{"alternative_title":["LNCS"],"quality_controlled":"1","title":"Automated tail bound analysis for probabilistic recurrence relations","status":"public","related_material":{"link":[{"url":"https://github.com/boyvolcano/PRR","relation":"software"}]},"publisher":"Springer Nature","scopus_import":"1","oa_version":"Published Version","oa":1,"date_created":"2023-09-10T22:01:12Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"KrCh"}],"abstract":[{"lang":"eng","text":"Probabilistic recurrence relations (PRRs) are a standard formalism for describing the runtime of a randomized algorithm. Given a PRR and a time limit κ, we consider the tail probability Pr[T≥κ], i.e., the probability that the randomized runtime T of the PRR exceeds κ. Our focus is the formal analysis of tail bounds that aims at finding a tight asymptotic upper bound u≥Pr[T≥κ]. To address this problem, the classical and most well-known approach is the cookbook method by Karp (JACM 1994), while other approaches are mostly limited to deriving tail bounds of specific PRRs via involved custom analysis.\r\nIn this work, we propose a novel approach for deriving the common exponentially-decreasing tail bounds for PRRs whose preprocessing time and random passed sizes observe discrete or (piecewise) uniform distribution and whose recursive call is either a single procedure call or a divide-and-conquer. We first establish a theoretical approach via Markov’s inequality, and then instantiate the theoretical approach with a template-based algorithmic approach via a refined treatment of exponentiation. Experimental evaluation shows that our algorithmic approach is capable of deriving tail bounds that are (i) asymptotically tighter than Karp’s method, (ii) match the best-known manually-derived asymptotic tail bound for QuickSelect, and (iii) is only slightly worse (with a loglogn factor) than the manually-proven optimal asymptotic tail bound for QuickSort. Moreover, our algorithmic approach handles all examples (including realistic PRRs such as QuickSort, QuickSelect, DiameterComputation, etc.) in less than 0.1 s, showing that our approach is efficient in practice."}],"date_published":"2023-07-17T00:00:00Z","publication_status":"published","month":"07","article_processing_charge":"Yes (in subscription journal)","date_updated":"2023-09-20T08:25:57Z","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"type":"conference","file":[{"date_updated":"2023-09-20T08:24:47Z","file_size":624647,"date_created":"2023-09-20T08:24:47Z","success":1,"file_id":"14348","checksum":"42917e086f8c7699f3bccf84f74fe000","relation":"main_file","creator":"dernst","access_level":"open_access","file_name":"2023_LNCS_Sun.pdf","content_type":"application/pdf"}],"day":"17","project":[{"call_identifier":"H2020","grant_number":"863818","_id":"0599E47C-7A3F-11EA-A408-12923DDC885E","name":"Formal Methods for Stochastic Models: Algorithms and Applications"}],"ddc":["000"],"file_date_updated":"2023-09-20T08:24:47Z","acknowledgement":"We thank Prof. Bican Xia for valuable information on the exponential theory of reals. The work is partially supported by the National Natural Science Foundation of China (NSFC) with Grant No. 62172271, ERC CoG 863818 (ForM-SMArt), the Hong Kong Research Grants Council ECS Project Number 26208122, the HKUST-Kaisa Joint Research Institute Project Grant HKJRI3A-055 and the HKUST Startup Grant R9272.","year":"2023","intvolume":"     13966","volume":13966,"_id":"14318","language":[{"iso":"eng"}],"conference":{"end_date":"2023-07-22","location":"Paris, France","name":"CAV: Computer Aided Verification","start_date":"2023-07-17"},"citation":{"mla":"Sun, Yican, et al. “Automated Tail Bound Analysis for Probabilistic Recurrence Relations.” <i>Computer Aided Verification</i>, vol. 13966, Springer Nature, 2023, pp. 16–39, doi:<a href=\"https://doi.org/10.1007/978-3-031-37709-9_2\">10.1007/978-3-031-37709-9_2</a>.","ieee":"Y. Sun, H. Fu, K. Chatterjee, and A. K. Goharshady, “Automated tail bound analysis for probabilistic recurrence relations,” in <i>Computer Aided Verification</i>, Paris, France, 2023, vol. 13966, pp. 16–39.","ama":"Sun Y, Fu H, Chatterjee K, Goharshady AK. Automated tail bound analysis for probabilistic recurrence relations. In: <i>Computer Aided Verification</i>. Vol 13966. Springer Nature; 2023:16-39. doi:<a href=\"https://doi.org/10.1007/978-3-031-37709-9_2\">10.1007/978-3-031-37709-9_2</a>","ista":"Sun Y, Fu H, Chatterjee K, Goharshady AK. 2023. Automated tail bound analysis for probabilistic recurrence relations. Computer Aided Verification. CAV: Computer Aided Verification, LNCS, vol. 13966, 16–39.","chicago":"Sun, Yican, Hongfei Fu, Krishnendu Chatterjee, and Amir Kafshdar Goharshady. “Automated Tail Bound Analysis for Probabilistic Recurrence Relations.” In <i>Computer Aided Verification</i>, 13966:16–39. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/978-3-031-37709-9_2\">https://doi.org/10.1007/978-3-031-37709-9_2</a>.","apa":"Sun, Y., Fu, H., Chatterjee, K., &#38; Goharshady, A. K. (2023). Automated tail bound analysis for probabilistic recurrence relations. In <i>Computer Aided Verification</i> (Vol. 13966, pp. 16–39). Paris, France: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-031-37709-9_2\">https://doi.org/10.1007/978-3-031-37709-9_2</a>","short":"Y. Sun, H. Fu, K. Chatterjee, A.K. Goharshady, in:, Computer Aided Verification, Springer Nature, 2023, pp. 16–39."},"ec_funded":1,"publication_identifier":{"isbn":["9783031377082"],"eissn":["1611-3349"],"issn":["0302-9743"]},"author":[{"full_name":"Sun, Yican","first_name":"Yican","last_name":"Sun"},{"last_name":"Fu","first_name":"Hongfei","full_name":"Fu, Hongfei"},{"orcid":"0000-0002-4561-241X","full_name":"Chatterjee, Krishnendu","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","last_name":"Chatterjee","first_name":"Krishnendu"},{"orcid":"0000-0003-1702-6584","last_name":"Goharshady","first_name":"Amir Kafshdar","id":"391365CE-F248-11E8-B48F-1D18A9856A87","full_name":"Goharshady, Amir Kafshdar"}],"publication":"Computer Aided Verification","has_accepted_license":"1","doi":"10.1007/978-3-031-37709-9_2","page":"16-39"}