37,848 research outputs found
Computing Groebner Fans
This paper presents algorithms for computing the Groebner fan of an arbitrary
polynomial ideal. The computation involves enumeration of all reduced Groebner
bases of the ideal. Our algorithms are based on a uniform definition of the
Groebner fan that applies to both homogeneous and non-homogeneous ideals and a
proof that this object is a polyhedral complex. We show that the cells of a
Groebner fan can easily be oriented acyclically and with a unique sink,
allowing their enumeration by the memory-less reverse search procedure. The
significance of this follows from the fact that Groebner fans are not always
normal fans of polyhedra in which case reverse search applies automatically.
Computational results using our implementation of these algorithms in the
software package Gfan are included.Comment: 26 page
GTRACE-RS: Efficient Graph Sequence Mining using Reverse Search
The mining of frequent subgraphs from labeled graph data has been studied
extensively. Furthermore, much attention has recently been paid to frequent
pattern mining from graph sequences. A method, called GTRACE, has been proposed
to mine frequent patterns from graph sequences under the assumption that
changes in graphs are gradual. Although GTRACE mines the frequent patterns
efficiently, it still needs substantial computation time to mine the patterns
from graph sequences containing large graphs and long sequences. In this paper,
we propose a new version of GTRACE that enables efficient mining of frequent
patterns based on the principle of a reverse search. The underlying concept of
the reverse search is a general scheme for designing efficient algorithms for
hard enumeration problems. Our performance study shows that the proposed method
is efficient and scalable for mining both long and large graph sequence
patterns and is several orders of magnitude faster than the original GTRACE
Constant Amortized Time Enumeration of Eulerian trails
In this paper, we consider enumeration problems for edge-distinct and
vertex-distinct Eulerian trails. Here, two Eulerian trails are
\emph{edge-distinct} if the edge sequences are not identical, and they are
\emph{vertex-distinct} if the vertex sequences are not identical. As the main
result, we propose optimal enumeration algorithms for both problems, that is,
these algorithm runs in total time, where is the number of
solutions. Our algorithms are based on the reverse search technique introduced
by [Avis and Fukuda, DAM 1996], and the push out amortization technique
introduced by [Uno, WADS 2015]
All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs
We describe algorithms, based on Avis and Fukuda's reverse search paradigm,
for listing all maximal independent sets in a sparse graph in polynomial time
and delay per output. For bounded degree graphs, our algorithms take constant
time per set generated; for minor-closed graph families, the time is O(n) per
set, and for more general sparse graph families we achieve subquadratic time
per set. We also describe new data structures for maintaining a dynamic vertex
set S in a sparse or minor-closed graph family, and querying the number of
vertices not dominated by S; for minor-closed graph families the time per
update is constant, while it is sublinear for any sparse graph family. We can
also maintain a dynamic vertex set in an arbitrary m-edge graph and test the
independence of the maintained set in time O(sqrt m) per update. We use the
domination data structures as part of our enumeration algorithms.Comment: 10 page
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