2017/02/19 by Daniel Kane, Shachar Lovett, Kane, Daniel +3
Computer Science · Mathematics · #05C15 #05C22 #05C35 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #cs.DM #math.CO #msc:05C15 #msc:05C22 #msc:05C35
paper · pdf · doi:10.48550/arxiv.1702.05773
arxiv created 2017/03/31 · arxiv updated 2017/04/04
Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph Kn,n with labels coming from F2d , that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n x n grid topologies. Prior to the current work, it was known that d is between (log n)2 and nlog n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.