Boolean circuits, tensor ranks, and communication complexity

SIAM Journal on Computing

Volumn 26, Issue 3, 1997, Pages 605-633

  Pudlák, Pavel ^a   Rödl, Vojtěch ^b   Sgall, Jiří ^a,c  

^a MATHEMATICAL INSTITUTE   (Czech Republic)

^b EMORY UNIVERSITY   (United States)

^c Carnegie Mellon University ^*  (United States)

Author keywords

Circuit; Communication complexity; Random graph; Tensor rank

Indexed keywords

EID: 0000362179     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539794264809     Document Type: Article

References (34)

1. M. AJTAI, J. KOMLÓS, AND E. SZEMERÉDI, A note on Ramsey numbers, J. Combin. Theory Ser. A, 29 (1980), pp. 354-360.
2. N. ALON AND P. PUDLÁK, Superconcentrators of depth 2 and 3: Odd levels help (rarely), J. Comput. System Sci., 48 (1994), pp. 194-202.
3. N. ALON AND J. H. SPENCER, The Probabilistic Method, John Wiley, New York, 1992.
4. A. AMBAINIS, Upper bounds on multiparty communication complexity of shifts, in Proc. 13th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 1046, Springer-Verlag, Berlin, 1996, pp. 631-642.
5. K. AZUMA, Weighted sums of certain dependent random variables, Tôhoku Math. J., 3 (1967), pp. 357-367.
6. L. BABAI, P. KIMMEL, AND S. V. LOKAM, Simultaneous messages vs. communication, in Proc. 12th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 900, Springer-Verlag, Berlin, 1995, pp. 361-372.
7. L. BABAI, N. NISAN, AND M. SZEGEDY, Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs, J. Comput. System Sci., 45 (1992), pp. 204-232.
8. R. BEIGEL AND J. TARUI, On ACC, Comput. Complexity, 4 (1994), pp. 350-366.
9. B. BOLLOBÁS, The chromatic number of random graphs, Combinatorica, 8 (1988), pp. 49-55.
10. B. BOLLOBÁS, Random Graphs, Academic Press, New York, 1985.
11. C. DAMM, S. JUKNA, AND J. SGALL, Some bounds for multiparty communication complexity of pointer jumping, Comput. Complexity, to appear.
12. H. F. DE GROOTE, Lectures on the Complexity of Bilinear Problems, Lecture Notes in Comput. Sci. 245, Springer-Verlag, Berlin, 1987.
13. D. DOLEV, C. DWORK, N. PIPPENGER, AND A. WIGDERSON, Superconcentrators, generalizers and generalized connectors with limited depth, in Proc. 15th Annual ACM Symposium on Theory of Computing, ACM, New York, 1983.
14. J. FRIEDMAN, A note on matrix rigidity, Combinatorica, 13 (1993), pp. 235-239.
15. J. HÅSTAD AND M. GOLDMANN, On the power of small-depth threshold circuits, Comput. Complexity, 1 (1991), pp. 113-129.
16. A. K. CHANDRA, M. L. FURST, AND R. J. LIPTON, Multi-party protocols, in Proc. 15th Annual ACM Symposium on Theory of Computing, ACM, New York, 1983, pp. 94-99.
17. D. V. CHUDNOVSKY AND G. V. CHUDNOVSKY, Algebraic complexities and algebraic curves over finite fields, Proc. Nat. Acad. Sci. U.S.A., 84 (1987), pp. 1739-1743.
18. S. JANSON, T. LUCZAK, AND A. RUCIŃSKI, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, in Random Graphs '87, M. Karoński et al., eds., John Wiley, New York, 1990, pp. 73-87.
19. E. KUSHILEVITZ AND N. NISAN, Communication Complexity, Cambridge University Press, Cambridge, UK, to appear.
20. N. NISAN AND A. WIGDERSON, Rounds in communication complexity revisited, SIAM J. Comput., 22 (1993), pp. 211-219.
21. N. PIPPENGER, The complexity of computations by networks, IBM J. Res. Develop., 31 (1987), pp. 235-243.
22. N. PIPPENGER AND L. G. VALIANT, Shifting graphs and their applications, J. Assoc. Comput. Mach., 23 (1976), pp. 423-432.
23. N. PIPPENGER AND A. C.-C. YAO, Rearrangeable networks with limited depth, SIAM J. Algebraic Discrete Meth., 3 (1982), pp. 411-417.
24. P. PUDLÁK, Communication in bounded depth circuits, Combinatorica, 14 (1994), pp. 203-216.
25. P. PUDLÁK AND V. RÖDL, Modified ranks of tensors and the size of circuits, in Proc. 25th Annual ACM Symposium on Theory of Computing, ACM, New York, 1993, pp. 523-531.
26. P. PUDLÁK AND P. SAVICKÝ, On shifting networks, Theoret. Comput. Sci., 116 (1993), pp. 415-419.
27. A. A. RAZBOROV, On rigid matrices, preprint, Mathematical Institute, Academy of Sciences of USSR, Moscow, 1989 (in Russian).
28. J. SGALL, A note on multiparty communication complexity of shifts, manuscript, 1992.
29. M. A. SHOKROLLAHI, Beiträge zur codierungs- und komplexitätstheorie mittels algebraischer funktionkörper, Bayreuth. Math. Schr., 39 (1991).
30. V. SHOUP AND R. SMOLENSKY, Lower bounds for polynomial evaluation and interpolation, in Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1991, pp. 378-383.
31. V. STRASSEN, Vermeidung von divisionen, Crelles J. Reine Angew. Math., 264 (1973), pp. 184-202.
32. L. G. VALIANT, Graph-theoretic arguments in low level complexity, in Proc. 6th Mathematical Foundations of Computer Sci., Lecture Notes in Comput. Sci. 53, Springer-Verlag, Berlin, 1977, pp. 162-176.
33. L. G. VALIANT, Why is Boolean complexity theory difficult?, in Boolean Function Complexity, M. S. Paterson, ed., Cambridge University Press, Cambridge, UK, 1992, pp. 84-94.
34. A. C.-C. YAO, On ACC and threshold circuits, in Proc. 31st Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1990, pp. 619-627.