169 research outputs found
Arboricity, h-Index, and Dynamic Algorithms
In this paper we present a modification of a technique by Chiba and Nishizeki
[Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J.
Comput. 14(1), pp. 210--223 (1985)]. Based on it, we design a data structure
suitable for dynamic graph algorithms. We employ the data structure to
formulate new algorithms for several problems, including counting subgraphs of
four vertices, recognition of diamond-free graphs, cop-win graphs and strongly
chordal graphs, among others. We improve the time complexity for graphs with
low arboricity or h-index.Comment: 19 pages, no figure
Fast Quasi-Threshold Editing
We introduce Quasi-Threshold Mover (QTM), an algorithm to solve the
quasi-threshold (also called trivially perfect) graph editing problem with edge
insertion and deletion. Given a graph it computes a quasi-threshold graph which
is close in terms of edit count. This edit problem is NP-hard. We present an
extensive experimental study, in which we show that QTM is the first algorithm
that is able to scale to large real-world graphs in practice. As a side result
we further present a simple linear-time algorithm for the quasi-threshold
recognition problem.Comment: 26 pages, 4 figures, submitted to ESA 201
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
Dynamic Set Intersection
Consider the problem of maintaining a family of dynamic sets subject to
insertions, deletions, and set-intersection reporting queries: given , report every member of in any order. We show that in the word
RAM model, where is the word size, given a cap on the maximum size of
any set, we can support set intersection queries in
expected time, and updates in expected time. Using this algorithm
we can list all triangles of a graph in
expected time, where and
is the arboricity of . This improves a 30-year old triangle enumeration
algorithm of Chiba and Nishizeki running in time.
We provide an incremental data structure on that supports intersection
{\em witness} queries, where we only need to find {\em one} .
Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected
time, where . Finally, we provide time/space tradeoffs for
the fully dynamic set intersection reporting problem. Using words of space,
each update costs expected time, each reporting query
costs expected time where
is the size of the output, and each witness query costs expected time.Comment: Accepted to WADS 201
A New Tool for Rectangular Dualization
OcORD is a software tool for rectangular dualization. Rectangular dualization is a dual representation of a plane graph introduced in the early seventies. It proved to be effective in applications such as architectural space planning and VLSI floorplanning. However, not all plane graphs admit a rectangular dual, which imposes severe limitations on its use in other applications. OcORD aims at freeing rectangular dualization from such restrictions and proving its effectiveness in graph visualization. This is achieved in two ways. Firstly, OcORD features a new linear-time algorithm creating a rectangular dual of any plane graph. Secondly, it shows how nice drawings of a graph can be easily obtained from its rectangular dual. Finally, the automatic generation of a Virtual World through rectangular dualization is described.
[DOI: 10.1685/CSC09301] About DO
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
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