Algorithms for Stable Matching and Clustering in a Grid

We study a discrete version of a geometric stable marriage problem originally proposed in a continuous setting by Hoffman, Holroyd, and Peres, in which points in the plane are stably matched to cluster centers, as prioritized by their distances, so that each cluster center is apportioned a set of points of equal area. We show that, for a discretization of the problem to an $n\times n$ grid of pixels with $k$ centers, the problem can be solved in time $O(n^2 \log^5 n)$, and we experiment with two slower but more practical algorithms and a hybrid method that switches from one of these algorithms to the other to gain greater efficiency than either algorithm alone. We also show how to combine geometric stable matchings with a $k$-means clustering algorithm, so as to provide a geometric political-districting algorithm that views distance in economic terms, and we experiment with weighted versions of stable $k$-means in order to improve the connectivity of the resulting clusters.Comment: 23 pages, 12 figures. To appear (without the appendices) at the 18th International Workshop on Combinatorial Image Analysis, June 19-21, 2017, Plovdiv, Bulgari

4MOST: 4-metre Multi-Object Spectroscopic Telescope

1134MOST is a wide-field, high-multiplex spectroscopic survey facility under development for the VISTA telescope of the European Southern Observatory (ESO). Its main science drivers are in the fields of galactic archeology, high-energy physics, galaxy evolution and cosmology. 4MOST will in particular provide the spectroscopic complements to the large area surveys coming from space missions like Gaia, eROSITA, Euclid, and PLATO and from ground-based facilities like VISTA, VST, DES, LSST and SKA. The 4MOST baseline concept features a 2.5 degree diameter field-of-view with ~2400 fibres in the focal surface that are configured by a fibre positioner based on the tilting spine principle. The fibres feed two types of spectrographs; ~1600 fibres go to two spectrographs with resolution R<5000 (λ~390-930 nm) and ~800 fibres to a spectrograph with R>18,000 (λ~392-437 nm and 515-572 nm and 605-675 nm). Both types of spectrographs are fixed-configuration, three-channel spectrographs. 4MOST will have an unique operations concept in which 5 year public surveys from both the consortium and the ESO community will be combined and observed in parallel during each exposure, resulting in more than 25 million spectra of targets spread over a large fraction of the southern sky. The 4MOST Facility Simulator (4FS) was developed to demonstrate the feasibility of this observing concept. 4MOST has been accepted for implementation by ESO with operations expected to start by the end of 2020. This paper provides a top-level overview of the 4MOST facility, while other papers in these proceedings provide more detailed descriptions of the instrument concept[1], the instrument requirements development[2], the systems engineering implementation[3], the instrument model[4], the fibre positioner concepts[5], the fibre feed[6], and the spectrographs[7].nonenonede Jong Roelof S.; Barden Sam; Bellido-Tirado Olga; Brynnel Joar; Chiappini Cristina; Depagne Éric; Haynes Roger; Johl Diana; Phillips Daniel P.; Schnurr Olivier; Schwope Axel D.; Walcher Jakob; Bauer Svend M.; Cescutti G; Cioni Maria-Rosa L.; Dionies Frank; Enke Harry; Haynes Dionne M.; Kelz Andreas; Kitaura Francisco S.; Lamer Georg; Minchev Ivan; Müller Volker; Nuza Sebastián. E.; Olaya Jean-Christophe; Piffl Tilmann; Popow Emil; Saviauk Allar; Steinmetz Matthias; Ural Uǧur; Valentini Monica; Winkler Roland; Wisotzki Lutz; Ansorge Wolfgang R.; Banerji Manda; Gonzalez Solares Eduardo; Irwin Mike; Kennicutt Robert C.; King David M. P.; McMahon Richard; Koposov Sergey; Parry Ian R.; Sun Xiaowei; Walton Nicholas A.; Finger Gert; Iwert Olaf; Krumpe Mirko; Lizon Jean-Louis; Mainieri Vincenzo; Amans Jean-Philippe; Bonifacio Piercarlo; Cohen Matthieu; François Patrick; Jagourel Pascal; Mignot Shan B.; Royer Frédéric; Sartoretti Paola; Bender Ralf; Hess Hans-Joachim; Lang-Bardl Florian; Muschielok Bernard; Schlichter Jörg; Böhringer Hans; Boller Thomas; Bongiorno Angela; Brusa Marcella; Dwelly Tom; Merloni Andrea; Nandra Kirpal; Salvato Mara; Pragt Johannes H.; Navarro Ramón; Gerlofsma Gerrit; Roelfsema Ronald; Dalton Gavin B.; Middleton Kevin F.; Tosh Ian A.; Boeche Corrado; Caffau Elisabetta; Christlieb Norbert; Grebel Eva K.; Hansen Camilla J.; Koch Andreas; Ludwig Hans-G.; Mandel Holger; Quirrenbach Andreas; Sbordone Luca; Seifert Walter; Thimm Guido; Helmi Amina; trager Scott C.; Bensby Thomas; Feltzing Sofia; Ruchti Gregory; Edvardsson Bengt; Korn Andreas; Lind Karin; Boland Wilfried; Colless Matthew; Frost Gabriella; Gilbert James; Gillingham Peter; Lawrence Jon; Legg Neville; Saunders Will; Sheinis Andrew; Driver Simon; Robotham Aaron; Bacon Roland; Caillier Patrick; Kosmalski Johan; Laurent Florence; Richard Johande Jong Roelof, S.; Barden, Sam; Bellido-Tirado, Olga; Brynnel, Joar; Chiappini, Cristina; Depagne, Éric; Haynes, Roger; Johl, Diana; Phillips Daniel, P.; Schnurr, Olivier; Schwope Axel, D.; Walcher, Jakob; Bauer Svend, M.; Cescutti, G; Cioni Maria-Rosa, L.; Dionies, Frank; Enke, Harry; Haynes Dionne, M.; Kelz, Andreas; Kitaura Francisco, S.; Lamer, Georg; Minchev, Ivan; Müller, Volker; Nuza, Sebastián. E.; Olaya, Jean-Christophe; Piffl, Tilmann; Popow, Emil; Saviauk, Allar; Steinmetz, Matthias; Ural, Uǧur; Valentini, Monica; Winkler, Roland; Wisotzki, Lutz; Ansorge Wolfgang, R.; Banerji, Manda; Gonzalez Solares, Eduardo; Irwin, Mike; Kennicutt Robert, C.; King David, M. P.; Mcmahon, Richard; Koposov, Sergey; Parry Ian, R.; Sun, Xiaowei; Walton Nicholas, A.; Finger, Gert; Iwert, Olaf; Krumpe, Mirko; Lizon, Jean-Louis; Mainieri, Vincenzo; Amans, Jean-Philippe; Bonifacio, Piercarlo; Cohen, Matthieu; François, Patrick; Jagourel, Pascal; Mignot Shan, B.; Royer, Frédéric; Sartoretti, Paola; Bender, Ralf; Hess, Hans-Joachim; Lang-Bardl, Florian; Muschielok, Bernard; Schlichter, Jörg; Böhringer, Hans; Boller, Thomas; Bongiorno, Angela; Brusa, Marcella; Dwelly, Tom; Merloni, Andrea; Nandra, Kirpal; Salvato, Mara; Pragt Johannes, H.; Navarro, Ramón; Gerlofsma, Gerrit; Roelfsema, Ronald; Dalton Gavin, B.; Middleton Kevin, F.; Tosh Ian, A.; Boeche, Corrado; Caffau, Elisabetta; Christlieb, Norbert; Grebel Eva, K.; Hansen Camilla, J.; Koch, Andreas; Ludwig, Hans-G.; Mandel, Holger; Quirrenbach, Andreas; Sbordone, Luca; Seifert, Walter; Thimm, Guido; Helmi, Amina; trager Scott, C.; Bensby, Thomas; Feltzing, Sofia; Ruchti, Gregory; Edvardsson, Bengt; Korn, Andreas; Lind, Karin; Boland, Wilfried; Colless, Matthew; Frost, Gabriella; Gilbert, James; Gillingham, Peter; Lawrence, Jon; Legg, Neville; Saunders, Will; Sheinis, Andrew; Driver, Simon; Robotham, Aaron; Bacon, Roland; Caillier, Patrick; Kosmalski, Johan; Laurent, Florence; Richard, Joha