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Beyond Set Disjointness : The Communication Complexity of Finding the Intersection

Beyond Set Disjointness : The Communication Complexity of Finding the Intersection. Grigory Yaroslavtsev http://grigory.us. Joint with Brody, Chakrabarti , Kondapally and Woodruff. Communication Complexity [Yao’79]. Shared randomness. Bob: . Alice: . ….

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Beyond Set Disjointness : The Communication Complexity of Finding the Intersection

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  1. Beyond Set Disjointness: The Communication Complexity of Finding the Intersection GrigoryYaroslavtsev http://grigory.us Joint with Brody, Chakrabarti, Kondapally and Woodruff

  2. Communication Complexity [Yao’79] Shared randomness Bob: Alice: … • = min. communication (error ) • min. -round communication (error )

  3. Set Intersection = ? (-Intersection) = ? is big, n is huge, where huge big

  4. Our results Let • (-Intersection) = [Brody, Chakrabarti, Kondapally, Woodruff, Y.; PODC’14] • (-Intersection) = [Saglam-Tardos FOCS’13; Brody, Chakrabarti, Kondapally, Woodruff, Y.’; RANDOM’14] { times (-Intersection) = for

  5. Applications • ExactJaccard index (for -approximate use MinHash[Broder’98; Li-Konig’11; Path-Strokel-Woodruff’14]) • Rarity, distinct elements, joins,… • Multi-party set intersection (later) • Contrast:

  6. 1-round -protocol

  7. Hashing Expected # of elements =# of buckets

  8. Secondary Hashing where = # of hash functions

  9. 2-Round -protocol Total communication = = O()

  10. Collisions

  11. Collisions Key fact: If then also =

  12. Collisions • Second round: • For each bucket send -bit equality check (total -communication) • Correct intersection computed in buckets where • Expected # items in incorrect buckets • Use 1-round protocol for incorrect buckets • Total communication

  13. Main protocol Expected # of elements =# of buckets

  14. Verification tree -degree … buckets = leaves of the verification tree

  15. Verification bottom-up

  16. Verification bottom-up Incorrect Incorrect Correct EQUALITY CHECK Incorrect Correct

  17. Verification bottom-up EQUALITY CHECK FAILS => RESTART THE SUBTREE Incorrect Correct Correct Correct Incorrect Correct

  18. Verification bottom-up … … …

  19. Analysis of Stage • = [node at stage computed correctly] • Set = • Run equality checks and basic intersection protocols with success probability • Key lemma: [# of restarts per leaf=> Cost of Intersection in leafs = • Cost of Equality = • [protocol succeeds] =

  20. Multi-partyextensions players: , where • Boost error probability of 2-player protocol to • Average per player (using coordinator): in rounds • Worst-case per player (using a tournament) in rounds

  21. Open Problems • (-Intersection) =? • Better protocols for the multi-party setting?

  22. -Disjointness • , iff • [Razborov’92; Hastad-Wigderson’96] [Folklore + Dasgupta, Kumar, Sivakumar; Buhrman’12, Garcia-Soriano, Matsliah, De Wolf’12] • [Saglam, Tardos’13] • [Braverman, Garg, Pankratov, Weinstein’13]

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