No abstract

Given a planar graph G on n vertices and an integer parameter r < n, an r-division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O( √ r). We provide a linear-time algorithm for computing r-divisions with few holes. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r-divisions for essentially all values of r. In particular, given an increasing sequence r = (r1, r2, ...), our algorithm can produce a recursive r-division with few holes in linear time.r-divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st-cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs).

We consider the point-to-point (approximate) shortest-path query problem , which is the following generalization of the classical single-source (SSSP) and all-pairs shortest-path (APSP) problems: we are first presented with a network (graph) . A so-called preprocessing algorithm may compute certain information (a data structure or index) to prepare for the next phase. After this preprocessing step, applications may ask shortest-path or distance queries, which should be answered as fast as possible. Due to its many applications in areas such as transportation, networking, and social science, this problem has been considered by researchers from various communities (sometimes under different names): algorithm engineers construct fast route planning methods; database and information systems researchers investigate materialization tradeoffs , query processing on spatial networks , and reachability queries ; and theoretical computer scientists analyze distance oracles and sparse spanners . Related problems are considered for compact routing and distance labeling schemes in networking and distributed computing and for metric embeddings in geometry as well. In this survey, we review selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time. We survey methods for general graphs as well as specialized methods for restricted graph classes, in particular for those classes with arguable practical significance such as planar graphs and complex networks.

Mountain glaciers are known to be strongly affected by global climate change. Here we compute temporally consistent changes in glacier area, surface elevation and ice mass over the entire European Alps between 2000 and 2014. We apply remote sensing techniques on an extensive database of optical and radar imagery covering 93% of the total Alpine glacier volume. Our results reveal rapid glacier retreat across the Alps (−39 km² a −1) with regionally variable ice thickness changes (−0.5 to −0.9 m a −1). The strongest downwasting is observed in the Swiss Glarus and Lepontine Alps with specific mass change rates up to −1.03 m.w.e. a −1. For the entire Alps a mass loss of 1.3 ± 0.2 Gt a −1 (2000-2014) is estimated. Compared to previous studies, our estimated mass changes are similar for the central Alps, but less negative for the lower mountain ranges. These observations provide important information for future research on various socioeconomic impacts like water resource management, risk assessments and tourism.

Supraglacial debris affects glacier mass balance as a thin layer enhances surface melting, while a thick layer reduces it. While many glaciers are debris-covered, global glacier models do not account for debris because its thickness is unknown. We provide the first globally distributed debris thickness estimates using a novel approach combining sub-debris melt and surface temperature inversion methods. Results are evaluated against observations from 22 glaciers. We find the median global debris thickness is ~0.15 ± 0.06 m. In all regions, the net effect of accounting for debris is a reduction in sub-debris melt, on average, by 37%, which can impact regional mass balance by up to 0.40 m water equivalent (w.e.) yr-1. We also find recent observations of similar thinning rates over debris-covered and clean ice glacier tongues is primarily due to differences in ice dynamics. Our results demonstrate the importance of accounting for debris in glacier modeling efforts.