More Like this

1. Near-Optimal Averaging Samplers and Matrix Samplers

   https://doi.org/10.4230/lipics.ccc.2025.6  

   Xun, Zhiyang; Zuckerman, David (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Srinivasan, Srikanth (Ed.)

   Abstract: We present the first efficient averaging sampler that achieves asymptotically optimal randomness complexity and near-optimal sample complexity. For any δ < ε and any constant α > 0, our sampler uses m + O(log (1 / δ)) random bits to output t = O((1/ε² log 1/δ)^{1 + α}) samples Z_1, … , Z_t ∈ {0, 1}^m such that for any function f: {0, 1}^m → [0, 1], Pr[|1/t∑_{i=1}^t f(Z_i) - 𝔼[f]| ≤ ε] ≥ 1 - δ. The randomness complexity is optimal up to a constant factor, and the sample complexity is optimal up to the O((1/(ε²) log 1/(δ))^α) factor. Our technique generalizes to matrix samplers. A matrix sampler is defined similarly, except that f: {0, 1}^m → ℂ^{d×d} and the absolute value is replaced by the spectral norm. Our matrix sampler achieves randomness complexity m + Õ(log(d / δ)) and sample complexity O((1/ε² log d/δ)^{1 + α}) for any constant α > 0, both near-optimal with only a logarithmic factor in randomness complexity and an additional α exponent on the sample complexity. We use known connections with randomness extractors and list-decodable codes to give applications to these objects. Specifically, we give the first extractor construction with optimal seed length up to an arbitrarily small constant factor above 1, when the min-entropy k = β n for a large enough constant β < 1. Finally, we generalize the definition of averaging sampler to any normed vector space. 

   more » « less
2. Online Spanners in Metric Spaces

   Bhore, Sujoy; Filtser, Arnold; Khodabandeh, Hadi; Toth, Csaba D. (September 2022, 30th Annual European Symposium on Algorithms (ESA 2022))

   Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness. 

   more » « less
3. Online Spanners in Metric Spaces

   Bhore, Sujoy; Filtser, Arnold; Khodabandeh, Hadi; Toth, Csaba D. (September 2022, Proc. 30th Annual European Symposium on Algorithms (ESA))

   Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness. 

   more » « less
4. Almost Sharp Bounds on the Number of Discrete Chains in the Plane

   Frankl, N; Kupavskii, A. (June 2020, Proceedings of 36th International Symposium on Computational Geometry (SoCG 2020))

   The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ=(δ₁,… ,δ_k) of k distances, a (k+1)-tuple (p₁,… ,p_{k+1}) of distinct points in ℝ^d is called a (k,δ)-chain if ‖p_j-p_{j+1}‖ = δ_j for every 1 ≤ j ≤ k. What is the maximum number C_k^d(n) of (k,δ)-chains in a set of n points in ℝ^d, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3. 

   more » « less
5. Negative Moments of L –Functions With Small Shifts Over Function Fields

   https://doi.org/10.1093/imrn/rnad118  

   Florea, Alexandra (June 2023, International Mathematics Research Notices)

   Abstract We consider negative moments of quadratic Dirichlet $$L$$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $$\mathbb{F}_{q}[x]$$, we obtain an asymptotic formula for the $$k^{\textrm{th}}$$ shifted negative moment of $$L(1/2+\beta ,\chi _{D})$$, in certain ranges of $$\beta $$ (e.g., when roughly $$\beta \gg \log g/g $$ and $k<1$). We also obtain non-trivial upper bounds for the $$k^{\textrm{th}}$$ shifted negative moment when $$\log (1/\beta ) \ll \log g$$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $$\beta \gg g^{-\frac{1}{2k}+\epsilon }$$. 

   more » « less