12,193 research outputs found
Complexity of Multi-Value Byzantine Agreement
In this paper, we consider the problem of maximizing the throughput of
Byzantine agreement, given that the sum capacity of all links in between nodes
in the system is finite. We have proposed a highly efficient Byzantine
agreement algorithm on values of length l>1 bits. This algorithm uses error
detecting network codes to ensure that fault-free nodes will never disagree,
and routing scheme that is adaptive to the result of error detection. Our
algorithm has a bit complexity of n(n-1)l/(n-t), which leads to a linear cost
(O(n)) per bit agreed upon, and overcomes the quadratic lower bound
(Omega(n^2)) in the literature. Such linear per bit complexity has only been
achieved in the literature by allowing a positive probability of error. Our
algorithm achieves the linear per bit complexity while guaranteeing agreement
is achieved correctly even in the worst case. We also conjecture that our
algorithm can be used to achieve agreement throughput arbitrarily close to the
agreement capacity of a network, when the sum capacity is given
Low-density MDS codes and factors of complete graphs
We present a class of array code of size n×l, where l=2n or 2n+1, called B-Code. The distances of the B-Code and its dual are 3 and l-1, respectively. The B-Code and its dual are optimal in the sense that i) they are maximum-distance separable (MDS), ii) they have an optimal encoding property, i.e., the number of the parity bits that are affected by change of a single information bit is minimal, and iii) they have optimal length. Using a new graph description of the codes, we prove an equivalence relation between the construction of the B-Code (or its dual) and a combinatorial problem known as perfect one-factorization of complete graphs, thus obtaining constructions of two families of the B-Code and its dual, one of which is new. Efficient decoding algorithms are also given, both for erasure correcting and for error correcting. The existence of perfect one-factorizations for every complete graph with an even number of nodes is a 35 years long conjecture in graph theory. The construction of B-Codes of arbitrary odd length will provide an affirmative answer to the conjecture
String Matching: Communication, Circuits, and Learning
String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings.
- Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol.
- Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k.
- Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem
Invariant Synthesis for Incomplete Verification Engines
We propose a framework for synthesizing inductive invariants for incomplete
verification engines, which soundly reduce logical problems in undecidable
theories to decidable theories. Our framework is based on the counter-example
guided inductive synthesis principle (CEGIS) and allows verification engines to
communicate non-provability information to guide invariant synthesis. We show
precisely how the verification engine can compute such non-provability
information and how to build effective learning algorithms when invariants are
expressed as Boolean combinations of a fixed set of predicates. Moreover, we
evaluate our framework in two verification settings, one in which verification
engines need to handle quantified formulas and one in which verification
engines have to reason about heap properties expressed in an expressive but
undecidable separation logic. Our experiments show that our invariant synthesis
framework based on non-provability information can both effectively synthesize
inductive invariants and adequately strengthen contracts across a large suite
of programs
Subspace subcodes of Reed-Solomon codes
We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed ν-dimensional vector subspace S of GF (2m). SSRS codes are constructed using properties of the Galois field GF(2m). They are not linear over the field GF(2ν), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2ν) is a subfield of GF(2m), it may not be the best ν-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems
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