988 research outputs found
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
Listing Subgraphs by Cartesian Decomposition
We investigate a decomposition technique for listing problems in graphs and set systems. It is based on the Cartesian product of some iterators, which list the solutions of simpler problems. Our ideas applies to several problems, and we illustrate one of them in depth, namely, listing all minimum spanning trees of a weighted graph G. Here iterators over the spanning trees for unweighted graphs can be obtained by a suitable modification of the listing algorithm by [Shioura et al., SICOMP 1997], and the decomposition of G is obtained by suitably partitioning its edges according to their weights. By combining these iterators in a Cartesian product scheme that employs Gray coding, we give the first algorithm which lists all minimum spanning trees of G in constant delay, where the delay is the time elapsed between any two consecutive outputs. Our solution requires polynomial preprocessing time and uses polynomial space
An Output Sensitive Algorithm for Maximal Clique Enumeration in Sparse Graphs
The degeneracy of a graph G is the smallest integer k such that every subgraph of G contains a vertex of degree at most k. Given an n-order k-degenerate graph G, we present an algorithm for enumerating all its maximal cliques. Assuming that c is the number of maximal cliques of G, our algorithm has setup time O(n(k^2+s(k+1))) and enumeration time cO((k+1)f(k+1)) where s(k+1) (resp. f(k+1)) is the preprocessing time (resp. enumeration time) for maximal clique enumeration in a general (k+1)-order graph. This is the first output sensitive algorithm whose enumeration time depends only on the degeneracy of the graph
Efficient Enumeration of Bipartite Subgraphs in Graphs
Subgraph enumeration problems ask to output all subgraphs of an input graph
that belongs to the specified graph class or satisfy the given constraint.
These problems have been widely studied in theoretical computer science. As
far, many efficient enumeration algorithms for the fundamental substructures
such as spanning trees, cycles, and paths, have been developed. This paper
addresses the enumeration problem of bipartite subgraphs. Even though bipartite
graphs are quite fundamental and have numerous applications in both theory and
application, its enumeration algorithms have not been intensively studied, to
the best of our knowledge. We propose the first non-trivial algorithms for
enumerating all bipartite subgraphs in a given graph. As the main results, we
develop two efficient algorithms: the one enumerates all bipartite induced
subgraphs of a graph with degeneracy in time per solution. The other
enumerates all bipartite subgraphs in time per solution
Sublinear-Space Bounded-Delay Enumeration for Massive Network Analytics: Maximal Cliques
Due to the sheer size of real-world networks, delay and space become quite relevant measures for the cost of enumeration in network analytics. This paper presents efficient algorithms for listing maximum cliques in networks, providing the first sublinear-space bounds with guaranteed delay per enumerated clique, thus comparing favorably with the known literature
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