2,006 research outputs found
A Family of matroid intersection algorithms for the computation of approximated symbolic network functions
In recent years, the technique of simplification during generation has turned out to be very promising for the efficient computation of approximate symbolic network functions for large transistor circuits. In this paper it is shown how symbolic network functions can be simplified during their generation with any well-known symbolic network analysis method. The underlying algorithm for the different techniques is always a matroid intersection algorithm. It is shown that the most efficient technique is the two-graph method. An implementation of the simplification during generation technique with the two-graph method illustrates its benefits for the symbolic analysis of large analog circuits
Parametric shortest-path algorithms via tropical geometry
We study parameterized versions of classical algorithms for computing
shortest-path trees. This is most easily expressed in terms of tropical
geometry. Applications include shortest paths in traffic networks with variable
link travel times.Comment: 24 pages and 8 figure
Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph
In this paper, we address the problem of enumerating all induced subtrees in
an input k-degenerate graph, where an induced subtree is an acyclic and
connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for
any its induced subgraph has a vertex whose degree is less than or equal to k,
and many real-world graphs have small degeneracies, or very close to small
degeneracies. Although, the studies are on subgraphs enumeration, such as
trees, paths, and matchings, but the problem addresses the subgraph
enumeration, such as enumeration of subgraphs that are trees. Their induced
subgraph versions have not been studied well. One of few example is for
chordless paths and cycles. Our motivation is to reduce the time complexity
close to O(1) for each solution. This type of optimal algorithms are proposed
many subgraph classes such as trees, and spanning trees. Induced subtrees are
fundamental object thus it should be studied deeply and there possibly exist
some efficient algorithms. Our algorithm utilizes nice properties of
k-degeneracy to state an effective amortized analysis. As a result, the time
complexity is reduced to O(k) time per induced subtree. The problem is solved
in constant time for each in planar graphs, as a corollary
Counting and Enumerating Crossing-free Geometric Graphs
We describe a framework for counting and enumerating various types of
crossing-free geometric graphs on a planar point set. The framework generalizes
ideas of Alvarez and Seidel, who used them to count triangulations in time
where is the number of points. The main idea is to reduce the
problem of counting geometric graphs to counting source-sink paths in a
directed acyclic graph.
The following new results will emerge. The number of all crossing-free
geometric graphs can be computed in time for some .
The number of crossing-free convex partitions can be computed in time
. The number of crossing-free perfect matchings can be computed in
time . The number of convex subdivisions can be computed in time
. The number of crossing-free spanning trees can be computed in time
for some . The number of crossing-free spanning cycles
can be computed in time for some .
With the same bounds on the running time we can construct data structures
which allow fast enumeration of the respective classes. For example, after
time of preprocessing we can enumerate the set of all crossing-free
perfect matchings using polynomial time per enumerated object. For
crossing-free perfect matchings and convex partitions we further obtain
enumeration algorithms where the time delay for each (in particular, the first)
output is bounded by a polynomial in .
All described algorithms are comparatively simple, both in terms of their
analysis and implementation
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
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