137 research outputs found
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
HyperANF: Approximating the Neighbourhood Function of Very Large Graphs on a Budget
The neighbourhood function N(t) of a graph G gives, for each t, the number of
pairs of nodes such that y is reachable from x in less that t hops. The
neighbourhood function provides a wealth of information about the graph (e.g.,
it easily allows one to compute its diameter), but it is very expensive to
compute it exactly. Recently, the ANF algorithm (approximate neighbourhood
function) has been proposed with the purpose of approximating NG(t) on large
graphs. We describe a breakthrough improvement over ANF in terms of speed and
scalability. Our algorithm, called HyperANF, uses the new HyperLogLog counters
and combines them efficiently through broadword programming; our implementation
uses overdecomposition to exploit multi-core parallelism. With HyperANF, for
the first time we can compute in a few hours the neighbourhood function of
graphs with billions of nodes with a small error and good confidence using a
standard workstation. Then, we turn to the study of the distribution of the
shortest paths between reachable nodes (that can be efficiently approximated by
means of HyperANF), and discover the surprising fact that its index of
dispersion provides a clear-cut characterisation of proper social networks vs.
web graphs. We thus propose the spid (Shortest-Paths Index of Dispersion) of a
graph as a new, informative statistics that is able to discriminate between the
above two types of graphs. We believe this is the first proposal of a
significant new non-local structural index for complex networks whose
computation is highly scalable
Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions
AbstractIn 1964, Sarkovskii defined a certain linear ordering ⩽s of the positive integers and proved that m⩽sn if every continuous f:R→R having an orbit of size n also has an orbit of size m. This idea is extended to get a partial (but not linear) ordering in which the pattern of the orbit is taken into account. For example if x1<x2<x3<x4, then x1→x2→x3→x4x→1 and x1→x3→x2→x4→x1 are both orbits of size 4 but are considered to have distinct patterns in this paper. A combinatorial algorithm which decides the status of any two patterns with respect to the partial ordering is derived, and examples are given for patterns of small size
On the Existence of Hamiltonian Paths for History Based Pivot Rules on Acyclic Unique Sink Orientations of Hypercubes
An acyclic USO on a hypercube is formed by directing its edges in such as way
that the digraph is acyclic and each face of the hypercube has a unique sink
and a unique source. A path to the global sink of an acyclic USO can be modeled
as pivoting in a unit hypercube of the same dimension with an abstract
objective function, and vice versa. In such a way, Zadeh's 'least entered rule'
and other history based pivot rules can be applied to the problem of finding
the global sink of an acyclic USO. In this paper we present some theoretical
and empirical results on the existence of acyclic USOs for which the various
history based pivot rules can be made to follow a Hamiltonian path. In
particular, we develop an algorithm that can enumerate all such paths up to
dimension 6 using efficient pruning techniques. We show that Zadeh's original
rule admits Hamiltonian paths up to dimension 9 at least, and prove that most
of the other rules do not for all dimensions greater than 5
About Correctness of Graph-Based Social Network Analysis
Social network analysis widely uses graph techniques. Together with correct applications, in some cases, results are obtained from the graphs using paths longer than one, and due to intransitivity of relationships, several metrics and results are not applicable backward to objects in the investigated domain in a meaningful way. The author provides several examples and tries to recover roots of an incorrect application of graphs
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