6 research outputs found
A Simple Proof of the Shallow Packing Lemma
International audienceWe show that the shallow packing lemma follows from a simple modification of the standard proof, due to Haussler and simplified by Chazelle, of the packing lemma
A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension ∗
Let (X,S) be a set system on an n-point set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S ∈ S and two-colorings χ on X. We consider the scenario where, for any subset X ′ ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S to X ′ of size at most k is only O(m d1 k d−d1), for fixed integers d> 0 and 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case we show that there exists a coloring χ with discrepancy bound O ∗ (|S | 1/2−d1/(2d) n (d1−1)/(2d)), for each S ∈ S, where O ∗ (·) hides a polylogarithmic factor in n. This bound is tight up to a polylogarithmic factor [25, 27] and the corresponding coloring χ can be computed in expected polynomial time using the very recent machinery of Lovett and Meka for constructive discrepancy minimization [24]. Our bound improves and generalizes the bounds obtained from the machinery of Har-Peled and Sharir [19] (and the follow-up work in [32]) for points and halfspaces in d-space for d ≥ 3. Last but not least, we show that our bound yields improved bounds for the size of relative (ε,δ)-approximations for set systems of the above kind
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size