10,674 research outputs found
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
ARPA Whitepaper
We propose a secure computation solution for blockchain networks. The
correctness of computation is verifiable even under malicious majority
condition using information-theoretic Message Authentication Code (MAC), and
the privacy is preserved using Secret-Sharing. With state-of-the-art multiparty
computation protocol and a layer2 solution, our privacy-preserving computation
guarantees data security on blockchain, cryptographically, while reducing the
heavy-lifting computation job to a few nodes. This breakthrough has several
implications on the future of decentralized networks. First, secure computation
can be used to support Private Smart Contracts, where consensus is reached
without exposing the information in the public contract. Second, it enables
data to be shared and used in trustless network, without disclosing the raw
data during data-at-use, where data ownership and data usage is safely
separated. Last but not least, computation and verification processes are
separated, which can be perceived as computational sharding, this effectively
makes the transaction processing speed linear to the number of participating
nodes. Our objective is to deploy our secure computation network as an layer2
solution to any blockchain system. Smart Contracts\cite{smartcontract} will be
used as bridge to link the blockchain and computation networks. Additionally,
they will be used as verifier to ensure that outsourced computation is
completed correctly. In order to achieve this, we first develop a general MPC
network with advanced features, such as: 1) Secure Computation, 2) Off-chain
Computation, 3) Verifiable Computation, and 4)Support dApps' needs like
privacy-preserving data exchange
Theories for TC0 and Other Small Complexity Classes
We present a general method for introducing finitely axiomatizable "minimal"
two-sorted theories for various subclasses of P (problems solvable in
polynomial time). The two sorts are natural numbers and finite sets of natural
numbers. The latter are essentially the finite binary strings, which provide a
natural domain for defining the functions and sets in small complexity classes.
We concentrate on the complexity class TC^0, whose problems are defined by
uniform polynomial-size families of bounded-depth Boolean circuits with
majority gates. We present an elegant theory VTC^0 in which the provably-total
functions are those associated with TC^0, and then prove that VTC^0 is
"isomorphic" to a different-looking single-sorted theory introduced by
Johannsen and Pollet. The most technical part of the isomorphism proof is
defining binary number multiplication in terms a bit-counting function, and
showing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc
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