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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.

We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: 1• Given a desired space allocation S ∈ [n lg lg n, n 2 ], we show how to construct inÕ(S) time a data structure of size O(S) that answers distance queries inÕ(n/ √ S) time per query. The best distance oracles for planar graphs until the current work are due to Cabello (SODA 2006), Chen and Xu (STOC 2000), Djidjev (WG 1996), and Fakcharoenphol and Rao (FOCS 2001). For σ ∈ (1, 4/3) and space S = n σ , we essentially improve the query time from n 2 /S to p n 2 /S.• As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever k ∈ [ √ n, n).• We provide a linear-space exact distance oracle for planar graphs with query time O(n 1/2+ ) for any constant > 0. This is the first such data structure with provable sublinear query time.• For edge lengths ≥ 1, we provide an exact distance oracle of spaceÕ(n) such that for any pair of nodes at distance the query time isÕ(min{ , √ n}). Comparable query performance had been observed experimentally but could not be explained theoretically.Our data structures with superlinear space are based on the following new tool: given a non-self-crossing cycle C with c = O( √ n) nodes, we can preprocess G inÕ(n) time to produce a data structure of size O(n lg lg c) that can answer the following queries inÕ(c) time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and provides an alternative for a related data structure of Klein (SODA 2005), which reports distances to the boundary of a face, rather than a cycle.

Abstract. Glaciers in tropical regions are very sensitive to climatic variations and thus strongly affected by climate change. The majority of the tropical glaciers worldwide are located in the Peruvian Andes, which have shown significant ice loss in the last century. Here, we present the first multi-temporal, region-wide survey of geodetic mass balances and glacier area fluctuations throughout Peru covering the period 2000–2016. Glacier extents are derived from Landsat imagery by performing automatic glacier delineation based on a combination of the NDSI and band ratio method and final manual inspection and correction. The mapping of debris-covered glacier extents is supported by synthetic aperture radar (SAR) coherence information. A total glacier area loss of -548.5±65.7 km2 (−29 %, −34.3 km2 a−1) is obtained for the study period. Using interferometric satellite SAR acquisitions, bi-temporal geodetic mass balances are derived. An average specific mass balance of -296±41 kg m−2 a−1 is found throughout Peru for the period 2000–2016. However, there are strong regional and temporal differences in the mass budgets ranging from 45±97 to -752±452 kg m−2 a−1. The ice loss increased towards the end of the observation period. Between 2013 and 2016, a retreat of the glacierized area of -203.8±65.7 km2 (−16 %, −101.9 km2 a−1) is mapped and the average mass budget amounts to -660±178 kg m−2 a−1. The glacier changes revealed can be attributed to changes in the climatic settings in the study region, derived from ERA-Interim reanalysis data and the Oceanic Nino Index. The intense El Niño activities in 2015/16 are most likely the trigger for the increased change rates in the time interval 2013–2016. Our observations provide fundamental information on the current dramatic glacier changes for local authorities and for the calibration and validation of glacier change projections.

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