The cost of the missing bit: communication complexity with help
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
Randomized Simultaneous Messages: Solution Of A Problem Of Yao In Communication Complexity
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
On distributing symmetric streaming computations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Lower bounds for depth-2 and depth-3 Boolean circuits with arbitrary gates
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
On distributing symmetric streaming computations
ACM Transactions on Algorithms (TALG)
Min-rank conjecture for log-depth circuits
Journal of Computer and System Sciences
Hadamard tensors and lower bounds on multiparty communication complexity
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
NOF-multiparty information complexity bounds for pointer jumping
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Hadamard tensors and lower bounds on multiparty communication complexity
Computational Complexity
Optimal Collapsing Protocol for Multiparty Pointer Jumping
Theory of Computing Systems
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We investigate two methods for proving lower bounds on the size of small-depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that our main contribution concerns the algebraic concepts used in this area (tensor ranks and rigidity). Our main results are following.(i) An $o(n)$-bit protocol for a communication game for computing shifts, which also gives an upper bound of $o(n^2)$ on the contact rank of the tensor of multiplication of polynomials; this disproves some earlier conjectures. A related probabilistic construction gives an $o(n)$ upper bound for computing all permutations and an $O(n\log\log n)$ upper bound on the communication complexity of pointer jumping with permutations. (ii) A lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of logarithmic-depth circuits; this bound has interpretations both as a lower bound on the rigidity of the tensor of multiplication of polynomials and as a lower bound on the communication needed to compute the shift function in a restricted model. (iii) An upper bound on Boolean circuits of depth 2 for computing shifts and, more generally, all permutations; this shows that such circuits are more efficient than the model based on sending bits along vertex-disjoint paths.