32,461 research outputs found
Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance
We present two on-line algorithms for maintaining a topological order of a
directed -vertex acyclic graph as arcs are added, and detecting a cycle when
one is created. Our first algorithm handles arc additions in
time. For sparse graphs (), this bound improves the best previous
bound by a logarithmic factor, and is tight to within a constant factor among
algorithms satisfying a natural {\em locality} property. Our second algorithm
handles an arbitrary sequence of arc additions in time. For
sufficiently dense graphs, this bound improves the best previous bound by a
polynomial factor. Our bound may be far from tight: we show that the algorithm
can take time by relating its performance to a
generalization of the -levels problem of combinatorial geometry. A
completely different algorithm running in time was given
recently by Bender, Fineman, and Gilbert. We extend both of our algorithms to
the maintenance of strong components, without affecting the asymptotic time
bounds.Comment: 31 page
One machine, one minute, three billion tetrahedra
This paper presents a new scalable parallelization scheme to generate the 3D
Delaunay triangulation of a given set of points. Our first contribution is an
efficient serial implementation of the incremental Delaunay insertion
algorithm. A simple dedicated data structure, an efficient sorting of the
points and the optimization of the insertion algorithm have permitted to
accelerate reference implementations by a factor three. Our second contribution
is a multi-threaded version of the Delaunay kernel that is able to concurrently
insert vertices. Moore curve coordinates are used to partition the point set,
avoiding heavy synchronization overheads. Conflicts are managed by modifying
the partitions with a simple rescaling of the space-filling curve. The
performances of our implementation have been measured on three different
processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we
have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds
to a generation rate of over 55 million tetrahedra per second. We finally show
how this very efficient parallel Delaunay triangulation can be integrated in a
Delaunay refinement mesh generator which takes as input the triangulated
surface boundary of the volume to mesh
A review of associative classification mining
Associative classification mining is a promising approach in data mining that utilizes the
association rule discovery techniques to construct classification systems, also known as
associative classifiers. In the last few years, a number of associative classification algorithms
have been proposed, i.e. CPAR, CMAR, MCAR, MMAC and others. These algorithms
employ several different rule discovery, rule ranking, rule pruning, rule prediction and rule
evaluation methods. This paper focuses on surveying and comparing the state-of-the-art associative
classification techniques with regards to the above criteria. Finally, future directions in associative
classification, such as incremental learning and mining low-quality data sets, are also
highlighted in this paper
On dynamic breadth-first search in external-memory
We provide the first non-trivial result on dynamic breadth-first search (BFS) in external-memory: For general sparse undirected graphs of initially nodes and O(n) edges and monotone update sequences of either edge insertions or edge deletions, we prove an amortized high-probability bound of O(n/B^{2/3}+\sort(n)\cdot \log B) I/Os per update. In contrast, the currently best approach for static BFS on sparse undirected graphs requires \Omega(n/B^{1/2}+\sort(n)) I/Os. 1998 ACM Subject Classification: F.2.2. Key words and phrases: External Memory, Dynamic Graph Algorithms, BFS, Randomization
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
Regular triangulations of dynamic sets of points
The Delaunay triangulations of a set of points are a class of
triangulations which play an important role in a variety of
different disciplines of science. Regular triangulations are a
generalization of Delaunay triangulations that maintain both their
relationship with convex hulls and with Voronoi diagrams. In regular
triangulations, a real value, its weight, is assigned to each point.
In this paper a simple data structure is presented that allows
regular triangulations of sets of points to be dynamically updated,
that is, new points can be incrementally inserted in the set and old
points can be deleted from it. The algorithms we propose for
insertion and deletion are based on a geometrical interpretation of
the history data structure in one more dimension and use lifted
flips as the unique topological operation. This results in rather
simple and efficient algorithms. The algorithms have been
implemented and experimental results are given.Postprint (published version
An O(n^{2.75}) algorithm for online topological ordering
We present a simple algorithm which maintains the topological order of a
directed acyclic graph with n nodes under an online edge insertion sequence in
O(n^{2.75}) time, independent of the number of edges m inserted. For dense
DAGs, this is an improvement over the previous best result of O(min(m^{3/2}
log(n), m^{3/2} + n^2 log(n)) by Katriel and Bodlaender. We also provide an
empirical comparison of our algorithm with other algorithms for online
topological sorting. Our implementation outperforms them on certain hard
instances while it is still competitive on random edge insertion sequences
leading to complete DAGs.Comment: 20 pages, long version of SWAT'06 pape
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