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The message is optimally (respectively, sub-optimally) delivered in at most r (respectively, 2r+1) hops.
This ensures local repair of each code symbol by contacting at most r other code symbols.
Nodes that are separated by a distance of at most R can communicate directly with each other.
A code is said to have all-symbol locality r if every the code symbol is a function of at most r other code symbols.
For (h>0) and (rgeq1), (X_{h}^{r}subset H_{0}^{1}(Omega)) is a finite-dimensional subspace consisting of all piecewise polynomials of degree at most r.
Note that Definition 1 ensures that any ℓ code symbols can be cooperatively repaired from at most r other code symbols.
This allows for cooperative local repair of code symbols, where any group of ℓ failed code symbols is repaired by contacting at most r other code symbols.
We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite matrix of rank at most r, where r is the rank of the matrix involved in the objective function of the SDP.
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