Graph Packing over a Rooted Tree

This paper investigates the computational complexity of the graph packing problem over a rooted tree (GPT) as a generalization of the one dimensional bin packing problem, where both the bins and the set of items to be packed are rooted trees. GPT is defined under two problem settings, edge GPT (EPT) and node GPT (NPT). In EPT, the items packed in a bin cannot share any edge but can share some node, while in NPT, the items can share neither node nor edge. We first prove that these problems are in general NP-complete, which strongly suggests that these problems are computationally intractable. However, for the case where the number k of different kinds of items is fixed, we derive a recursive formula of dynamic programming for the minimum number of bins required to pack all the items. This formula can be solved in polynomial time, if the bins and items are all uniform trees and/or comb-shaped trees in which each non-leaf node has the same number of sons. Furthermore, for GPT's with bins of uniform (d, H) trees and only one kind of item, of uniform (d, h) trees, we derive explicit formulas for the number of bins required

Path (or cycle)-trees with Graph Equations involving Line and Split Graphs

H-trees generalizes the existing notions of trees, higher dimensional trees and k-ctrees. The characterizations and properties of both Pk-trees for k at least 4 and Cn-trees for n at least 5 and their hamiltonian property, dominations, planarity, chromatic and b-chromatic numbers are established. The conditions under which Pk-trees for k at least 3 (resp. Cn-trees for n at least 4), are the line graphs are determined. The relationship between path-trees and split graphs are developed

Carbon allocation during defoliation : testing a defense-growth trade-off in balsam fir

During repetitive defoliation events, carbon can become limiting for trees. To maintain growth and survival, the resources have to be shared more efficiently, which could result in a trade-off between the different physiological processes of a plant. The objective of this study was to assess the effect of defoliation in carbon allocation of balsam fir [Abies balsamea (L.) Mill.] to test the presence of a trade-off between allocation to growth, carbon storage, and defense. Three defoliation intensities [control (C-trees, 0% defoliation), moderately (M-trees, 41-60%), and heavily (H-trees, 61-80%) defoliated] were selected in order to monitor several variables related to stem growth (wood formation in xylem), carbon storage in stem and needle (non-structural soluble sugars and starch), and defense components in needles (terpenoids compound) from May to October 2011. The concentration of starch was drastically reduced in both wood and leaves of H-trees with a quasi-absence of carbon partitioning to storage in early summer. Fewer kinds of monoterpenes and sesquiterpenes were formed with an increasing level of defoliation indicating a lower carbon allocation for the production of defense. The carbon allocation to wood formation gradually reduced at increasing defoliation intensities, with a lower growth rate and fewer tracheids resulting in a reduced carbon sequestration in cell walls. The hypothesis of a trade-off between the allocations to defense components and to non-structural (NCS) and structural (growth) carbon was rejected as most of the measured variables decreased with increasing defoliation. The starch amount was highly indicative of the tree carbon status at different defoliation intensity and future research should focus on the mechanism of starch utilization for survival and growth following an outbreak

Markov convexity and local rigidity of distorted metrics

It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p-convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.Comment: 47 pages, full version, replacing the previous version which was an announcemen

On the design and implementation of broadcast and global combine operations using the postal model

There are a number of models that were proposed in recent years for message passing parallel systems. Examples are the postal model and its generalization the LogP model. In the postal model a parameter λ is used to model the communication latency of the message-passing system. Each node during each round can send a fixed-size message and, simultaneously, receive a message of the same size. Furthermore, a message sent out during round r will incur a latency of hand will arrive at the receiving node at round r + λ - 1. Our goal in this paper is to bridge the gap between the theoretical modeling and the practical implementation. In particular, we investigate a number of practical issues related to the design and implementation of two collective communication operations, namely, the broadcast operation and the global combine operation. Those practical issues include, for example, 1) techniques for measurement of the value of λ on a given machine, 2) creating efficient broadcast algorithms that get the latency hand the number of nodes n as parameters and 3) creating efficient global combine algorithms for parallel machines with λ which is not an integer. We propose solutions that address those practical issues and present results of an experimental study of the new algorithms on the Intel Delta machine. Our main conclusion is that the postal model can help in performance prediction and tuning, for example, a properly tuned broadcast improves the known implementation by more than 20%

Synthèse de réseaux de distribution d'horloges en présence de variations du procédé de fabrication

Design of clock distributions networks in presence of process variations -- Importance des variations spatiales de la constante de temps du transistor MOS -- Pipelined H-trees for high-speed clocking of large integrated systems in presence of process variations -- Conception de réseaux de distribution d'horloges fiables et à faible consommation de puissance -- Design of low-power and reliable logic-based H-trees -- Sources des variations spatiales de la constante de temps du transistor MOS -- Spatial characterization of process variations via MOS transistor time constants in VLSI & WSI -- Techniques de minimisation du biais de synchronisation par calibration de délai -- Minimizing process-induced skew using delay tuning