We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R^3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(n^(1/2)). A matching Omega(n^(1/2)) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log(n)) upper bound of Aronov et al. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) polylog n) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

Let P be a k-ary predicate over a finite alphabet. Consider a random CSP(P) instance I over n variables with m constraints. When m ≫ n the instance will be unsatisfiable with high probability and we want to find a certificate of unsatisfiability. When P is the 3-ary OR predicate, this is the well-studied problem of refuting random 3-SAT formulas and an efficient algorithm is known only when m ≫ n3/2. Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a k-ary predicate, having m ≫≫n[k/2]constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if P fails to support a t-wise uniform distribution, then there is an efficient algorithm that refutes random CSP(P) instances whp when m ≫ nt/2. Indeed, our algorithm will “somewhat strongly” refute the instance I, certifying Opt(I) ≤ 1 - Ωk(1). If t = k then we get the strongest possible refutation, certifying Opt(I) ≤ E[P] + o(1). This last result is new even for random k-SAT. Prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP(P) may be impossible when m ≫ nt/2; thus there is an indication that our algorithm's dependence on m is optimal for every P, at least in the context of SDP hierarchies. As an application of our result, we falsify assumptions used to show hardness-of-learning in recent work of Daniely, Linial, and Shalev-Shwartz.

Let X1, ..., Xnbe joint {±1}-valued random variables. It is known that conditioning on a random subset of O(1/ε2) of them reduces their average pairwise covariance to below ε (in expectation). We conjecture that O(1/ε2) can be improved to O(1/ε). The motivation for the problem and our conjectured improvement comes from the theory of global correlation rounding for convex relaxation hierarchies. We suggest attempting the conjecture in the case that X1, ..., Xnare the leaves of an information flow tree. We prove the conjecture in the case that the information flow tree is a caterpillar graph (similar to a two-state hidden Markov model).