We show optimal lower bounds for spanning forest computation in two different models:

* One wants a data structure for fully dynamic spanning forest in which updates can insert or delete edges amongst a base set of $n$ vertices. The sole allowed query asks for a spanning forest, which the ... more >>>

Recent work of [Dasgupta-Kumar-Sarl\'{o}s, STOC 2010] gave a sparse Johnson-Lindenstrauss transform and left as a main open question whether their construction could be efficiently derandomized. We answer their question affirmatively by giving an alternative proof of their result requiring only bounded independence hash functions. Furthermore, the sparsity bound obtained in ... more >>>

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of ... more >>>

In the set cover problem we are given a collection of $m$ sets whose union covers $[n] = \{1,\ldots,n\}$ and must find a minimum-sized subcollection whose union still covers $[n]$. We investigate the approximability of set cover by an approximation ratio that depends only on $m$ and observe that, for ... more >>>