Communication complexity of approximate matching in distributed graphs

In this paper we consider the communication complexity of approximation algorithms for maximum matching in a graph in the message-passing model of distributed computation. The input graph consists of n vertices and edges partitioned over a set of k sites. The output is an alpha-approximate maximum matching in the input graph which has to be reported by one of the sites. We show a lower bound on the communication complexity of Omega(alpha^2 k n) and show that it is tight up to poly-logarithmic factors. This lower bound also applies to other combinatorial problems on graphs in the message-passing computation model, including max-flow and graph sparsification

Atrial size and function in athletes

We wanted to explore if change in LA size would influence LA function, or increase regurgitation in the atrioventricular valves. 595 male elite football players and 47 non-athletic controls were included. End-systolic LA volume and RA area and end-diastolic LV volume and RV area were measured by two-dimensional (2D) echo. Pulsed and colour Doppler were used to estimate tricuspid and mitral regurgitations. 2D longitudinal strain of the 50 football players with the largest LA volumes were compared with the 50 players with the smallest LA volumes. The LA volumes in some athletes were more than tripled, compared to athletes with small atria. 2D strain however, could not reveal any impairment of LA function in the players with the largest atria, compared to those with the smallest LA. Tricuspid valve regurgitation was found in 343 (58%) of the athletes, compared to 17 (36%) of the controls (p<0.01), while mitral regurgitation was found in 116 (20%) football players and seven (15%) controls (NS). Furthermore, the RA area was significantly larger in athletes with tricuspid regurgitation compared to athletes without. The present study demonstrated a huge variation in atrial size between the athletes. This variation, however, had no impact on LA function. Tricuspid regurgitation was significantly more prevalent among the athletes, than among the controls