3,519 research outputs found
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
Algorithms for the Problems of Length-Constrained Heaviest Segments
We present algorithms for length-constrained maximum sum segment and maximum
density segment problems, in particular, and the problem of finding
length-constrained heaviest segments, in general, for a sequence of real
numbers. Given a sequence of n real numbers and two real parameters L and U (L
<= U), the maximum sum segment problem is to find a consecutive subsequence,
called a segment, of length at least L and at most U such that the sum of the
numbers in the subsequence is maximum. The maximum density segment problem is
to find a segment of length at least L and at most U such that the density of
the numbers in the subsequence is the maximum. For the first problem with
non-uniform width there is an algorithm with time and space complexities in
O(n). We present an algorithm with time complexity in O(n) and space complexity
in O(U). For the second problem with non-uniform width there is a combinatorial
solution with time complexity in O(n) and space complexity in O(U). We present
a simple geometric algorithm with the same time and space complexities.
We extend our algorithms to respectively solve the length-constrained k
maximum sum segments problem in O(n+k) time and O(max{U, k}) space, and the
length-constrained maximum density segments problem in O(n min{k, U-L})
time and O(U+k) space. We present extensions of our algorithms to find all the
length-constrained segments having user specified sum and density in O(n+m) and
O(nlog (U-L)+m) times respectively, where m is the number of output.
Previously, there was no known algorithm with non-trivial result for these
problems. We indicate the extensions of our algorithms to higher dimensions.
All the algorithms can be extended in a straight forward way to solve the
problems with non-uniform width and non-uniform weight.Comment: 21 pages, 12 figure
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