15,632 research outputs found
Efficient Optimally Lazy Algorithms for Minimal-Interval Semantics
Minimal-interval semantics associates with each query over a document a set
of intervals, called witnesses, that are incomparable with respect to inclusion
(i.e., they form an antichain): witnesses define the minimal regions of the
document satisfying the query. Minimal-interval semantics makes it easy to
define and compute several sophisticated proximity operators, provides snippets
for user presentation, and can be used to rank documents. In this paper we
provide algorithms for computing conjunction and disjunction that are linear in
the number of intervals and logarithmic in the number of operands; for
additional operators, such as ordered conjunction and Brouwerian difference, we
provide linear algorithms. In all cases, space is linear in the number of
operands. More importantly, we define a formal notion of optimal laziness, and
either prove it, or prove its impossibility, for each algorithm. We cast our
results in a general framework of antichains of intervals on total orders,
making our algorithms directly applicable to other domains.Comment: 24 pages, 4 figures. A preliminary (now outdated) version was
presented at SPIRE 200
Compressed Subsequence Matching and Packed Tree Coloring
We present a new algorithm for subsequence matching in grammar compressed
strings. Given a grammar of size compressing a string of size and a
pattern string of size over an alphabet of size , our algorithm
uses space and or time. Here
is the word size and is the number of occurrences of the pattern. Our
algorithm uses less space than previous algorithms and is also faster for
occurrences. The algorithm uses a new data structure
that allows us to efficiently find the next occurrence of a given character
after a given position in a compressed string. This data structure in turn is
based on a new data structure for the tree color problem, where the node colors
are packed in bit strings.Comment: To appear at CPM '1
Minimal bounds and members of effectively closed sets
We show that there exists a non-empty class, with no recursive
element, in which no member is a minimal cover for any Turing degree.Comment: 15 pages, 4 figures, 1 acknowledgemen
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