11,958 research outputs found
Communication tree problems
In this paper, we consider random communication
requirements and several cost
measures for a particular model of tree routing on a
complete network. First
we show that a random tree does not give any approximation.
Then give
approximation algorithms for the case for two random models
of requirements.Postprint (published version
Network correlated data gathering with explicit communication: NP-completeness and algorithms
We consider the problem of correlated data gathering by a network with a sink node and a tree-based communication structure, where the goal is to minimize the total transmission cost of transporting the information collected by the nodes, to the sink node. For source coding of correlated data, we consider a joint entropy-based coding model with explicit communication where coding is simple and the transmission structure optimization is difficult. We first formulate the optimization problem definition in the general case and then we study further a network setting where the entropy conditioning at nodes does not depend on the amount of side information, but only on its availability. We prove that even in this simple case, the optimization problem is NP-hard. We propose some efficient, scalable, and distributed heuristic approximation algorithms for solving this problem and show by numerical simulations that the total transmission cost can be significantly improved over direct transmission or the shortest path tree. We also present an approximation algorithm that provides a tree transmission structure with total cost within a constant factor from the optimal
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Minimum spanning trees on random networks
We show that the geometry of minimum spanning trees (MST) on random graphs is
universal. Due to this geometric universality, we are able to characterise the
energy of MST using a scaling distribution () found using uniform
disorder. We show that the MST energy for other disorder distributions is
simply related to . We discuss the relationship to invasion
percolation (IP), to the directed polymer in a random media (DPRM) and the
implications for the broader issue of universality in disordered systems.Comment: 4 pages, 3 figure
The complexity of Free-Flood-It on 2xn boards
We consider the complexity of problems related to the combinatorial game
Free-Flood-It, in which players aim to make a coloured graph monochromatic with
the minimum possible number of flooding operations. Our main result is that
computing the length of an optimal sequence is fixed parameter tractable (with
the number of colours present as a parameter) when restricted to rectangular
2xn boards. We also show that, when the number of colours is unbounded, the
problem remains NP-hard on such boards. This resolves a question of Clifford,
Jalsenius, Montanaro and Sach (2010)
- âŠ