6 research outputs found

    A Simple Proof of the Shallow Packing Lemma

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    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 ∗

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    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

    A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension

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    Output-Sensitive Tools for Range Searching in Higher Dimensions

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    Let PP be a set of nn points in Rd{\mathbb R}^{d}. A point pPp \in P is kk\emph{-shallow} if it lies in a halfspace which contains at most kk points of PP (including pp). We show that if all points of PP are kk-shallow, then PP can be partitioned into Θ(n/k)\Theta(n/k) subsets, so that any hyperplane crosses at most O((n/k)11/(d1)log2/(d1)(n/k))O((n/k)^{1-1/(d-1)} \log^{2/(d-1)}(n/k)) 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 PP, with crossing number O(n11/(d1)k1/d(d1)log2/(d1)(n/k))O(n^{1-1/(d-1)}k^{1/d(d-1)} \log^{2/(d-1)}(n/k)). 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 nn points in Rd{\mathbb R}^{d} (without the shallowness assumption), a spanning tree TT with {\em small relative crossing number}. That is, any hyperplane which contains wn/2w \leq n/2 points of PP on one side, crosses O(n11/(d1)w1/d(d1)log2/(d1)(n/w))O(n^{1-1/(d-1)}w^{1/d(d-1)} \log^{2/(d-1)}(n/w)) edges of TT. Using a similar mechanism, we also obtain a data structure for halfspace range counting, which uses O(nloglogn)O(n \log \log n) space (and somewhat higher preprocessing cost), and answers a query in time O(n11/(d1)k1/d(d1)(log(n/k))O(1))O(n^{1-1/(d-1)}k^{1/d(d-1)} (\log (n/k))^{O(1)}), where kk is the output size
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