Tomer Adar

Tomer Adar

We present an algorithm for simulating a distribution using prefix conditional samples (Adar, Fischer and Levi, 2024), as well as ``prefix-compatible'' conditional models such as the interval model (Cannone, Ron and Servedio, 2015) and the subcube model (CRS15, Bhattacharyya and Chakraborty, 2018). The sample complexity is $O(\log^2 N / \varepsilon^2)$ prefix conditional samples per query, which improves on the previously known $\tilde{O}(\log^3 N / \varepsilon^2)$ (Kumar, Meel and Pote, 2025). Moreover, our simulating distribution is $O(\varepsilon^2)$-close to the input distribution with respect to the Kullback-Leibler divergence, which is stricter than the usual guarantee of being $O(\varepsilon)$-close with respect to the total-variation distance. We show that our algorithm is tight with respect to the highly-related task of estimation: every algorithm that is able to estimate the mass of individual elements within $(1 \pm \varepsilon)$-multiplicative error must make $Ω(\log^2 N / \varepsilon^2)$ prefix conditional samples per element.

Tomer Adar

The $L_2$-norm, or collision norm, is a core entity in the analysis of distributions and probabilistic algorithms. Batu and Canonne (FOCS 2017) presented an extensive analysis of algorithmic aspects of the $L_2$-norm and its connection to uniformity testing. However, when it comes to estimating the $L_2$-norm itself, their algorithm is not always optimal compared to the instance-specific second-moment bounds, $O(1/(\varepsilon\|μ\|_2) + t_μ/\varepsilon^2)$, for $t_μ= \|μ\|_3^3 / \|μ\|_2^4 - 1$, as stated by Batu (WoLA 2025, open problem session). In this paper, we present an unbiased $L_2$-estimation algorithm whose sample complexity matches the instance-specific second-moment analysis. Additionally, we show that $Ω(1/(\varepsilon \|μ\|_2) + t_μ/ \varepsilon^2)$ is indeed the per-instance lower bound for estimating the norm of a distribution $μ$ by sampling (even for non-unbiased estimators).

Tomer Adar, Amit Levi

A central theme in sublinear graph algorithms is the relationship between counting and sampling: can the ability to approximately count a combinatorial structure be leveraged to sample it nearly uniformly at essentially the same cost? We study (i) independent-set (IS) queries, which return whether a vertex set $S$ is edge-free, and (ii) two standard local queries: degree and neighbor queries. Eden and Rosenbaum (SOSA `18) proved that in the local-query model, uniform edge sampling is no harder than approximate edge counting. We extend this phenomenon to new settings. We establish sampling-counting equivalence for the hybrid model that combines IS and local queries, matching the complexity of edge-count estimation achieved by Adar, Hotam and Levi (2026), and an analogous equivalence for IS queries, matching the complexity of edge-count estimation achieved by Xi, Levi and Waingarten (SODA `20). For each query model, we show lower bounds for uniform edge sampling that essentially coincide with the known bounds for approximate edge counting.