A seminal result of Ahn, Guha, and McGregor (PODS'12) showed that a cut sparsifier of an unweighted graph can be computed from a near-linear number of linear measurements. Later work extended this to spectral sparsifiers (Kapralov et al., FOCS'14) and to near-linear recovery time (Kapralov et al., SODA'20). For weighted graphs, these sketches rely on grouping edges by weight, which general turnstile streams do not allow. Chen, Khanna, and Li (FOCS'22) removed this limitation, using $\tilde{O}(n/\varepsilon^3)$ bits of space. We improve the space to $\tilde{O}(n/\varepsilon^2)$ bits. Our algorithm makes one pass over a general turnstile stream of a weighted graph, processes each update in $\tilde{O}(1)$ time, and recovers, in $\tilde{O}(n/\varepsilon^2)$ time, a reweighted subgraph with $O(n\varepsilon^{-2}\log^2 n)$ edges that preserves every cut within a factor of $1\pm\varepsilon$. Its space, update time, and recovery time are all optimal up to polylogarithmic factors.
Perfect $L_2$ sampling is a powerful primitive for turnstile data streams, allowing algorithms to extract an index with probability proportional to its squared frequency, with applications to dynamic join size estimation, query optimization, heavy-hitters, sampling from self-joins, and real-time monitoring of evolving datasets. The setting $p=2$ is also the largest $p$ for which $L_p$ sampling can be achieved in polylogarithmic space. However, existing space-optimal perfect samplers rely on explicit polynomial duplication, requiring inefficient polynomial update times per stream arrival. Resolving a open question left by Swartworth, Woodruff, and Zhou [FOCS 2025], we present a perfect $L_2$ sampler that achieves an expected $\text{polylog}(n)$ update time while maintaining an optimal $\tilde{\mathcal{O}}(\log^3 n)$ bits of space.
David P. Woodruff, Vincent Cohen-Addad, Lalit Jain, Jieming Mao, Song Zuo, MohammadHossein Bateni, Simina Branzei, Michael P. Brenner 等 36 位作者
Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.