage some structure in the losses (such as strong convexity), typically achieving O(log(T )) regret. Algorithms that focus on...

We propose a distributed algorithm based on Alternating Direction Method of Multipliers (ADMM) to minimize the sum of locally known convex functions. This optimization problem captures many applications in distributed machine learning and statistical estimation. We provide a novel analysis that shows if the functions are strongly convex and have Lipschitz gradients, then an -optimal solution ca...

In this thesis we develop optimal-time algorithms for computing shortest paths on polyhedral surfaces in three dimensions. Specifically: • We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm, which settles a major problem that has been open for more...

We study the $k$ nearest neighbors problem in plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals under a general family distance functions including $L_p$ norms additively weighted Euclidean distances. compose static data structure this setting with nearly optimal $O(n\log\log n)$ space, $O(\log n+k)$ query time...

We study competitive economy equilibrium computation. We show that, for the first time, the equilibrium sets of the following two markets: 1. A mixed Fisher and ArrowDebreu market with homogeneous and log-concave utility functions; 2. The Fisher and Arrow-Debreu markets with several classes of concave non-homogeneous utility functions; are convex or log-convex. Furthermore, an equilibrium can b...

Let S be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γi that the query point lies in a polygon Pi of S. We show how to preprocess S such that the query time for a point p ∈ Pi depends on γi and, in addition, on the distance from p to the boundary of Pi—the further away from the boundary, the ...

Some new Jensen’s type inequalities for di¤erentiable log-convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided. 1. Introduction Jensen’s inequality for convex functions is one of the most important result in the Theory of Inequalities due to the fact that many ...