Lessons from the Congested Clique Applied to MapReduce

The main results of this paper are (I) a simulation algorithm which, under quite general constraints, transforms algorithms running on the Congested Clique into algorithms running in the MapReduce model, and (II) a distributed $O(\Delta)$-coloring algorithm running on the Congested Clique which has an expected running time of (i) $O(1)$ rounds, if $\Delta \geq \Theta(\log^4 n)$; and (ii) $O(\log \log n)$ rounds otherwise. Applying the simulation theorem to the Congested-Clique $O(\Delta)$-coloring algorithm yields an $O(1)$-round $O(\Delta)$-coloring algorithm in the MapReduce model. Our simulation algorithm illustrates a natural correspondence between per-node bandwidth in the Congested Clique model and memory per machine in the MapReduce model. In the Congested Clique (and more generally, any network in the $\mathcal{CONGEST}$ model), the major impediment to constructing fast algorithms is the $O(\log n)$ restriction on message sizes. Similarly, in the MapReduce model, the combined restrictions on memory per machine and total system memory have a dominant effect on algorithm design. In showing a fairly general simulation algorithm, we highlight the similarities and differences between these models.Comment: 15 page

Venous injection of a triphasic calcium-based implant in a sheep model of pulmonary embolism demonstrates minimal acute systemic effects.

PURPOSE Implant leakage is the most common complication of vertebral augmentation. Alternative injectable materials must demonstrate intravascular safety comparable to or better than polymethyl methacrylate (PMMA). This study assessed the systemic effects of a triphasic calcium-based implant or PMMA injected directly into the femoral vein in a large animal model designed to mimic severe intravascular implant leakage. METHODS Six skeletally mature female sheep were randomly assigned (n = 3) to either the PMMA or the triphasic implant (AGN1, composition: calcium sulfate, β-tricalcium phosphate, brushite) treatment group. Femoral veins of each sheep were directly injected with 0.5 mL of implant material to mimic leakage volumes reported during PMMA vertebroplasty. To compare acute systemic effects of the materials, cardiovascular parameters, laboratory coagulation markers, and calcium and sulfate serum levels were monitored for 60 min after implant injection. Thrombotic and embolic events were evaluated by radiologic imaging, necropsy, and histopathology. RESULTS Heart rate, systemic arterial blood pressure, arterial oxygenation, arterial carbon dioxide content, and coagulation markers remained within physiological range after either AGN1 or PMMA injection. No blood flow interruption in the larger pulmonary vessels was observed in either group. Lung histopathology revealed that the severity of thrombotic changes after AGN1 injection was minimal to slight, while changes after PMMA injection were minimal to massive. CONCLUSION Acute systemic effects of intravascular AGN1 appeared to be comparable to or less than that of intravascular PMMA. Furthermore, in this preliminary study, the severity and incidence of pulmonary histological changes were lower for AGN1 compared to PMMA