Motivated by the desire to cope with data imprecision [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of whi...

In this paper, we present parallel algorithms for the coarse grained multicomputer (CGM) and bulk synchronous parallel computer (BSP) for solving two well known graph problems: (1) determining whether a graph G is bipartite, and (2) determining whether a bipartite graph G is convex. Our algorithms require O(log p) and O(log p) communication rounds, respectively, and linear sequential work per r...

This note gives a simple method for obtaining inequalities for ratios involving 3log-convex functions. As an example, an inequality for Wallis’s ratio of Gautchi-Kershaw type is obtained. Inequalities for generalized means are also considered.

Log-concave and Log-convex sequences arise often in combinatorics, algebra, probability and statistics. There has been a considerable amount of research devoted to this topic in recent years. Let {xi}i≥0 be a sequence of non-negative real numbers. We say that {xi} is Log-concave ( Log-convex resp.) if and only if xi−1xi+1 ≤ xi (xi−1xi+1 ≥ xi resp.) for all i ≥ 1 (relevant results can see [2] an...

In this paper, we consider an integer convex optimization problem where the objective function is the sum of separable convex functions (that is, of the form Σ(i,j)∈Q ij ij F (w ) + Σi∈P i i B ( ) μ ), the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the form μi μj ≤ wij, (i, j) ∈ Q), with lower and upper bounds on variables. Let n = |P|, m = ...

Given a set of n points P in the plane, the first layer L1 of P is formed by the points that appear on P ’s convex hull. In general, a point belongs to layer Li, if it lies on the convex hull of the set P \ ⋃ j<i{Lj}. The convex layers problem is to compute the convex layers Li. Existing algorithms for this problem either do not achieve the optimal O (n log n) runtime and linear space, or are o...

We motivate this study from a recent work on a stochastic gradient descent (SGD) method with only one projection (Mahdavi et al., 2012), which aims at alleviating the computational bottleneck of the standard SGD method in performing the projection at each iteration, and enjoys an O(log T/T ) convergence rate for strongly convex optimization. In this paper, we make further contributions along th...

Let μ = ρdx be a Borel measure on Rd. A Borel set A ⊂ R is a solution of the isoperimetric problem if for any B ⊂ R satisfying μ(A) = μ(B) one has μ(∂A) ≤ μ(∂B), where μ(∂A) = ∫ ∂A ρ dHd−1 is the corresponding surface measure. There exists only a small number of examples where the isoperimetric problem has an exact solution. The most important case is given by Lebesgue measure λ on R, the solut...

We give novel algorithms for stochastic strongly-convex optimization in the gradient oracle model which return a O( 1 T )-approximate solution after T iterations. The first algorithm is deterministic, and achieves this rate via gradient updates and historical averaging. The second algorithm is randomized, and is based on pure gradient steps with a random step size. This rate of convergence is o...