Advanced Algorithms for Big Data: Streaming Slides Summary
Explore advanced algorithms for big data in Lecture 4 focusing on streaming data, 0-sampling, sparse recovery, and Count-Min technique reevaluation. Dive into proofs, algorithms, and unique elements in the big data landscape.
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CIS 700: algorithms for Big Data Lecture 4: Streaming Slides at http://grigory.us/big-data-class.html Grigory Yaroslavtsev http://grigory.us
0-sampling Maintain Hash items using ?: ? 0,2? 1 for ? log? For each ?, maintain: ??= 1 0.1 |{?| ?? = 0}| ?0, and 1 0.1 -approximation to ?0. ??= ???? ?, ? ?=0 ??= ?? ?, ?? =0 ?0 there is a unique element Lemma: At level ? = 2 + log in the streams that maps to 0 (with constant probability) Uniqueness is verified if ??= 1 0.1. If so, then output ??/?? as the index and ?? as the count.
Proof of Lemma Let ? = log For any ?,Pr ?? = 0 = Probability there exists a unique ? such that ?? = 0, ?0 and note that 2?0< 2?< 12 ?0 1 2? Pr ?? = 0 ??? ? ?, ?? 0 ? = Pr ?? = 0 Pr ? ?, ?? 0 ? ? = 0] ? Pr ?? = 0 1 Pr ?? = 0 ?? = 0 ? ? ? 1 2?1 ?0 1 24 = Pr ?? = 0 1 Pr ?? = 0 2? ? ? ? ? Holds even if ? are only 2-wise independent
Sparse Recovery Goal: Find ? such that ? ? among ? s with at most ? non-zero entries. Definition: ????? = 1 is minimized min g: g ? ? 1 0 ? Exercise: ????? = ? ??? where ? are indices of ? largest ?? Using ? ? 1?log? space we can find ? such that ? 0 ? and ? ? 1 1 + ? ????(?)
Count-Min Revisited Use Count-Min with ? = ? log? ,? = 4?/? For ? ? , let ??= ??, ?? for some row ? ? Let ? = {?1, ,??} be the indices with max. frequencies. Let ?? be the event there doesn t exist ? ?/? with ?? = ?(?) Then for ? ? : Pr ?? ?? ?????? = ? Pr not ?? Pr ?? ?? ?????? ??? ?? + ? Pr ?? Pr ?? ?? ?????? ?? ? ?? ? ?+1 4 1 Pr not ?? + Pr ?? ?? ?????? ? 2 Because ? = ?(log ?) w.h.p. all ?? s approx . up to ?????? ?
Sparse Recovery Algorithm Use Count-Min with ? = ? log? ,? = 4?/? Let ? = ( ?1, ?2, , ??) be frequency estimates: ?? ?? ?????(?) ? Let ? be ? with all but the k-th largest entries replaced by 0. Lemma: ? ? 1 1 + 3 ? ????(?)
1 1 + 3 ? ????(?) ? ? Let ?,? ? be indices corresponding to k largest values of ? and ? . For a vector ? ? and ? [?] denote as ?? the vector formed by zeroing out all entries of ? except for those in ?. ? ?? ? ?? = ? 1 ?? = ? 1 ?? ?? ? 1 ?? ? 1+ 2 ?? ?? ? 1 ?? ?? ? ?? ????? + ? ?????? 1 + 3 ? ????(?) 1 1+ ?? ?? 1+ ?? ?? 1 ?? 1+ 2 ?? ?? 1 1 1+ 1+ ?? ?? 1 1 1+ 2 ?? ?? 1+ 2 ?? ?? ????? 1 1 ?? 1+ 1+ ?? ?? 1 ?? 1 1 + 2? ? ? ?
Count Sketch [Charikar, Chen, Farach-Colton] In addition to ??: ? [?] use random signs ??? 1,1 ??,?= ??? ?? ?:??? =? Estimate: ??= ??????(?1(?)?1,?1?, ,??? ??,??(?)) Parameters: ? = ? log1 ? ,? = Pr |?? ??| + ?| ? |2 1 ? Lemma: E[??(?)??,???] = ?? Lemma: Var[??(?)??,???] ?2 By Chebyshev: Pr |??? ??,??? ?? ? ?2] 1/3 By Chernoff with ? = ? log1 3 ?? ? ? error prob. 1 ?.
Count Sketch Analysis Fix ? and ?. Let ??= ? ? ? = ? ? : ? ? ??(?)= ? ? ? ? ???? ? Lemma: E[??(?)??,???] = ?? E[? ? ??(?)] = E[fx+ ? ?? ? ? ? ? ? ?? ] = ?? Lemma: Var[??(?)??,???] ?2 Var[? ? ??(?)] ?[( ?? ? ? ? ????)2] =?[ ???2?? ?2/? ? 2+ ( ? ?? ? ? ? ????????) ] =
