{"paper":{"title":"Improved Bounds and Schemes for the Declustering Problem","arxiv_id":"cs/0603012","primary_cat":"cs.DM","submitted_at":"2006-03-02T15:51:09+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"The declustering problem is to allocate given data on parallel working storage devices in such a manner that typical requests find their data evenly distributed on the devices.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.05,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.76,"id":"s1","own":true,"role":"contribution","text":"Using deep results from discrepancy theory, we improve previous work of several authors concerning range queries to higher-dimensional data.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.22,"result":0.01,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.76,"interpretation":0.0}},{"i":2,"p":0.67,"id":"s2","own":true,"role":"result","text":"We give a declustering scheme with an additive error of $O_d(\\log^{d-1} M)$ independent of the data size, where $d$ is the dimension, $M$ the number of storage devices and $d-1$ does not exceed the smallest prime power in the canonical decomposition of $M$ into prime powers.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.69,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.67,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.33,"interpretation":0.0}},{"i":3,"p":0.81,"id":"s3","own":true,"role":"result","text":"In particular, our schemes work for arbitrary $M$ in dimensions two and three.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.74,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.81,"background":0.0,"limitation":0.04,"future_work":0.0,"contribution":0.09,"interpretation":0.06}},{"i":4,"p":0.64,"id":"s4","own":true,"role":"result","text":"For general $d$, they work for all $M\\geq d-1$ that are powers of two.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.92,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.64,"background":0.0,"limitation":0.3,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":5,"p":0.72,"id":"s5","own":true,"role":"result","text":"Concerning lower bounds, we show that a recent proof of a $\\Omega_d(\\log^{\\frac{d-1}{2}} M)$ bound contains an error.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.03,"method":0.0,"result":0.72,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.25,"interpretation":0.0}},{"i":6,"p":0.53,"id":"s6","own":true,"role":"contribution","text":"We close the gap in the proof and thus establish the bound.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.17,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.01,"method":0.02,"result":0.44,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.53,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.78,"text":"We give a declustering scheme with an additive error of $O_d(\\log^{d-1} M)$ independent of the data size, where $d$ is the dimension, $M$ the number of storage devices and $d-1$ does not exceed the smallest prime power in the canonical decomposition of $M$ into prime 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