766 research outputs found
Compressed Data Structures for Dynamic Sequences
We consider the problem of storing a dynamic string over an alphabet
in compressed form. Our representation
supports insertions and deletions of symbols and answers three fundamental
queries: returns the -th symbol in ,
counts how many times a symbol occurs among the
first positions in , and finds the position
where a symbol occurs for the -th time. We present the first
fully-dynamic data structure for arbitrarily large alphabets that achieves
optimal query times for all three operations and supports updates with
worst-case time guarantees. Ours is also the first fully-dynamic data structure
that needs only bits, where is the -th order
entropy and is the string length. Moreover our representation supports
extraction of a substring in optimal time
A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics
From the output produced by a memoryless deletion channel from a uniformly
random input of known length , one obtains a posterior distribution on the
channel input. The difference between the Shannon entropy of this distribution
and that of the uniform prior measures the amount of information about the
channel input which is conveyed by the output of length , and it is natural
to ask for which outputs this is extremized. This question was posed in a
previous work, where it was conjectured on the basis of experimental data that
the entropy of the posterior is minimized and maximized by the constant strings
and and the alternating strings
and respectively. In the present
work we confirm the minimization conjecture in the asymptotic limit using
results from hidden word statistics. We show how the analytic-combinatorial
methods of Flajolet, Szpankowski and Vall\'ee for dealing with the hidden
pattern matching problem can be applied to resolve the case of fixed output
length and , by obtaining estimates for the entropy in
terms of the moments of the posterior distribution and establishing its
minimization via a measure of autocorrelation.Comment: 11 pages, 2 figure
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