A Practical Parallel Algorithm for Diameter Approximation of Massive Weighted Graphs

We present a space and time efficient practical parallel algorithm for approximating the diameter of massive weighted undirected graphs on distributed platforms supporting a MapReduce-like abstraction. The core of the algorithm is a weighted graph decomposition strategy generating disjoint clusters of bounded weighted radius. Theoretically, our algorithm uses linear space and yields a polylogarithmic approximation guarantee; moreover, for important practical classes of graphs, it runs in a number of rounds asymptotically smaller than those required by the natural approximation provided by the state-of-the-art $\Delta$-stepping SSSP algorithm, which is its only practical linear-space competitor in the aforementioned computational scenario. We complement our theoretical findings with an extensive experimental analysis on large benchmark graphs, which demonstrates that our algorithm attains substantial improvements on a number of key performance indicators with respect to the aforementioned competitor, while featuring a similar approximation ratio (a small constant less than 1.4, as opposed to the polylogarithmic theoretical bound)

Evolution of populations expanding on curved surfaces

The expansion of a population into new habitat is a transient process that leaves its footprints in the genetic composition of the expanding population. How the structure of the environment shapes the population front and the evolutionary dynamics during such a range expansion is little understood. Here, we investigate the evolutionary dynamics of populations consisting of many selectively neutral genotypes expanding on curved surfaces. Using a combination of individual-based off-lattice simulations, geometrical arguments, and lattice-based stepping-stone simulations, we characterise the effect of individual bumps on an otherwise flat surface. Compared to the case of a range expansion on a flat surface, we observe a transient relative increase, followed by a decrease, in neutral genetic diversity at the population front. In addition, we find that individuals at the sides of the bump have a dramatically increased expected number of descendants, while their neighbours closer to the bump's centre are far less lucky. Both observations can be explained using an analytical description of straight paths (geodesics) on the curved surface. Complementing previous studies of heterogeneous flat environments, the findings here build our understanding of how complex environments shape the evolutionary dynamics of expanding populations.Comment: This preprint has also been posted to http://www.biorxiv.org with doi: 10.1101/406280. Seven pages with 5 figures, plus an appendix containing 3 pages with 1 figur

Privaatsust säilitavad paralleelarvutused graafiülesannete jaoks

Turvalisel mitmeosalisel arvutusel põhinevate reaalsete privaatsusrakenduste loomine on SMC-protokolli arvutusosaliste ümmarguse keerukuse tõttu keeruline. Privaatsust säilitavate tehnoloogiate uudsuse ja nende probleemidega kaasnevate suurte arvutuskulude tõttu ei ole paralleelseid privaatsust säilitavaid graafikualgoritme veel uuritud. Graafikalgoritmid on paljude arvutiteaduse rakenduste selgroog, nagu navigatsioonisüsteemid, kogukonna tuvastamine, tarneahela võrk, hüperspektraalne kujutis ja hõredad lineaarsed lahendajad. Graafikalgoritmide suurte privaatsete andmekogumite töötlemise kiirendamiseks ja kõrgetasemeliste arvutusnõuete täitmiseks on vaja privaatsust säilitavaid paralleelseid algoritme. Seetõttu esitleb käesolev lõputöö tipptasemel protokolle privaatsuse säilitamise paralleelarvutustes erinevate graafikuprobleemide jaoks, ühe allika lühima tee, kõigi paaride lühima tee, minimaalse ulatuva puu ja metsa ning algebralise tee arvutamise. Need uued protokollid on üles ehitatud kombinatoorsete ja algebraliste graafikualgoritmide põhjal lisaks SMC protokollidele. Nende protokollide koostamiseks kasutatakse ka ühe käsuga mitut andmeoperatsiooni, et vooru keerukust tõhusalt vähendada. Oleme väljapakutud protokollid juurutanud Sharemind SMC platvormil, kasutades erinevaid graafikuid ja võrgukeskkondi. Selles lõputöös kirjeldatakse uudseid paralleelprotokolle koos nendega seotud algoritmide, tulemuste, kiirendamise, hindamiste ja ulatusliku võrdlusuuringuga. Privaatsust säilitavate ühe allika lühimate teede ja minimaalse ulatusega puuprotokollide tegelike juurutuste tulemused näitavad tõhusat meetodit, mis vähendas tööaega võrreldes varasemate töödega sadu kordi. Lisaks ei ole privaatsust säilitavate kõigi paaride lühima tee protokollide hindamine ja ulatuslik võrdlusuuringud sarnased ühegi varasema tööga. Lisaks pole kunagi varem käsitletud privaatsust säilitavaid metsa ja algebralise tee arvutamise protokolle.Constructing real-world privacy applications based on secure multiparty computation is challenging due to the round complexity of the computation parties of SMC protocol. Due to the novelty of privacy-preserving technologies and the high computational costs associated with these problems, parallel privacy-preserving graph algorithms have not yet been studied. Graph algorithms are the backbone of many applications in computer science, such as navigation systems, community detection, supply chain network, hyperspectral image, and sparse linear solvers. In order to expedite the processing of large private data sets for graphs algorithms and meet high-end computational demands, privacy-preserving parallel algorithms are needed. Therefore, this Thesis presents the state-of-the-art protocols in privacy-preserving parallel computations for different graphs problems, single-source shortest path (SSSP), All-pairs shortest path (APSP), minimum spanning tree (MST) and forest (MSF), and algebraic path computation. These new protocols have been constructed based on combinatorial and algebraic graph algorithms on top of the SMC protocols. Single-instruction-multiple-data (SIMD) operations are also used to build those protocols to reduce the round complexities efficiently. We have implemented the proposed protocols on the Sharemind SMC platform using various graphs and network environments. This Thesis outlines novel parallel protocols with their related algorithms, the results, speed-up, evaluations, and extensive benchmarking. The results of the real implementations of the privacy-preserving single-source shortest paths and minimum spanning tree protocols show an efficient method that reduced the running time hundreds of times compared with previous works. Furthermore, the evaluation and extensive benchmarking of privacy-preserving All-pairs shortest path protocols are not similar to any previous work. Moreover, the privacy-preserving minimum spanning forest and algebraic path computation protocols have never been addressed before.https://www.ester.ee/record=b555865

Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings

This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S from the tangent space at v. We characterize the `cut locus' (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R^2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo

A Faster Distributed Single-Source Shortest Paths Algorithm

We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of $\tilde \Omega (\sqrt{n} + D)$ rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where $n$ is the number of nodes in the network and $D$ is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs $\tilde O(n^{2/3} D^{1/3} + n^{5/6})$ rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing $\tilde O (\sqrt{n D})$ rounds and the second performing $\tilde O (\sqrt{n} D^{1/4} + n^{3/5} + D)$ rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a $(1 + \epsilon)$-approximation $\tilde O ((\sqrt{n} D^{1/4} + D) / \epsilon)$-round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018

JGraphT -- A Java library for graph data structures and algorithms

Mathematical software and graph-theoretical algorithmic packages to efficiently model, analyze and query graphs are crucial in an era where large-scale spatial, societal and economic network data are abundantly available. One such package is JGraphT, a programming library which contains very efficient and generic graph data-structures along with a large collection of state-of-the-art algorithms. The library is written in Java with stability, interoperability and performance in mind. A distinctive feature of this library is the ability to model vertices and edges as arbitrary objects, thereby permitting natural representations of many common networks including transportation, social and biological networks. Besides classic graph algorithms such as shortest-paths and spanning-tree algorithms, the library contains numerous advanced algorithms: graph and subgraph isomorphism; matching and flow problems; approximation algorithms for NP-hard problems such as independent set and TSP; and several more exotic algorithms such as Berge graph detection. Due to its versatility and generic design, JGraphT is currently used in large-scale commercial, non-commercial and academic research projects. In this work we describe in detail the design and underlying structure of the library, and discuss its most important features and algorithms. A computational study is conducted to evaluate the performance of JGraphT versus a number of similar libraries. Experiments on a large number of graphs over a variety of popular algorithms show that JGraphT is highly competitive with other established libraries such as NetworkX or the BGL.Comment: Major Revisio