52 research outputs found
Tribes Is Hard in the Message Passing Model
We consider the point-to-point message passing model of communication in
which there are processors with individual private inputs, each -bit
long. Each processor is located at the node of an underlying undirected graph
and has access to private random coins. An edge of the graph is a private
channel of communication between its endpoints. The processors have to compute
a given function of all their inputs by communicating along these channels.
While this model has been widely used in distributed computing, strong lower
bounds on the amount of communication needed to compute simple functions have
just begun to appear. In this work, we prove a tight lower bound of
on the communication needed for computing the Tribes function,
when the underlying graph is a star of nodes that has leaves with
inputs and a center with no input. Lower bound on this topology easily implies
comparable bounds for others. Our lower bounds are obtained by building upon
the recent information theoretic techniques of Braverman et.al (FOCS'13) and
combining it with the earlier work of Jayram, Kumar and Sivakumar (STOC'03).
This approach yields information complexity bounds that is of independent
interest
Graph Isomorphism is not AC^0 reducible to Group Isomorphism
We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with depth and nondeterministic bits,
where is the number of group elements. This improves the existing upper bound from cite{Wolf 94} for the problems. In the previous upper bound the circuits have bounded fan-in but depth and also nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC0 reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC0 reductions
The Power of Super-logarithmic Number of Players
In the `Number-on-Forehead\u27 (NOF) model of multiparty communication,
the input is a k times m boolean matrix A (where k is the number of players) and Player i sees all bits except those in the i-th row, and the players communicate by broadcast in order to evaluate a specified function f at A.
We discover new computational power when k exceeds log m. We give a protocol with communication cost poly-logarithmic in m, for block composed functions with limited block width. These are functions
of the form f o g where f is a symmetric b-variate function, and g is a (kr)-variate function and (f o g)(A) is defined, for a k times (br) matrix to be f(g(A-1),...,g(A-b)) where A-i is the i-th (k times r) block of A. Our protocol works provided that k > 1+ ln b + (2 to the power of r).
Ada et al. (ICALP\u272012) previously obtained simultaneous and deterministic efficient protocols for composed functions of block-width one. The new protocol is the first to work for block composed functions with block-width greather than one. Moreover, it is simultaneous, with vanishingly small error probability, if public coin randomness is allowed. The deterministic and zero-error version barely uses interaction
Query Complexity of Search Problems
We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we provide the following improvements upon the known relationship between pseudo-deterministic and deterministic query complexity for total search problems:
- We show that deterministic query complexity is at most the third power of its pseudo-deterministic query complexity. Previously, a fourth-power relation was shown by Goldreich, Goldwasser and Ron (ITCS\u2713).
- We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) we exhibit an exp(??(n^{1/4})) separation for the SearchCNF relation for random k-CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of SearchCNF associated with random k-CNFs. (2) we exhibit an exp(?(n)) separation for the ApproxHamWt relation. The previous best known separation for any relation was exp(?(n^{1/2})). We also separate pseudo-determinism from randomness in And and (And,Or) decision trees, and determinism from pseudo-determinism in Parity decision trees. For a hypercube colouring problem, that was introduced by Goldwasswer, Impagliazzo, Pitassi and Santhanam (CCC\u2721) to analyze the pseudo-deterministic complexity of a complete problem in TFNP^{dt}, we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is ?(n^{1/3}); Goldwasser et al. showed an ?(n^{1/2}) bound for general block-sensitivity
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