Search

Directed acyclic graphs are often used to model situations and problems in real life. If we consider the topological ordering of the graph as a process of arranging the vertices in the best possible way considering the constraints caused by the direction of edges, then it makes sense to try to optimize this process by minimizing the distances between vertices in the ordering. For this purpose, we define measures based on distances between vertices in the topological ordering that allow us to construct a graph with optimal topological ordering regarding a specific measure thus minimizing the complexity of the system represented by the graph. We explore minimal and maximal values of the defined measures and comment on the topology of graphs for which maximal and minimal values are obtained. Potentially, the proved bounds could be used to benchmark existing algorithms, devise new approximation algorithms or branch and bound schemas for some scheduling problems that are usually of hard computational complexity.

The effect of biomass dynamics on the estimation of watercolumn primary production is analyzed, by coupling a primary production model to a simple growth equation for phytoplankton. The production model is formulated with depth- and time-resolved biomass, and placed in the context of earlier models, with emphasis on the canonical solution for watercolumn production. A relation between the canonical solution and the general solution for the case of an arbitrary depth-dependent biomass profile was derived, together with an analytical solution for watercolumn production in case of a depth dependent biomass profile described with the shifted Gaussian function. The analysis was further extended to the case of a time-dependent, mixed-layer biomass, and two additional analytical solutions to this problem were derived, the first in case of increasing mixed-layer biomass and the second in case of declining biomass. The solutions were tested with Hawaii Ocean Time-series data. The canonical solution for mixed-layer production has proven to be a good model for this data set. The shifted Gaussian function was demonstrated to be an accurate model for the measured biomass profiles and the shifted Gaussian parameters extracted from the measured profiles were further used in the analytical solution for watercolumn production and results compared with data. The influence of time-dependent biomass on mixed-layer production was studied through analytical solutions. Re-examining the Critical Depth Hypothesis we derived an expression for the daily increase in mixed-layer biomass. Finally, the work was placed in a remote sensing context and the time-dependent model for biomass related to the remotely sensed-biomass.