The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log ∆) rounds with high probability, where n is the number of nodes, ∆ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O(log ∆) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.

Understanding groundwater storage (GWS) changes is vital to the utilization and control of water resources in the Tibetan Plateau. However, well level observations are rare in this big area, and reliable hydrology models including GWS are not available. We use hydro-geodesy to quantitate GWS changes in the Tibetan Plateau and surroundings from 2003 to 2009 using a combined analysis of satellite gravity and satellite altimetry data, hydrology models as well as a model of glacial isostatic adjustment (GIA). Release-5 GRACE gravity data are jointly used in a mascon fitting method to estimate the terrestrial water storage (TWS) changes during the period, from which the hydrology contributions and the GIA effects are effectively deducted to give the estimates of GWS changes for 12 selected regions of interest. The hydrology contributions are carefully calculated from glaciers and lakes by ICESat-1 satellite altimetry data, permafrost degradation by an Active-Layer Depth (ALD) model, soil moisture and snow water equivalent by multiple hydrology models, and the GIA effects are calculated with the new ICE-6G_C (VM5a) model. Taking into account the measurement errors and the variability of the models, the uncertainties are rigorously estimated for the TWS changes, the hydrology contributions (including GWS changes) and the GIA effect. For the first time, we show explicitly separated GWS changes in the Tibetan Plateau and adjacent areas except for those to the south of the Himalayas. We find increasing trend rates for eight basins: +2.46 ± 2.24 Gt/yr for the Jinsha River basin, +1.77 ± 2.09 Gt/yr for the Nujiang-Lancangjiang Rivers Source Region, +1.86 ± 1.69 Gt/yr for the Yangtze River Source Region, +1.14 ± 1.39 Gt/yr for the Yellow River Source Region, +1.52 ± 0.95 Gt/yr for the Qaidam basin, +1.66 ± 1.52 Gt/yr for the central Qiangtang Nature Reserve, +5.37 ± 2.17 Gt/yr for the Upper Indus basin and +2.77 ± 0.99 Gt/yr for the Aksu River basin. All these increasing trends are most likely caused by increased runoff recharges from melt water and/or precipitation in the surroundings. We also find that the administrative actions such as the Chinese Ecological Protection and Construction Project help to store more groundwater in the Three Rivers Source Region, and suggest that seepages from the Endorheic basin to the west of it are a possible source for GWS increase in this region. In addition, our estimates for GWS changes basically confirm previous results along Afghanistan, Pakistan, north India and Bangladesh, and clearly reflect the excessive use of groundwater. Our results will benefit the water resource management in the study area, and are of particular significance for the ecological restoration in the Tibetan Plateau.

No abstract