American Institute of Mathematical Sciences

February  2017, 11(1): 203-223. doi: 10.3934/amc.2017013

Information retrieval and the average number of input clues

 Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland

Received  August 2015 Revised  January 2016 Published  February 2017

Information retrieval in an associative memory was introduced in a recent paper by Yaakobi and Bruck. The associative memory is represented by a graph where the vertices correspond to the stored information units and the edges to associations between them. The goal is to find a stored information unit in the memory using input clues. In this paper, we study the minimum average number of input clues needed to find the sought information unit in the infinite king grid. We provide a geometric approach to determine the minimum number of input clues. Using this approach we are able to find optimal results and bounds on the number of input clues. The model by Yaakobi and Bruck has also applications to sensor networks monitoring and Levenshtein's sequence reconstruction problem.

Citation: Tero Laihonen. Information retrieval and the average number of input clues. Advances in Mathematics of Communications, 2017, 11 (1) : 203-223. doi: 10.3934/amc.2017013
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The code consists of the black vertices
(a) The example, (b) All the vertices form $B_1({\bf{u}})$. The gray nodes belong to $\mathcal{S}_1^1({\bf{u}})$ and the white ones to $L_1^1({\bf{u}})$
Part of the infinite king grid and the patterns $H$, $H'$, $J$ and $J'$ for $t=1.$ The code $C$ consists of the black vertices.
In these figures, the black nodes are codewords, white nodes are non-codewords and the gray ones are unknown: (a) The configuration (b) The rule R2
Optimal codes giving an $\mathcal{SAM}_\mathcal{K}(1;N)$ (a) for $N=3$ (b) for $N=2$.
In these figures, the black nodes are codewords, white nodes are non-codewords and the gray ones are unknown: (a) The first configuration (b) The second configuration
In these figures, a node with 3 in it belongs to $Z_{\ge 3}$: (a) The rule R2 (b) The rule R3
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