Proceedings ... annual Symposium on Foundations of Computer Science. Symposium on Foundations of Computer Science最新文献

{"title":"Pseudorandom Hashing for Space-bounded Computation with Applications in Streaming.","authors":"Praneeth Kacham, Rasmus Pagh, Mikkel Thorup, David P Woodruff","doi":"10.1109/focs57990.2023.00093","DOIUrl":"10.1109/focs57990.2023.00093","url":null,"abstract":"<p><p>We revisit Nisan's classical pseudorandom generator (PRG) for space-bounded computation (STOC 1990) and its applications in streaming algorithms. We describe a new generator, HashPRG, that can be thought of as a symmetric version of Nisan's generator over larger alphabets. Our generator allows a trade-off between seed length and the time needed to compute a given block of the generator's output. HashPRG can be used to obtain derandomizations with much better update time and <i>without sacrificing space</i> for a large number of data stream algorithms, for example: Andoni's <math> <mrow><msub><mi>F</mi> <mi>p</mi></msub> </mrow> </math> estimation algorithm for constant <math><mrow><mi>p</mi> <mo>></mo> <mn>2</mn></mrow> </math> (ICASSP, 2017) assumes a random oracle, but achieves optimal space and constant update time. Using HashPRG's time-space trade-off we eliminate the random oracle assumption while preserving the other properties. Previously no time-optimal derandomization was known. Using similar techniques, we give an algorithm for a relaxed version of <math> <mrow><msub><mo>ℓ</mo> <mi>p</mi></msub> </mrow> </math> sampling in a turnstile stream. Both of our algorithms use <math> <mrow><mover><mi>O</mi> <mo>˜</mo></mover> <mfenced> <mrow><msup><mi>d</mi> <mrow><mn>1</mn> <mo>-</mo> <mn>2</mn> <mo>/</mo> <mi>p</mi></mrow> </msup> </mrow> </mfenced> </mrow> </math> bits of space and have <math><mrow><mi>O</mi> <mfenced><mn>1</mn></mfenced> </mrow> </math> update time.For <math><mrow><mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn></mrow> </math> , the <math><mrow><mn>1</mn> <mo>±</mo> <mi>ε</mi></mrow> </math> approximate <math> <mrow><msub><mi>F</mi> <mi>p</mi></msub> </mrow> </math> estimation algorithm of Kane et al., (STOC, 2011) uses an optimal <math><mrow><mi>O</mi> <mfenced> <mrow><msup><mi>ε</mi> <mrow><mo>-</mo> <mn>2</mn></mrow> </msup> <mspace></mspace> <mtext>log</mtext> <mspace></mspace> <mi>d</mi></mrow> </mfenced> </mrow> </math> bits of space but has an update time of <math><mrow><mi>O</mi> <mfenced> <mrow> <msup><mrow><mtext>log</mtext></mrow> <mn>2</mn></msup> <mfenced><mrow><mn>1</mn> <mo>/</mo> <mi>ε</mi></mrow> </mfenced> <mtext>log</mtext> <mspace></mspace> <mtext>log</mtext> <mfenced><mrow><mn>1</mn> <mo>/</mo> <mi>ε</mi></mrow> </mfenced> </mrow> </mfenced> </mrow> </math> . Using HashPRG, we show that if <math><mrow><mn>1</mn> <mo>/</mo> <msqrt><mi>d</mi></msqrt> <mo>≤</mo> <mi>ε</mi> <mo>≤</mo> <mn>1</mn> <mo>/</mo> <msup><mi>d</mi> <mi>c</mi></msup> </mrow> </math> for an arbitrarily small constant <math><mrow><mi>c</mi> <mo>></mo> <mn>0</mn></mrow> </math> , then we can obtain a <math><mrow><mn>1</mn> <mo>±</mo> <mi>ε</mi></mrow> </math> approximate <math> <mrow><msub><mi>F</mi> <mi>p</mi></msub> </mrow> </math> estimation algorithm that uses the optimal <math><mrow><mi>O</mi> <mfenced> <mrow><msup><mi>ε</mi> <mrow><mo>-</mo> <mn>2</mn></mrow> </msup> <mspace></mspace> <mtext>log</mtext> <mspace></mspac","PeriodicalId":93353,"journal":{"name":"Proceedings ... annual Symposium on Foundations of Computer Science. Symposium on Foundations of Computer Science","volume":"2023 ","pages":"1515-1550"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12309723/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144755343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Near Optimal Linear Algebra in the Online and Sliding Window Models.","authors":"Vladimir Braverman,&nbsp;Petros Drineas,&nbsp;Cameron Musco,&nbsp;Christopher Musco,&nbsp;Jalaj Upadhyay,&nbsp;David P Woodruff,&nbsp;Samson Zhou","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>We initiate the study of numerical linear algebra in the sliding window model, where only the most recent <i>W</i> updates in a stream form the underlying data set. Although many existing algorithms in the sliding window model use or borrow elements from the smooth histogram framework (Braverman and Ostrovsky, FOCS 2007), we show that many interesting linear-algebraic problems, including spectral and vector induced matrix norms, generalized regression, and lowrank approximation, are not amenable to this approach in the row-arrival model. To overcome this challenge, we first introduce a unified row-sampling based framework that gives <i>randomized</i> algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and <i>ℓ</i> ₁-subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime. Our algorithms are based on \"reverse online\" versions of offline sampling distributions such as (ridge) leverage scores, <i>ℓ</i> ₁ sensitivities, and Lewis weights to quantify both the importance and the recency of a row; our structural results on these distributions may be of independent interest for future algorithmic design. Although our techniques initially address numerical linear algebra in the sliding window model, our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation. Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara <i>et al.</i> (FOCS 2019). We also give the first online algorithm for <i>ℓ</i> ₁-subspace embeddings. We further formalize the connection between the online model and the sliding window model by introducing an <i>additional</i> unified framework for <i>deterministic</i> algorithms using a merge and reduce paradigm and the concept of online coresets, which we define as a weighted subset of rows of the input matrix that can be used to compute a good approximation to some given function on all of its prefixes. Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and <i>ℓ</i> _1-subspace embeddings in the sliding window model that use nearly optimal space.</p>","PeriodicalId":93353,"journal":{"name":"Proceedings ... annual Symposium on Foundations of Computer Science. Symposium on Foundations of Computer Science","volume":"1 ","pages":"517-528"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8375632/pdf/nihms-1696963.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39333871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}