{"id":147974,"date":"2007-07-01T00:00:00","date_gmt":"2007-07-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/estimating-sum-by-weighted-sampling\/"},"modified":"2018-10-16T20:32:46","modified_gmt":"2018-10-17T03:32:46","slug":"estimating-sum-by-weighted-sampling","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/estimating-sum-by-weighted-sampling\/","title":{"rendered":"Estimating Sum by Weighted Sampling"},"content":{"rendered":"

We study the classic problem of estimating the sum of n<\/em> variables. The traditional uniform sampling approach requires a linear number of samples to provide any non-trivial guarantees on the estimated sum. In this paper we consider various sampling methods besides uniform sampling, in particular sampling a variable with probability proportional to its value, referred to as linear weighted sampling<\/em>. If only linear weighted sampling is allowed, we show an algorithm for estimating sum with O<\/mi>~<\/mo><\/mover><\/mrow>(<\/mo>n<\/mi><\/msqrt>)<\/mo><\/math>\">O<\/span>~<\/span><\/span><\/span><\/span>(<\/span>n<\/span><\/span>\u221a<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span> samples, and it is almost optimal in the sense that Ω<\/mi>(<\/mo>n<\/mi><\/msqrt>)<\/mo><\/math>\">\u03a9<\/span>(<\/span>n<\/span><\/span>\u221a<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span> samples are necessary for any reasonable sum estimator. If both uniform sampling and linear weighted sampling are allowed, we show a sum estimator with O<\/mi>~<\/mo><\/mover><\/mrow>(<\/mo>n<\/mi>3<\/mn><\/mroot>)<\/mo><\/math>\">O<\/span>~<\/span><\/span><\/span><\/span>(<\/span>n<\/span>\u221a3<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span> samples. More generally, we may allow general weighted sampling where the probability of sampling a variable is proportional to any function of its value. We prove a lower bound of Ω<\/mi>(<\/mo>n<\/mi>3<\/mn><\/mroot>)<\/mo><\/math>\">\u03a9<\/span>(<\/span>n<\/span>\u221a3<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span> samples for any reasonable sum estimator using general weighted sampling, which implies that our algorithm combining uniform and linear weighted sampling is an almost optimal sum estimator.<\/p>\n","protected":false},"excerpt":{"rendered":"

We study the classic problem of estimating the sum of n variables. The traditional uniform sampling approach requires a linear number of samples to provide any non-trivial guarantees on the estimated sum. In this paper we consider various sampling methods besides uniform sampling, in particular sampling a variable with probability proportional to its value, referred […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"International Colloquium on Automata, Languages and Programming, (ICALP)","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"53-64","msr_page_range_start":"53","msr_page_range_end":"64","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"International Colloquium on Automata, Languages and Programming, (ICALP)","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"Rajeev Motwani, Ying Xu 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