NDLI: Local Monotonicity Reconstruction

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Saks, Michael Seshadhri, C.
Copyright Year	2010
Abstract	We investigate the problem of monotonicity reconstruction, as defined by Ailon et al. (2004) in a localized setting. We have oracle access to a nonnegative real-valued function f defined on the domain $[n]^d=\{1,\dots,n\}^d$ (where d is viewed as a constant). We would like to closely approximate f by a monotone function g. This should be done by a procedure (a filter) that given as input a point $x\in[n]^d$ outputs the value of $g(x)$, and runs in time that is polylogarithmic in n. The procedure can (indeed must) be randomized, but we require that all of the randomness be specified in advance by a single short random seed. We construct such an implementation where the time and space per query is $(\log n)^{O(1)}$ and the size of the seed is polynomial in $\log n$ and d. Furthermore, with high probability, the ratio of the (Hamming) distance between g and f to the minimum possible Hamming distance between a monotone function and f is bounded above by a function of d (independent of n). This allows for a local implementation: one can initialize many copies of the filter with the same short random seed, and they can autonomously handle queries, while producing outputs that are consistent with the same approximating function g.
Starting Page	2897
Ending Page	2926
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/080728561
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	7
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-05-05
Access Restriction	Subscribed
Subject Keyword	sublinear algorithms Randomized algorithms monotonicity property testing local reconstruction Analysis of algorithms Combinatorics Parallel algorithms
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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