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Nearly optimal separations between Communication (or query) complexity and partitions

Nearly optimal separations between Communication (or query) complexity and partitions. Andris Ambainis (Latvia), Martins Kokainis (Latvia), Robin Kothari (MIT). communication complexity. Alice holds x X , Bob holds y Y .

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Nearly optimal separations between Communication (or query) complexity and partitions

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  1. Nearly optimal separations between Communication (or query) complexity and partitions Andris Ambainis (Latvia), Martins Kokainis (Latvia), Robin Kothari (MIT)

  2. communication complexity • Alice holds xX, Bob holds yY. • Dcc(f) - minimum amount of communication to compute F(x, y).

  3. communication complexity • k bit protocol  2k possible transcripts. • Transcript = rectangle XiYi, Xi, Yi– sets of inputs for Alice and Bob. k bit protocol  partition of XY into 2k rectangles, F constant on each rectangle

  4. Not all partitions into 2k rectangles correspond to k bit communication protocols. Partition  communication protocol?

  5. Communication vs. Partition number • (F) – min number of rectangles XiYi in a partition with F constant on every rectangle. • Dcc(F)  log (F) [Yao, 1979]. • Dcc(F) = O(log2(F)) [Aho, Ullman, Yannakakis, 1983]. F: Dcc(F)  2 log (F) [Kushilevitz, Linial, Ostrovsky, 1999] F: Dcc(F) = (log1.5(F)) [Goos, Pitassi, Watson, 2015]

  6. x2 1 0 1 x3 1 0 0 1 QUERY COMPLEXITY x1 • Task: compute F(x1, ..., xN), xi{0, 1}. • Operation: queries to variables xi. 1 0 1

  7. x2 1 0 1 x3 1 0 0 1 QUERY COMPLEXITY Decision tree with k queries x1 1 0 1 Partition of {0, 1}n into subcubes Si of dim  n-k with F constant on each Si

  8. Queries vs. partitions • D(F) – deterministic query complexity; • UC(F) – smallest k with a partition of {0, 1}n into subcubes defined by fixing k bits. • k  D(F)  k2; F: D(F) = (UC1.5(F)) [Goos, Pitassi, Watson, 2015]

  9. Our separations • Deterministic query: D(F)=(UC2-o(1)(F)); • Deterministic communication: Dcc(F) = (log2-o(1)(F)); • Query result + lifting [GPW15]. • Randomized query: R(F)=(UC2-o(1)(F)); • Quantum query: Q(F)=(UC1.5-o(1)(F));

  10. certificates • Certificate = partial assignment C such that: • x1 ... xnsatisfies C  F(x1... xn)=a. • OR(x1, x2, ... xN): • 1-certificates: 1***, *1**, ..., ***1. • 0-certificates: 0000.

  11. Subcubes vs. certificates • Certificate  subcube consisting of all x1 ... xn that satisfy C. • Partition of {0, 1}ninto subcubes  collection of certificates with each x1 ... xnsatisfying exactly one C. Unambiguous certificates.

  12. Notation • UC(F) – smallest k such that there is an unambiguous collection of C covering {0, 1}n. • UCa(F) – smallest k such that there is an unambiguous collection of C covering x:F(x)=a. • C(F) – smallest k such that there is a collection of C covering {0, 1}n.

  13. Cheat-sheets • [Aaronson, Ben-David, Kothari, STOC’16]. • Show R(F) = (Q2.5-o(1)(F)). • Number of other separations.

  14. Cheat-sheets (Part 1) 0 1 0 ... 0 1 0 F(x(1)) F(x(2)) 0 0 0 ... 0 1 1 k a1a2... ak ... 1 1 1 ... 0 0 0 F(x(k))

  15. Cheat-sheets (part 2) C1, C2, ..., Ck 0...00 0 1 0 ... 0 1 0 C1, C2, ..., Ck 0...01 0 0 0 ... 0 1 1 2k ... Ci – certificate for F(x(i))=ai 1...11 1 1 1 ... 0 0 0

  16. Cheat-sheets • Fcs=1 if: • F(x(1))=a1, ..., F(x(k))=ak; • Cheat-sheet a1...akcontains descriptions of correct certificates for F(x(1))=a1, ..., F(x(k))=ak.

  17. Certificates for fcs=1 0 1 0 ... 0 1 0 0 0 0 ... 0 1 1 C1, C2, ..., Ck UC1 kC ... cheat sheet 1 1 1 ... 0 0 0 Cheat-sheets decrease UC1!

  18. Other complexity measures • UC1 (FCS)  k C(F); • UC0 (FCS)  k UC(F); • C(FCS)  k C(F); • D(FCS)  k D(F); • Typically, k  log N, can be omitted. decreases does not decrease

  19. STAGE 1 Cheat-sheet ORn ANDn ... ... ANDn ANDn

  20. STAGE k, K>1 Cheat-sheet ORn ... ANDn ... ANDn ANDn F F F F F F F F F

  21. D C0 C1 UC0 UC1 AND n 1 n n n OR n2 n n n2 n2 cheat-sheet n2 n n n2n

  22. D C0 C1 UC0 UC1 AND n2i+1 ni ni+1 ni+1 ni+1 cheat-sheet n2i ni ni ni+1ni OR n2i+2 ni+1 ni+1 ni+2 ni+2 cheat-sheet n2i+2 ni+1 ni+1 ni+2ni+1

  23. Result • D  n2i, UC  ni+1. • TheoremD(F) = (UC2-o(1)(F)). R(F) = (UC2-o(1)(F)) follows similarly.

  24. Quantum results • TheoremQ(F)=(UC1.5-o(1)(F)). • TheoremQ(F)=(UC12-o(1)(F)). • Cheat-sheet+OR+BKK (instead of AND).

  25. Conclusion • Almost quadratic gap between partitions and deterministic communication/query complexity. • Simpler construction? • Gap for randomized communication vs. partitions? • Quantum?

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