[{"department":[{"_id":"MoHe"},{"_id":"GradSch"}],"quality_controlled":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2512.13105","open_access":"1"}],"date_published":"2026-01-07T00:00:00Z","abstract":[{"lang":"eng","text":"We present an exact fully-dynamic minimum cut algorithm that runs in 𝑛𝑜⁡(1) deterministic update time when the minimum cut size is at most 2Θ⁡(log3/4−𝑐⁡𝑛) for any 𝑐 >0, improving on the previous algorithm of Jin, Sun, and Thorup (SODA 2024) whose minimum cut size limit is (log⁡𝑛)𝑜⁡(1). Combined with graph sparsification, we obtain the first (1 +𝜖)-approximate fully-dynamic minimum cut algorithm on weighted graphs, for any 𝜖 ≥2−Θ⁡(log3/4−𝑐⁡𝑛), in 𝑛𝑜⁡(1) randomized update time.\r\nOur main technical contribution is a deterministic local minimum cut algorithm, which replaces the randomized LocalKCut procedure from El-Hayek, Henzinger, and Li (SODA 2025)."}],"publisher":"Society for Industrial and Applied Mathematics","OA_place":"repository","citation":{"ista":"El-Hayek A, Henzinger M, Li J. 2026. Deterministic and exact fully-dynamic minimum cut of superpolylogarithmic size in subpolynomial time. Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms vol. 2026, 613–663.","ama":"El-Hayek A, Henzinger M, Li J. Deterministic and exact fully-dynamic minimum cut of superpolylogarithmic size in subpolynomial time. In: <i>Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms</i>. Vol 2026. Society for Industrial and Applied Mathematics; 2026:613-663. doi:<a href=\"https://doi.org/10.1137/1.9781611978971.25\">10.1137/1.9781611978971.25</a>","chicago":"El-Hayek, Antoine, Monika Henzinger, and Jason Li. “Deterministic and Exact Fully-Dynamic Minimum Cut of Superpolylogarithmic Size in Subpolynomial Time.” In <i>Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms</i>, 2026:613–63. Society for Industrial and Applied Mathematics, 2026. <a href=\"https://doi.org/10.1137/1.9781611978971.25\">https://doi.org/10.1137/1.9781611978971.25</a>.","apa":"El-Hayek, A., Henzinger, M., &#38; Li, J. (2026). Deterministic and exact fully-dynamic minimum cut of superpolylogarithmic size in subpolynomial time. In <i>Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms</i> (Vol. 2026, pp. 613–663). Vancouver, Canada: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611978971.25\">https://doi.org/10.1137/1.9781611978971.25</a>","short":"A. El-Hayek, M. Henzinger, J. Li, in:, Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2026, pp. 613–663.","mla":"El-Hayek, Antoine, et al. “Deterministic and Exact Fully-Dynamic Minimum Cut of Superpolylogarithmic Size in Subpolynomial Time.” <i>Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms</i>, vol. 2026, Society for Industrial and Applied Mathematics, 2026, pp. 613–63, doi:<a href=\"https://doi.org/10.1137/1.9781611978971.25\">10.1137/1.9781611978971.25</a>.","ieee":"A. El-Hayek, M. Henzinger, and J. Li, “Deterministic and exact fully-dynamic minimum cut of superpolylogarithmic size in subpolynomial time,” in <i>Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms</i>, Vancouver, Canada, 2026, vol. 2026, pp. 613–663."},"arxiv":1,"conference":{"location":"Vancouver, Canada","start_date":"2026-01-11","end_date":"2026-01-14","name":"SODA: Symposium on Discrete Algorithms"},"type":"conference","title":"Deterministic and exact fully-dynamic minimum cut of superpolylogarithmic size in subpolynomial time","date_created":"2026-04-12T22:01:51Z","date_updated":"2026-05-04T11:36:47Z","volume":2026,"doi":"10.1137/1.9781611978971.25","_id":"21720","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"language":[{"iso":"eng"}],"acknowledgement":"Funded by the European union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MoDynStruct, No. 101019564) and the Austrian Science Fund (FWF) grant DOI 10.55776/I5982. For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.","oa_version":"Preprint","page":"613-663","month":"01","publication_identifier":{"issn":["1071-9040"],"eisbn":["9781611978971"],"eissn":["1557-9468"]},"project":[{"call_identifier":"H2020","_id":"bd9ca328-d553-11ed-ba76-dc4f890cfe62","grant_number":"101019564","name":"The design and evaluation of modern fully dynamic data structures"},{"name":"Static and Dynamic Hierarchical Graph Decompositions","_id":"bda196b2-d553-11ed-ba76-8e8ee6c21103","grant_number":"I05982"}],"ec_funded":1,"day":"07","author":[{"first_name":"Antoine","last_name":"El-Hayek","orcid":"0000-0003-4268-7368","full_name":"El-Hayek, Antoine","id":"888a098e-fcac-11ee-aff7-d347be57b725"},{"full_name":"Henzinger, Monika H","id":"540c9bbd-f2de-11ec-812d-d04a5be85630","orcid":"0000-0002-5008-6530","last_name":"Henzinger","first_name":"Monika H"},{"last_name":"Li","first_name":"Jason","full_name":"Li, Jason"}],"intvolume":"      2026","year":"2026","publication_status":"published","OA_type":"green","publication":"Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms","scopus_import":"1","article_processing_charge":"No","external_id":{"arxiv":["2512.13105"]}},{"project":[{"name":"The design and evaluation of modern fully dynamic data structures","_id":"bd9ca328-d553-11ed-ba76-dc4f890cfe62","call_identifier":"H2020","grant_number":"101019564"},{"name":"Efficient algorithms","_id":"34def286-11ca-11ed-8bc3-da5948e1613c","grant_number":"Z00422"},{"grant_number":"I05982","_id":"bda196b2-d553-11ed-ba76-8e8ee6c21103","name":"Static and Dynamic Hierarchical Graph Decompositions"},{"_id":"bd9e3a2e-d553-11ed-ba76-8aa684ce17fe","grant_number":"P33775","name":"Fast Algorithms for a Reactive Network Layer"}],"publication_identifier":{"issn":["1071-9040"],"isbn":["979-833131200-8"]},"ec_funded":1,"author":[{"id":"540c9bbd-f2de-11ec-812d-d04a5be85630","full_name":"Henzinger, Monika H","first_name":"Monika H","orcid":"0000-0002-5008-6530","last_name":"Henzinger"},{"full_name":"Upadhyay, Jalaj","last_name":"Upadhyay","first_name":"Jalaj"}],"day":"20","year":"2025","publication_status":"published","intvolume":"         5","OA_type":"green","scopus_import":"1","publication":"Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms","external_id":{"arxiv":["2412.02840"]},"article_processing_charge":"No","date_published":"2025-01-20T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2412.02840"}],"quality_controlled":"1","department":[{"_id":"MoHe"}],"publisher":"Association for Computing Machinery","abstract":[{"lang":"eng","text":"Differentially private weighted prefix sum under continual observation is a crucial component in the production-level deployment of private next-word prediction for Gboard, which, according to Google, has over a billion users. More specifically, Google uses a differentially private mechanism to sum weighted gradients in its private follow-the-regularized leader algorithm. Apart from efficiency, the additive error of the private mechanism is crucial as multiplied with the square root of the model’s dimension d (with d ranging up to 10 trillion, for example, Switch Transformers or M6-10T), it determines the accuracy of the learning system. So, any improvement in leading constant matters significantly in practice. In this paper, we show a novel connection between mechanisms for continual weighted prefix sum and a concept in representation theory known as the group matrix introduced in correspondence between Dedekind and Frobenius (Sitzungsber. Preuss. Akad. Wiss. Berlin, 1897) and generalized by Schur (Journal für die reine und angewandte Mathematik, 1904). To the best of our knowledge, this is the first application of group algebra in the analysis of differentially private algorithms. Using this connection, we analyze a class of matrix norms known as factorization norms that give upper and lower bounds for the additive error under general ℓp-norms of the matrix mechanism. This allows us to give 1. the first efficient factorization that matches the best-known non-constructive upper bound on the factorization norm by Mathias (SIAM Journal of Matrix Analysis and Applications, 1993) for the matrix used in Google’s deployment, and also improves on the previous best-known constructive bound of Fichtenberger, Henzinger, and Upadhyay (ICML 2023) and Henzinger, Upadhyay, and Upadhyay (SODA 2023); thereby, partially resolving an open question in operator theory, 2. the first upper bound on the additive error for a large class of weight functions for weighted prefix sum problems, including the sliding window matrix (Bolot, Fawaz, Muthukrishnan, Nikolov, and Taft (ICDT 2013). We also improve the bound on factorizing the striped matrix used for outputting a synthetic graph that approximates all cuts (Fichtenberger, Henzinger, and Upadhyay (ICML 2023)); 3. a general improved upper bound on the factorization norms that depend on algebraic properties of the weighted sum matrices and that applies to a more general class of weighting functions than the ones considered in Henzinger, Upadhyay, and Upadhyay (SODA 2024). Using the known connection between these factorization norms and the ℓp-error of continual weighted sum, we give an upper bound on the ℓp-error for the continual weighted sum problem for p ≥ 2."}],"conference":{"location":"New Orleans, LA, United States","start_date":"2025-01-12","name":"SODA: Symposium on Discrete Algorithms","end_date":"2025-01-15"},"arxiv":1,"citation":{"ista":"Henzinger M, Upadhyay J. 2025. Improved differentially private continual observation using group algebra. Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms vol. 5, 2951–2970.","ama":"Henzinger M, Upadhyay J. Improved differentially private continual observation using group algebra. In: <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Vol 5. Association for Computing Machinery; 2025:2951-2970. doi:<a href=\"https://doi.org/10.1137/1.9781611978322.95\">10.1137/1.9781611978322.95</a>","chicago":"Henzinger, Monika, and Jalaj Upadhyay. “Improved Differentially Private Continual Observation Using Group Algebra.” In <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 5:2951–70. Association for Computing Machinery, 2025. <a href=\"https://doi.org/10.1137/1.9781611978322.95\">https://doi.org/10.1137/1.9781611978322.95</a>.","apa":"Henzinger, M., &#38; Upadhyay, J. (2025). Improved differentially private continual observation using group algebra. In <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i> (Vol. 5, pp. 2951–2970). New Orleans, LA, United States: Association for Computing Machinery. <a href=\"https://doi.org/10.1137/1.9781611978322.95\">https://doi.org/10.1137/1.9781611978322.95</a>","ieee":"M. Henzinger and J. Upadhyay, “Improved differentially private continual observation using group algebra,” in <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, New Orleans, LA, United States, 2025, vol. 5, pp. 2951–2970.","short":"M. Henzinger, J. Upadhyay, in:, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, 2025, pp. 2951–2970.","mla":"Henzinger, Monika, and Jalaj Upadhyay. “Improved Differentially Private Continual Observation Using Group Algebra.” <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, vol. 5, Association for Computing Machinery, 2025, pp. 2951–70, doi:<a href=\"https://doi.org/10.1137/1.9781611978322.95\">10.1137/1.9781611978322.95</a>."},"OA_place":"repository","title":"Improved differentially private continual observation using group algebra","type":"conference","status":"public","_id":"19038","doi":"10.1137/1.9781611978322.95","date_updated":"2025-04-14T13:50:49Z","volume":5,"date_created":"2025-02-17T09:31:03Z","language":[{"iso":"eng"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"Monika Henzinger: This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (Grantagreement No. 101019564) and the Austrian Science Fund (FWF) grant DOI 10.55776/Z422,grant DOI 10.55776/I5982, and grant DOI 10.55776/P33775 with additional funding from the netidee SCIENCEStiftung, 2020–2024.Jalaj Upadhyay’s research was funded by the Rutgers Decanal Grant no. 302918 and an unrestricted giftfrom Google. This work was done in part while visiting the Institute of Science and Technology Austria (ISTA).The authors would like to thank Sarvagya Upadhyay for the initial discussion and feedback on the early draft of the paper. The authors would like to thank the anonymous reviewers, Brendan McMahan and Abhradeep Thakurta for the discussions that helped improve the presentation of the final version of the paper.","oa_version":"Preprint","month":"01","page":"2951 - 2970"}]