469 research outputs found
On Finding Optimal Polytrees
Peer reviewe
Evolutionary branching in a stochastic population model with discrete mutational steps
Evolutionary branching is analysed in a stochastic, individual-based
population model under mutation and selection. In such models, the common
assumption is that individual reproduction and life career are characterised by
values of a trait, and also by population sizes, and that mutations lead to
small changes in trait value. Then, traditionally, the evolutionary dynamics is
studied in the limit of vanishing mutational step sizes. In the present
approach, small but non-negligible mutational steps are considered. By means of
theoretical analysis in the limit of infinitely large populations, as well as
computer simulations, we demonstrate how discrete mutational steps affect the
patterns of evolutionary branching. We also argue that the average time to the
first branching depends in a sensitive way on both mutational step size and
population size.Comment: 12 pages, 8 figures. Revised versio
Subjectively interesting connecting trees and forests
Consider a large graph or network, and a user-provided set of query vertices between which the user wishes to explore relations. For example, a researcher may want to connect research papers in a citation network, an analyst may wish to connect organized crime suspects in a communication network, or an internet user may want to organize their bookmarks given their location in the world wide web. A natural way to do this is to connect the vertices in the form of a tree structure that is present in the graph. However, in sufficiently dense graphs, most such trees will be large or somehow trivial (e.g. involving high degree vertices) and thus not insightful. Extending previous research, we define and investigate the new problem of mining subjectively interesting trees connecting a set of query vertices in a graph, i.e., trees that are highly surprising to the specific user at hand. Using information theoretic principles, we formalize the notion of interestingness of such trees mathematically, taking in account certain prior beliefs the user has specified about the graph. A remaining problem is efficiently fitting a prior belief model. We show how this can be done for a large class of prior beliefs. Given a specified prior belief model, we then propose heuristic algorithms to find the best trees efficiently. An empirical validation of our methods on a large real graphs evaluates the different heuristics and validates the interestingness of the given trees
Efficiently Computing Directed Minimum Spanning Trees
Computing a directed minimum spanning tree, called arborescence, is a
fundamental algorithmic problem, although not as common as its undirected
counterpart. In 1967, Edmonds discussed an elegant solution. It was refined to
run in by Tarjan which is optimal for very dense and
very sparse graphs. Gabow et al.~gave a version of Edmonds' algorithm that runs
in , thus asymptotically beating the Tarjan variant in the
regime between sparse and dense. Despite the attention the problem received
theoretically, there exists, to the best of our knowledge, no empirical
evaluation of either of these algorithms. In fact, the version by Gabow et
al.~has never been implemented and, aside from coding competitions, all readily
available Tarjan implementations run in . In this paper, we provide the
first implementation of the version by Gabow et al.~as well as five variants of
Tarjan's version with different underlying data structures. We evaluate these
algorithms and existing solvers on a large set of real-world and random graphs
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