7,835 research outputs found
Permutation graphs, fast forward permutations, and sampling the cycle structure of a permutation
A permutation P on {1,..,N} is a_fast_forward_permutation_ if for each m the
computational complexity of evaluating P^m(x)$ is small independently of m and
x. Naor and Reingold constructed fast forward pseudorandom cycluses and
involutions. By studying the evolution of permutation graphs, we prove that the
number of queries needed to distinguish a random cyclus from a random
permutation on {1,..,N} is Theta(N) if one does not use queries of the form
P^m(x), but is only Theta(1) if one is allowed to make such queries.
We construct fast forward permutations which are indistinguishable from
random permutations even when queries of the form P^m(x) are allowed. This is
done by introducing an efficient method to sample the cycle structure of a
random permutation, which in turn solves an open problem of Naor and Reingold.Comment: Corrected a small erro
Exact and fixed-parameter algorithms for metro-line crossing minimization problems
A metro-line crossing minimization problem is to draw multiple lines on an
underlying graph that models stations and rail tracks so that the number of
crossings of lines becomes minimum. It has several variations by adding
restrictions on how lines are drawn. Among those, there is one with a
restriction that line terminals have to be drawn at a verge of a station, and
it is known to be NP-hard even when underlying graphs are paths. This paper
studies the problem in this setting, and propose new exact algorithms. We first
show that a problem to decide if lines can be drawn without crossings is solved
in polynomial time, and propose a fast exponential algorithm to solve a
crossing minimization problem. We then propose a fixed-parameter algorithm with
respect to the multiplicity of lines, which implies that the problem is FPT.Comment: 19 pages, 15 figure
Hamilton paths with lasting separation
We determine the asymptotics of the largest cardinality of a set of Hamilton
paths in the complete graph with vertex set [n] under the condition that for
any two of the paths in the family there is a subpath of length k entirely
contained in only one of them and edge{disjoint from the other one
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