552 research outputs found
Partial fillup and search time in LC tries
Andersson and Nilsson introduced in 1993 a level-compressed trie (in short:
LC trie) in which a full subtree of a node is compressed to a single node of
degree being the size of the subtree. Recent experimental results indicated a
'dramatic improvement' when full subtrees are replaced by partially filled
subtrees. In this paper, we provide a theoretical justification of these
experimental results showing, among others, a rather moderate improvement of
the search time over the original LC tries. For such an analysis, we assume
that n strings are generated independently by a binary memoryless source with p
denoting the probability of emitting a 1. We first prove that the so called
alpha-fillup level (i.e., the largest level in a trie with alpha fraction of
nodes present at this level) is concentrated on two values with high
probability. We give these values explicitly up to O(1), and observe that the
value of alpha (strictly between 0 and 1) does not affect the leading term.
This result directly yields the typical depth (search time) in the alpha-LC
tries with p not equal to 1/2, which turns out to be C loglog n for an
explicitly given constant C (depending on p but not on alpha). This should be
compared with recently found typical depth in the original LC tries which is C'
loglog n for a larger constant C'. The search time in alpha-LC tries is thus
smaller but of the same order as in the original LC tries.Comment: 13 page
Identifying statistical dependence in genomic sequences via mutual information estimates
Questions of understanding and quantifying the representation and amount of
information in organisms have become a central part of biological research, as
they potentially hold the key to fundamental advances. In this paper, we
demonstrate the use of information-theoretic tools for the task of identifying
segments of biomolecules (DNA or RNA) that are statistically correlated. We
develop a precise and reliable methodology, based on the notion of mutual
information, for finding and extracting statistical as well as structural
dependencies. A simple threshold function is defined, and its use in
quantifying the level of significance of dependencies between biological
segments is explored. These tools are used in two specific applications. First,
for the identification of correlations between different parts of the maize
zmSRp32 gene. There, we find significant dependencies between the 5'
untranslated region in zmSRp32 and its alternatively spliced exons. This
observation may indicate the presence of as-yet unknown alternative splicing
mechanisms or structural scaffolds. Second, using data from the FBI's Combined
DNA Index System (CODIS), we demonstrate that our approach is particularly well
suited for the problem of discovering short tandem repeats, an application of
importance in genetic profiling.Comment: Preliminary version. Final version in EURASIP Journal on
Bioinformatics and Systems Biology. See http://www.hindawi.com/journals/bsb
Asymmetry and structural information in preferential attachment graphs
Graph symmetries intervene in diverse applications, from enumeration, to
graph structure compression, to the discovery of graph dynamics (e.g., node
arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically
asymmetric, real networks are highly symmetric. So a natural question is
whether preferential attachment graphs, where in each step a new node with
edges is added, exhibit any symmetry. In recent work it was proved that
preferential attachment graphs are symmetric for , and there is some
non-negligible probability of symmetry for . It was conjectured that these
graphs are asymmetric when . We settle this conjecture in the
affirmative, then use it to estimate the structural entropy of the model. To do
this, we also give bounds on the number of ways that the given graph structure
could have arisen by preferential attachment. These results have further
implications for information theoretic problems of interest on preferential
attachment graphs.Comment: 24 pages; to appear in Random Structures & Algorithm
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