2,676 research outputs found
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Large Graph Analysis in the GMine System
Current applications have produced graphs on the order of hundreds of
thousands of nodes and millions of edges. To take advantage of such graphs, one
must be able to find patterns, outliers and communities. These tasks are better
performed in an interactive environment, where human expertise can guide the
process. For large graphs, though, there are some challenges: the excessive
processing requirements are prohibitive, and drawing hundred-thousand nodes
results in cluttered images hard to comprehend. To cope with these problems, we
propose an innovative framework suited for any kind of tree-like graph visual
design. GMine integrates (a) a representation for graphs organized as
hierarchies of partitions - the concepts of SuperGraph and Graph-Tree; and (b)
a graph summarization methodology - CEPS. Our graph representation deals with
the problem of tracing the connection aspects of a graph hierarchy with sub
linear complexity, allowing one to grasp the neighborhood of a single node or
of a group of nodes in a single click. As a proof of concept, the visual
environment of GMine is instantiated as a system in which large graphs can be
investigated globally and locally
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
Loom: Query-aware Partitioning of Online Graphs
As with general graph processing systems, partitioning data over a cluster of
machines improves the scalability of graph database management systems.
However, these systems will incur additional network cost during the execution
of a query workload, due to inter-partition traversals. Workload-agnostic
partitioning algorithms typically minimise the likelihood of any edge crossing
partition boundaries. However, these partitioners are sub-optimal with respect
to many workloads, especially queries, which may require more frequent
traversal of specific subsets of inter-partition edges. Furthermore, they
largely unsuited to operating incrementally on dynamic, growing graphs.
We present a new graph partitioning algorithm, Loom, that operates on a
stream of graph updates and continuously allocates the new vertices and edges
to partitions, taking into account a query workload of graph pattern
expressions along with their relative frequencies.
First we capture the most common patterns of edge traversals which occur when
executing queries. We then compare sub-graphs, which present themselves
incrementally in the graph update stream, against these common patterns.
Finally we attempt to allocate each match to single partitions, reducing the
number of inter-partition edges within frequently traversed sub-graphs and
improving average query performance.
Loom is extensively evaluated over several large test graphs with realistic
query workloads and various orderings of the graph updates. We demonstrate
that, given a workload, our prototype produces partitionings of significantly
better quality than existing streaming graph partitioning algorithms Fennel and
LDG
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
Optimal Hierarchical Layouts for Cache-Oblivious Search Trees
This paper proposes a general framework for generating cache-oblivious
layouts for binary search trees. A cache-oblivious layout attempts to minimize
cache misses on any hierarchical memory, independent of the number of memory
levels and attributes at each level such as cache size, line size, and
replacement policy. Recursively partitioning a tree into contiguous subtrees
and prescribing an ordering amongst the subtrees, Hierarchical Layouts
generalize many commonly used layouts for trees such as in-order, pre-order and
breadth-first. They also generalize the various flavors of the van Emde Boas
layout, which have previously been used as cache-oblivious layouts.
Hierarchical Layouts thus unify all previous attempts at deriving layouts for
search trees.
The paper then derives a new locality measure (the Weighted Edge Product)
that mimics the probability of cache misses at multiple levels, and shows that
layouts that reduce this measure perform better. We analyze the various degrees
of freedom in the construction of Hierarchical Layouts, and investigate the
relative effect of each of these decisions in the construction of
cache-oblivious layouts. Optimizing the Weighted Edge Product for complete
binary search trees, we introduce the MinWEP layout, and show that it
outperforms previously used cache-oblivious layouts by almost 20%.Comment: Extended version with proofs added to the appendi
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