50 research outputs found
Repetition-free longest common subsequence of random sequences
A repetition free Longest Common Subsequence (LCS) of two sequences x and y
is an LCS of x and y where each symbol may appear at most once. Let R denote
the length of a repetition free LCS of two sequences of n symbols each one
chosen randomly, uniformly, and independently over a k-ary alphabet. We study
the asymptotic, in n and k, behavior of R and establish that there are three
distinct regimes, depending on the relative speed of growth of n and k. For
each regime we establish the limiting behavior of R. In fact, we do more, since
we actually establish tail bounds for large deviations of R from its limiting
behavior.
Our study is motivated by the so called exemplar model proposed by Sankoff
(1999) and the related similarity measure introduced by Adi et al. (2007). A
natural question that arises in this context, which as we show is related to
long standing open problems in the area of probabilistic combinatorics, is to
understand the asymptotic, in n and k, behavior of parameter R.Comment: 15 pages, 1 figur
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
Universal Geometric Graphs
We introduce and study the problem of constructing geometric graphs that have
few vertices and edges and that are universal for planar graphs or for some
sub-class of planar graphs; a geometric graph is \emph{universal} for a class
of planar graphs if it contains an embedding, i.e., a
crossing-free drawing, of every graph in .
Our main result is that there exists a geometric graph with vertices and
edges that is universal for -vertex forests; this extends to
the geometric setting a well-known graph-theoretic result by Chung and Graham,
which states that there exists an -vertex graph with edges
that contains every -vertex forest as a subgraph. Our bound on
the number of edges cannot be improved, even if more than vertices are
allowed.
We also prove that, for every positive integer , every -vertex convex
geometric graph that is universal for -vertex outerplanar graphs has a
near-quadratic number of edges, namely ; this almost
matches the trivial upper bound given by the -vertex complete
convex geometric graph.
Finally, we prove that there exists an -vertex convex geometric graph with
vertices and edges that is universal for -vertex
caterpillars.Comment: 20 pages, 8 figures; a 12-page extended abstracts of this paper will
appear in the Proceedings of the 46th Workshop on Graph-Theoretic Concepts in
Computer Science (WG 2020
Enumeration of noncrossing trees on a circle
AbstractWe consider several enumerative problems concerning labelled trees whose vertices lie on a circle and whose edges are rectilinear and do not cross