100,078 research outputs found
Some remarks on the Zarankiewicz problem
The Zarankiewicz problem asks for an estimate on z(m,n;s,t), the largest number of 1's in an m×n matrix with all entries 0 or 1 containing no s×t submatrix consisting entirely of 1's. We show that a classical upper bound for z(m,n;s,t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method
Some remarks on the Zarankiewicz problem
The Zarankiewicz problem asks for an estimate on , the largest
number of 's in an matrix with all entries or
containing no submatrix consisting entirely of 's. We show that
a classical upper bound for due to K\H{o}v\'ari, S\'os and
Tur\'an is tight up to the constant for a broad range of parameters. The proof
relies on a new quantitative variant of the random algebraic method.Comment: 6 page
Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes
We consider the bit-probe complexity of the set membership problem: represent an n-element subset S of an m-element universe as a succinct bit vector so that membership queries of the form "Is x ? S" can be answered using at most t probes into the bit vector. Let s(m,n,t) (resp. s_N(m,n,t)) denote the minimum number of bits of storage needed when the probes are adaptive (resp. non-adaptive). Lewenstein, Munro, Nicholson, and Raman (ESA 2014) obtain fully-explicit schemes that show that
s(m,n,t) = ?((2^t-1)m^{1/(t - min{2?log n?, n-3/2})}) for n ? 2,t ? ?log n?+1 .
In this work, we improve this bound when the probes are allowed to be superlinear in n, i.e., when t ? ?(nlog n), n ? 2, we design fully-explicit schemes that show that
s(m,n,t) = ?((2^t-1)m^{1/(t-{n-1}/{2^{t/(2(n-1))}})}),
asymptotically (in the exponent of m) close to the non-explicit upper bound on s(m,n,t) derived by Radhakrishan, Shah, and Shannigrahi (ESA 2010), for constant n.
In the non-adaptive setting, it was shown by Garg and Radhakrishnan (STACS 2017) that for a large constant n?, for n ? n?, s_N(m,n,3) ? ?{mn}. We improve this result by showing that the same lower bound holds even for storing sets of size 2, i.e., s_N(m,2,3) ? ?(?m)
Sparsified Block Elimination for Directed Laplacians
We show that the sparsified block elimination algorithm for solving
undirected Laplacian linear systems from [Kyng-Lee-Peng-Sachdeva-Spielman
STOC'16] directly works for directed Laplacians. Given access to a
sparsification algorithm that, on graphs with vertices and edges, takes
time to output a sparsifier with edges, our algorithm solves a directed Eulerian system on vertices
and edges to relative accuracy in time where the
notation hides factors. By previous results,
this implies improved runtimes for linear systems in strongly connected
directed graphs, PageRank matrices, and asymmetric M-matrices. When combined
with slower constructions of smaller Eulerian sparsifiers based on short cycle
decompositions, it also gives a solver that runs in time after pre-processing. At the core of
our analyses are constructions of augmented matrices whose Schur complements
encode error matrices
Constructions of skew-tolerant and skew-detecting codes
The paradigm of skew-tolerant parallel asynchronous communication was introduced by Blaum and Bruck (see ibid., vol. 39, 1993) along with constructions for codes that can tolerate or detect skew. Some of these constructions were improved by Khachatrian (1991). In this paper these constructions are improved upon further, and the authors prove that the new constructions are, in a certain sense, optimal
- …