@inproceedings{2043,
abstract = {Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically – as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems [1], we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 6GB of memory per node.},
author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan},
booktitle = {Proceedings of the Workshop on Algorithm Engineering and Experiments},
editor = { McGeoch, Catherine and Meyer, Ulrich},
location = {Portland, USA},
pages = {31 -- 38},
publisher = {Society of Industrial and Applied Mathematics},
title = {{Distributed computation of persistent homology}},
doi = {10.1137/1.9781611973198.4},
year = {2014},
}
@inproceedings{2156,
abstract = {We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s).},
author = {Bauer, Ulrich and Ge, Xiaoyin and Wang, Yusu},
booktitle = {Proceedings of the Annual Symposium on Computational Geometry},
location = {Kyoto, Japan},
pages = {464 -- 473},
publisher = {ACM},
title = {{Measuring distance between Reeb graphs}},
doi = {10.1145/2582112.2582169},
year = {2014},
}
@inproceedings{2210,
abstract = {A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon. In this paper, we ask the reverse question: Given the straight skeleton (in form of a tree with a drawing in the plane, but with the exact position of the leaves unspecified), can we reconstruct the polygon? We show that in most cases there exists at most one polygon; in the remaining case there is an infinite number of polygons determined by one angle that can range in an interval. We can find this (set of) polygon(s) in linear time in the Real RAM computer model.},
author = {Biedl, Therese and Held, Martin and Huber, Stefan},
booktitle = {29th European Workshop on Computational Geometry},
location = {Braunschweig, Germany},
pages = {95 -- 98},
publisher = {TU Braunschweig},
title = {{Reconstructing polygons from embedded straight skeletons}},
year = {2013},
}
@article{2304,
abstract = {This extended abstract is concerned with the irregularities of distribution of one-dimensional permuted van der Corput sequences that are generated from linear permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying earlier results of Faure and Niederreiter.},
author = {Pausinger, Florian},
journal = {Electronic Notes in Discrete Mathematics},
pages = {43 -- 50},
publisher = {Elsevier},
title = {{Van der Corput sequences and linear permutations}},
doi = {10.1016/j.endm.2013.07.008},
volume = {43},
year = {2013},
}
@article{2859,
abstract = {Given a continuous function f:X-R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X=R3 has ramifications in the fields of medical imaging and scientific visualization.},
author = {Bendich, Paul and Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
journal = {Homology, Homotopy and Applications},
number = {1},
pages = {51 -- 72},
publisher = {International Press},
title = {{Homology and robustness of level and interlevel sets}},
doi = {10.4310/HHA.2013.v15.n1.a3},
volume = {15},
year = {2013},
}
@inproceedings{2209,
abstract = {A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon or planar straight-line graph. In this paper, we ask the reverse question: Given the straight skeleton (in form of a planar straight-line graph, with some rays to infinity), can we reconstruct a planar straight-line graph for which this was the straight skeleton? We show how to reduce this problem to the problem of finding a line that intersects a set of convex polygons. We can find these convex polygons and all such lines in $O(nlog n)$ time in the Real RAM computer model, where $n$ denotes the number of edges of the input graph. We also explain how our approach can be used for recognizing Voronoi diagrams of points, thereby completing a partial solution provided by Ash and Bolker in 1985.
},
author = {Biedl, Therese and Held, Martin and Huber, Stefan},
location = {St. Petersburg, Russia},
pages = {37 -- 46},
publisher = {IEEE},
title = {{Recognizing straight skeletons and Voronoi diagrams and reconstructing their input}},
doi = {10.1109/ISVD.2013.11},
year = {2013},
}
@inproceedings{2812,
abstract = {We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H* (X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in ℝ3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.},
author = {Attali, Dominique and Bauer, Ulrich and Devillers, Olivier and Glisse, Marc and Lieutier, André},
booktitle = {Proceedings of the 29th annual symposium on Computational Geometry},
location = {Rio de Janeiro, Brazil},
pages = {117 -- 125},
publisher = {ACM},
title = {{Homological reconstruction and simplification in R3}},
doi = {10.1145/2462356.2462373},
year = {2013},
}
@inproceedings{2901,
abstract = { We introduce the M-modes problem for graphical models: predicting the M label configurations of highest probability that are at the same time local maxima of the probability landscape. M-modes have multiple possible applications: because they are intrinsically diverse, they provide a principled alternative to non-maximum suppression techniques for structured prediction, they can act as codebook vectors for quantizing the configuration space, or they can form component centers for mixture model approximation. We present two algorithms for solving the M-modes problem. The first algorithm solves the problem in polynomial time when the underlying graphical model is a simple chain. The second algorithm solves the problem for junction chains. In synthetic and real dataset, we demonstrate how M-modes can improve the performance of prediction. We also use the generated modes as a tool to understand the topography of the probability distribution of configurations, for example with relation to the training set size and amount of noise in the data. },
author = {Chen, Chao and Kolmogorov, Vladimir and Yan, Zhu and Metaxas, Dimitris and Lampert, Christoph},
location = {Scottsdale, AZ, United States},
pages = {161 -- 169},
publisher = {JMLR},
title = {{Computing the M most probable modes of a graphical model}},
volume = {31},
year = {2013},
}
@inproceedings{2906,
abstract = {Motivated by an application in cell biology, we describe an extension of the kinetic data structures framework from Delaunay triangulations to fixed-radius alpha complexes. Our algorithm is implemented
using CGAL, following the exact geometric computation paradigm. We report on several
techniques to accelerate the computation that turn our implementation applicable to the underlying biological
problem.},
author = {Kerber, Michael and Edelsbrunner, Herbert},
booktitle = {2013 Proceedings of the 15th Workshop on Algorithm Engineering and Experiments},
location = {New Orleans, LA, United States},
pages = {70 -- 77},
publisher = {Society of Industrial and Applied Mathematics},
title = {{3D kinetic alpha complexes and their implementation}},
doi = {10.1137/1.9781611972931.6},
year = {2013},
}
@inproceedings{2843,
abstract = {Mathematical objects can be measured unambiguously, but not so objects from our physical world. Even the total length of tubelike shapes has its difficulties. We introduce a combination of geometric, probabilistic, and topological methods to design a stable length estimate for tube-like shapes; that is: one that is insensitive to small shape changes.},
author = {Edelsbrunner, Herbert and Pausinger, Florian},
booktitle = {17th IAPR International Conference on Discrete Geometry for Computer Imagery},
location = {Seville, Spain},
pages = {XV -- XIX},
publisher = {Springer},
title = {{Stable length estimates of tube-like shapes}},
doi = {10.1007/978-3-642-37067-0},
volume = {7749},
year = {2013},
}
@article{2887,
abstract = {Root system growth and development is highly plastic and is influenced by the surrounding environment. Roots frequently grow in heterogeneous environments that include interactions from neighboring plants and physical impediments in the rhizosphere. To investigate how planting density and physical objects affect root system growth, we grew rice in a transparent gel system in close proximity with another plant or a physical object. Root systems were imaged and reconstructed in three dimensions. Root-root interaction strength was calculated using quantitative metrics that characterize the extent towhich the reconstructed root systems overlap each other. Surprisingly, we found the overlap of root systems of the same genotype was significantly higher than that of root systems of different genotypes. Root systems of the same genotype tended to grow toward each other but those of different genotypes appeared to avoid each other. Shoot separation experiments excluded the possibility of aerial interactions, suggesting root communication. Staggered plantings indicated that interactions likely occur at root tips in close proximity. Recognition of obstacles also occurred through root tips, but through physical contact in a size-dependent manner. These results indicate that root systems use two different forms of communication to recognize objects and alter root architecture: root-root recognition, possibly mediated through root exudates, and root-object recognition mediated by physical contact at the root tips. This finding suggests that root tips act as local sensors that integrate rhizosphere information into global root architectural changes.},
author = {Fang, Suqin and Clark, Randy and Zheng, Ying and Iyer Pascuzzi, Anjali and Weitz, Joshua and Kochian, Leon and Edelsbrunner, Herbert and Liao, Hong and Benfey, Philip},
journal = {PNAS},
number = {7},
pages = {2670 -- 2675},
publisher = {National Academy of Sciences},
title = {{Genotypic recognition and spatial responses by rice roots}},
doi = {10.1073/pnas.1222821110},
volume = {110},
year = {2013},
}
@inproceedings{2807,
abstract = {We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks, given topological spaces X; Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that X and Y are represented as finite simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map. In this generality the problem is undecidable, as follows from Novikov's result from the 1950s on uncomputability of the fundamental group π1(Y ). We thus study the problem under the assumption that, for some k ≥ 2, Y is (k - 1)-connected; informally, this means that Y has \no holes up to dimension k-1" (a basic example of such a Y is the sphere Sk). We prove that, on the one hand, this problem is still undecidable for dimX = 2k. On the other hand, for every fixed k ≥ 2, we obtain an algorithm that solves the extension problem in polynomial time assuming Y (k - 1)-connected and dimX ≤ 2k - 1. For dimX ≤ 2k - 2, the algorithm also provides a classification of all extensions up to homotopy (continuous deformation). This relies on results of our SODA 2012 paper, and the main new ingredient is a machinery of objects with polynomial-time homology, which is a polynomial-time analog of objects with effective homology developed earlier by Sergeraert et al. We also consider the computation of the higher homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, Anick proved in 1989 that computing πk(Y ) is #P-hard if k is a part of input, where Y is a cell complex with certain rather compact encoding. We strengthen his result to #P-hardness for Y given as a simplicial complex. },
author = {Čadek, Martin and Krcál, Marek and Matoušek, Jiří and Vokřínek, Lukáš and Wagner, Uli},
booktitle = {45th Annual ACM Symposium on theory of computing},
location = {Palo Alto, CA, United States},
pages = {595 -- 604},
publisher = {ACM},
title = {{Extending continuous maps: Polynomiality and undecidability}},
doi = {10.1145/2488608.2488683},
year = {2013},
}
@article{2939,
abstract = {In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any Γ > 0, it returns only those homology classes with persistence at least Γ. Instead of the classical reduction via column operations, our algorithm performs rank computations on submatrices of the boundary matrix. For an arbitrary constant δ ∈ (0, 1), the running time is O (C (1 - δ) Γ R d (n) log n), where C (1 - δ) Γ is the number of homology classes with persistence at least (1 - δ) Γ, n is the total number of simplices in the complex, d its dimension, and R d (n) is the complexity of computing the rank of an n × n matrix with O (d n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O (C (1 - δ) Γ n 2.376) algorithm, an O (C (1 - δ) Γ n 2.28) Las-Vegas algorithm, or an O (C (1 - δ) Γ n 2 + ε{lunate}) Monte-Carlo algorithm for an arbitrary ε{lunate} > 0. The space complexity of the Monte-Carlo version is bounded by O (d n) = O (n log n).},
author = {Chen, Chao and Kerber, Michael},
journal = {Computational Geometry: Theory and Applications},
number = {4},
pages = {435 -- 447},
publisher = {Elsevier},
title = {{An output sensitive algorithm for persistent homology}},
doi = {10.1016/j.comgeo.2012.02.010},
volume = {46},
year = {2013},
}
@article{2822,
abstract = {Identification of genes that control root system architecture in crop plants requires innovations that enable high-throughput and accurate measurements of root system architecture through time. We demonstrate the ability of a semiautomated 3D in vivo imaging and digital phenotyping pipeline to interrogate the quantitative genetic basis of root system growth in a rice biparental mapping population, Bala x Azucena. We phenotyped >1,400 3D root models and >57,000 2D images for a suite of 25 traits that quantified the distribution, shape, extent of exploration, and the intrinsic size of root networks at days 12, 14, and 16 of growth in a gellan gum medium. From these data we identified 89 quantitative trait loci, some of which correspond to those found previously in soil-grown plants, and provide evidence for genetic tradeoffs in root growth allocations, such as between the extent and thoroughness of exploration. We also developed a multivariate method for generating and mapping central root architecture phenotypes and used it to identify five major quantitative trait loci (r2 = 24-37%), two of which were not identified by our univariate analysis. Our imaging and analytical platform provides a means to identify genes with high potential for improving root traits and agronomic qualities of crops.},
author = {Topp, Christopher and Iyer Pascuzzi, Anjali and Anderson, Jill and Lee, Cheng and Zurek, Paul and Symonova, Olga and Zheng, Ying and Bucksch, Alexander and Mileyko, Yuriy and Galkovskyi, Taras and Moore, Brad and Harer, John and Edelsbrunner, Herbert and Mitchell Olds, Thomas and Weitz, Joshua and Benfey, Philip},
journal = {PNAS},
number = {18},
pages = {E1695 -- E1704},
publisher = {National Academy of Sciences},
title = {{3D phenotyping and quantitative trait locus mapping identify core regions of the rice genome controlling root architecture}},
doi = {10.1073/pnas.1304354110},
volume = {110},
year = {2013},
}
@article{2815,
abstract = {The fact that a sum of isotropic Gaussian kernels can have more modes than kernels is surprising. Extra (ghost) modes do not exist in ℝ1 and are generally not well studied in higher dimensions. We study a configuration of n+1 Gaussian kernels for which there are exactly n+2 modes. We show that all modes lie on a finite set of lines, which we call axes, and study the restriction of the Gaussian mixture to these axes in order to discover that there are an exponential number of critical points in this configuration. Although the existence of ghost modes remained unknown due to the difficulty of finding examples in ℝ2, we show that the resilience of ghost modes grows like the square root of the dimension. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.},
author = {Edelsbrunner, Herbert and Fasy, Brittany Terese and Rote, Günter},
journal = {Discrete & Computational Geometry},
number = {4},
pages = {797 -- 822},
publisher = {Springer},
title = {{Add isotropic Gaussian kernels at own risk: More and more resilient modes in higher dimensions}},
doi = {10.1007/s00454-013-9517-x},
volume = {49},
year = {2013},
}
@article{2912,
author = {Edelsbrunner, Herbert and Strelkova, Nataliya},
journal = { Uspekhi Mat. Nauk},
number = {6},
pages = {203 -- 204},
publisher = {Moscow Mathematical Society },
title = {{Configuration space for shortest networks }},
doi = {10.4213/rm9503},
volume = {67},
year = {2012},
}
@inproceedings{3129,
abstract = {Let K be a simplicial complex and g the rank of its p-th homology group Hp(K) defined with ℤ2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω ) time, where n is the size of K and ω < 2.376 is a quantity so that two n×n matrices can be multiplied in O(n ω ) time. The precomputed annotations permit answering queries about the independence or the triviality of p-cycles efficiently.
Using annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1 - dimensional homology. Specifically, for computing an optimal basis of H1(K) , we improve the previously known time complexity from O(n 4) to O(n ω + n 2 g ω − 1). Here n denotes the size of the 2-skeleton of K and g the rank of H1(K) . Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2 O(g) nlogn time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n ω ) + 2 O(g) n 2logn time using annotations.
},
author = {Busaryev, Oleksiy and Cabello, Sergio and Chen, Chao and Dey, Tamal and Wang, Yusu},
location = {Helsinki, Finland},
pages = {189 -- 200},
publisher = {Springer},
title = {{Annotating simplices with a homology basis and its applications}},
doi = {10.1007/978-3-642-31155-0_17},
volume = {7357},
year = {2012},
}
@article{3256,
abstract = {We use a distortion to define the dual complex of a cubical subdivision of ℝ n as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated by the topological analysis of high-dimensional digital image data, we consider such subdivisions defined by generalizations of quad- and oct-trees to n dimensions. Assuming the subdivision is balanced, we show that mapping each vertex to the center of the corresponding n-cube gives a geometric realization of the dual complex in ℝ n.},
author = {Edelsbrunner, Herbert and Kerber, Michael},
journal = {Discrete & Computational Geometry},
number = {2},
pages = {393 -- 414},
publisher = {Springer},
title = {{Dual complexes of cubical subdivisions of ℝn}},
doi = {10.1007/s00454-011-9382-4},
volume = {47},
year = {2012},
}
@article{3120,
abstract = {We introduce a strategy based on Kustin-Miller unprojection that allows us to construct many hundreds of Gorenstein codimension 4 ideals with 9 × 16 resolutions (that is, nine equations and sixteen first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds of Altinok's thesis. There are 115 cases whose numerical data (in effect, the Hilbert series) allow a Type I projection. In every case, at least one Tom and one Jerry construction works, providing at least two deformation families of quasismooth Fano 3-folds having the same numerics but different topology. © 2012 Copyright Foundation Compositio Mathematica.},
author = {Brown, Gavin and Kerber, Michael and Reid, Miles},
journal = {Compositio Mathematica},
number = {4},
pages = {1171 -- 1194},
publisher = {Cambridge University Press},
title = {{Fano 3 folds in codimension 4 Tom and Jerry Part I}},
doi = {10.1112/S0010437X11007226},
volume = {148},
year = {2012},
}
@article{3310,
abstract = {The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calculus, which we present for the family of interlevel sets of a real-valued function. Our account of the subject is expository, devoid of proofs, and written for non-experts in algebraic topology.},
author = {Bendich, Paul and Cabello, Sergio and Edelsbrunner, Herbert},
journal = {Pattern Recognition Letters},
number = {11},
pages = {1436 -- 1444},
publisher = {Elsevier},
title = {{A point calculus for interlevel set homology}},
doi = {10.1016/j.patrec.2011.10.007},
volume = {33},
year = {2012},
}