Achieving Maximum Speedup in Multi-level Acceleration for Massive Coronavirus Testing

Achieving Maximum Speedup in Multi-level Acceleration for Massive Coronavirus Testing | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Achieving Maximum Speedup in Multi-level Acceleration for Massive Coronavirus Testing Keqin Li, Bo Yang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2679304/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Note: Please see pdf for full abstract with equations. It is well and widely known that sample pooling could provide an effective and efficient way for fast coronavirus testing among massive asymptomatic individuals. The method of multi-level acceleration for asymptomatic COVID-19 screening has been introduced, and for one and two levels, the optimal group sizes have been obtained. However, there are still multiple challenges. First, it is not clear how to find the optimal group sizes for three or more levels. Second, there is lack of closed-form expressions for the optimal group sizes for two or more levels. Third, it is not clear how to determine the optimal number of levels. And last, it is not known what the maximum achievable speedup is. The motivation of this paper is to address all the above challenges. The optimization of a hierarchical pooling strategy includes its number of levels and the group size of each level. In this paper, based on multi-variable optimization and Taylor approximation, we are able to derive closed-form expressions for the optimal number of levels d* = ln(1/ln(1/q 0 )) − 1 , the optimal group sizes m 1 * = e d* = 1/(ep 0 ), m 2 * = e d*−1 = 1/(e 2 p 0 ), ..., m d* * = e = 1/(e d* p 0 ) , and the maximum possible speedup of a hierarchical pooling strategy of 1/(ep 0 ln(1/p 0 )) , where p 0 is the fraction of infected people. The above speedup is nearly a linear function of the reciprocal of p 0 , in the sense that it is asymptotically greater than any sub-linear function (1/p 0 ) 1−ε of the reciprocal of p 0 for any small ε > 0 . Using the results in this paper, we can quickly and easily predict the performance of an optimal hierarchical pooling strategy. For instance, if the fraction of infected people is 0.0001, an 8-level hierarchical pooling strategy can achieve speedup of nearly 400. Epidemiology Operations Research Systems Engineering Computer Architecture and Engineering Asymptomatic screening coronavirus group test hierarchical pooling strategy multi-level acceleration optimization speedup Full Text Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-2679304","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":182533571,"identity":"0705ba2f-cde5-44bd-b9ab-2168aba08aac","order_by":0,"name":"Keqin Li","email":"","orcid":"https://orcid.org/0000-0001-5224-4048","institution":"State University of New York","correspondingAuthor":false,"prefix":"","firstName":"Keqin","middleName":"","lastName":"Li","suffix":""},{"id":182534732,"identity":"b57a0106-c112-43bd-ae47-543529b96907","order_by":1,"name":"Bo Yang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzElEQVRIiWNgGAWjYBACAygtB2GwkaDFmHQtiRuI1mLO3sAm8XNHbfp26R4Dhg9lhxn4Zzfg12LZc4BNsvfM8dydc84YMM44d5hB4s4BAg67kcAmwdt2LHfDjRwDZt62wwwGEgmEtUj+bTuWbgDS8pdYLdK8bTUJYC2MxGgB+oXZWrbtgOGGG2kFB3vOpfNI3CCgBRhijDffttXJG9xI3vjgR5m1HP8MAloYGPi/SDAwHAYzDwAxDyH1IMD8gYGhjhiFo2AUjIJRMFIBAHDvQwU00cVoAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0003-3782-9093","institution":"Hunan University of Finance and Economics","correspondingAuthor":true,"prefix":"","firstName":"Bo","middleName":"","lastName":"Yang","suffix":""}],"badges":[],"createdAt":"2023-03-10 20:25:30","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2679304/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2679304/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":34245996,"identity":"49977e32-643b-4c23-920e-883c681f4025","added_by":"auto","created_at":"2023-03-14 16:45:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":192260,"visible":true,"origin":"","legend":"","description":"","filename":"preprint.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2679304/v1/d2fbe8f3a6943651f0dfc5de.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003eAchieving Maximum Speedup in Multi-level Acceleration for Massive Coronavirus Testing\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"State University of New York at New Paltz","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Asymptomatic screening, coronavirus, group test, hierarchical pooling strategy, multi-level acceleration, optimization, speedup","lastPublishedDoi":"10.21203/rs.3.rs-2679304/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2679304/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eNote: Please see pdf for full abstract with equations.\u003c/p\u003e\n\u003cp\u003eIt is well and widely known that sample pooling could provide an effective and efficient way for fast coronavirus testing among massive asymptomatic individuals. The method of multi-level acceleration for asymptomatic COVID-19 screening has been introduced, and for one and two levels, the optimal group sizes have been obtained. However, there are still multiple challenges. First, it is not clear how to find the optimal group sizes for three or more levels. Second, there is lack of closed-form expressions for the optimal group sizes for two or more levels. Third, it is not clear how to determine the optimal number of levels. And last, it is not known what the maximum achievable speedup is. The motivation of this paper is to address all the above challenges. The optimization of a hierarchical pooling strategy includes its number of levels and the group size of each level. In this paper, based on multi-variable optimization and Taylor approximation, we are able to derive closed-form expressions for the optimal number of levels \u003cem\u003ed* = ln(1/ln(1/q\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e)) − 1\u003c/em\u003e, the optimal group sizes \u003cem\u003em\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e* = e\u003c/em\u003e\u003csup\u003e\u003cem\u003ed* \u003c/em\u003e\u003c/sup\u003e\u003cem\u003e= 1/(ep\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e), m\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e* = e\u003c/em\u003e\u003csup\u003e\u003cem\u003ed*−1\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e = 1/(e\u003c/em\u003e\u003csup\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sup\u003e\u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e), ..., m\u003c/em\u003e\u003csub\u003e\u003cem\u003ed*\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e* = e = 1/(e\u003c/em\u003e\u003csup\u003e\u003cem\u003ed*\u003c/em\u003e\u003c/sup\u003e\u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e)\u003c/em\u003e, and the maximum possible speedup of a hierarchical pooling strategy of \u003cem\u003e1/(ep\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003eln(1/p\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e))\u003c/em\u003e, where \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e \u003c/em\u003eis the fraction of infected people. The above speedup is nearly a linear function of the reciprocal of \u003cem\u003ep\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e, in the sense that it is asymptotically greater than any sub-linear function \u003cem\u003e(1/p\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e)\u003c/em\u003e\u003csup\u003e\u003cem\u003e1−ε\u003c/em\u003e\u003c/sup\u003e\u003csup\u003e \u003c/sup\u003eof the reciprocal of \u003cem\u003ep\u003c/em\u003e\u003csub\u003e0 \u003c/sub\u003efor any small \u003cem\u003eε \u0026gt; 0\u003c/em\u003e. Using the results in this paper, we can quickly and easily predict the performance of an optimal hierarchical pooling strategy. For instance, if the fraction of infected people is 0.0001, an 8-level hierarchical pooling strategy can achieve speedup of nearly 400.\u003c/p\u003e","manuscriptTitle":"Achieving Maximum Speedup in Multi-level Acceleration for Massive Coronavirus Testing","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-03-14 16:44:59","doi":"10.21203/rs.3.rs-2679304/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"396bdeb3-2711-45bc-b10c-ff43b0efe5fa","owner":[],"postedDate":"March 14th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":19831570,"name":"Epidemiology"},{"id":19831571,"name":"Operations Research"},{"id":19831572,"name":"Systems Engineering"},{"id":19831573,"name":"Computer Architecture and Engineering"}],"tags":[],"updatedAt":"2023-03-14T16:44:59+00:00","versionOfRecord":[],"versionCreatedAt":"2023-03-14 16:44:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-2679304","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2679304","identity":"rs-2679304","version":["v1"]},"buildId":"J0_U0BvcaRcwD8yVFaRlm","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}