19,689 research outputs found
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly
Non-asymptotic Upper Bounds for Deletion Correcting Codes
Explicit non-asymptotic upper bounds on the sizes of multiple-deletion
correcting codes are presented. In particular, the largest single-deletion
correcting code for -ary alphabet and string length is shown to be of
size at most . An improved bound on the asymptotic
rate function is obtained as a corollary. Upper bounds are also derived on
sizes of codes for a constrained source that does not necessarily comprise of
all strings of a particular length, and this idea is demonstrated by
application to sets of run-length limited strings.
The problem of finding the largest deletion correcting code is modeled as a
matching problem on a hypergraph. This problem is formulated as an integer
linear program. The upper bound is obtained by the construction of a feasible
point for the dual of the linear programming relaxation of this integer linear
program.
The non-asymptotic bounds derived imply the known asymptotic bounds of
Levenshtein and Tenengolts and improve on known non-asymptotic bounds.
Numerical results support the conjecture that in the binary case, the
Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure
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