This paper presents the optimal compression for sequences with undefined values. Let we have $(N-m)$ undefined and $m$ defined positions in the boolean sequence $vv V$ of length $N$. The sequence code length can't be less then $m$ in general case, otherwise at least two sequences will have the same code. We present the coding algorithm which generates codes of almost $m$ length, i.e. almost equal to the lower bound. The paper presents the decoding circuit too. The circuit has low complexity which depends from the inverse density of defined values $D(vv V) = frac{N}{m}$. The decoding circuit includes RAM and random logic. It performs sequential decoding. The total RAM size is proportional to the $$logleft(D(vv V) ight) ,$$ the number of random logic cells is proportional to $$log logleft(D(vv V) ight) * left(log log logleft(D(vv V) ight) ight)^2 .$$ So the decoding circuit will be small enough even for the very low density sequences. The decoder complexity doesn't depend of the sequence length at all.
@InProceedings{andreev:DagSemProc.06111.19, author = {Andreev, Alexander E.}, title = {{The optimal sequence compression}}, booktitle = {Complexity of Boolean Functions}, pages = {1--11}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.19}, URN = {urn:nbn:de:0030-drops-6025}, doi = {10.4230/DagSemProc.06111.19}, annote = {Keywords: Compression, partial boolean function} }
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