9 Search Results for "Mémoli, Facundo"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.25
Meta-Diagrams for 2-Parameter Persistence

Authors: Nate Clause, Tamal K. Dey, Facundo Mémoli, and Bei Wang

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module M indexed by a bifiltration of n simplices in O(n³) time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has O(n⁴) time complexity. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of M from O(n⁴) to O(n³). In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting.
Cite as

Nate Clause, Tamal K. Dey, Facundo Mémoli, and Bei Wang. Meta-Diagrams for 2-Parameter Persistence. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{clause_et_al:LIPIcs.SoCG.2023.25,
  author =	{Clause, Nate and Dey, Tamal K. and M\'{e}moli, Facundo and Wang, Bei},
  title =	{{Meta-Diagrams for 2-Parameter Persistence}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{25:1--25:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.25},
  URN =		{urn:nbn:de:0030-drops-178754},
  doi =		{10.4230/LIPIcs.SoCG.2023.25},
  annote =	{Keywords: Multiparameter persistence modules, persistent homology, M\"{o}bius inversion, barcodes, computational topology, topological data analysis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.37
A Generalization of the Persistent Laplacian to Simplicial Maps

Authors: Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
The (combinatorial) graph Laplacian is a fundamental object in the analysis of, and optimization on, graphs. Via a topological view, this operator can be extended to a simplicial complex K and therefore offers a way to perform "signal processing" on p-(co)chains of K. Recently, the concept of persistent Laplacian was proposed and studied for a pair of simplicial complexes K ↪ L connected by an inclusion relation, further broadening the use of Laplace-based operators. In this paper, we significantly expand the scope of the persistent Laplacian by generalizing it to a pair of weighted simplicial complexes connected by a weight preserving simplicial map f: K → L. Such a simplicial map setting arises frequently, e.g., when relating a coarsened simplicial representation with an original representation, or the case when the two simplicial complexes are spanned by different point sets, i.e. cases in which it does not hold that K ⊂ L. However, the simplicial map setting is much more challenging than the inclusion setting since the underlying algebraic structure is much more complicated. We present a natural generalization of the persistent Laplacian to the simplicial setting. To shed insight on the structure behind it, as well as to develop an algorithm to compute it, we exploit the relationship between the persistent Laplacian and the Schur complement of a matrix. A critical step is to view the Schur complement as a functorial way of restricting a self-adjoint positive semi-definite operator to a given subspace. As a consequence of this relation, we prove that the qth persistent Betti number of the simplicial map f: K → L equals the nullity of the qth persistent Laplacian Δ_q^{K,L}. We then propose an algorithm for finding the matrix representation of Δ_q^{K,L} which in turn yields a fundamentally different algorithm for computing the qth persistent Betti number of a simplicial map. Finally, we study the persistent Laplacian on simplicial towers under weight-preserving simplicial maps and establish monotonicity results for their eigenvalues.
Cite as

Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang. A Generalization of the Persistent Laplacian to Simplicial Maps. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gulen_et_al:LIPIcs.SoCG.2023.37,
  author =	{G\"{u}len, Aziz Burak and M\'{e}moli, Facundo and Wan, Zhengchao and Wang, Yusu},
  title =	{{A Generalization of the Persistent Laplacian to Simplicial Maps}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{37:1--37:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.37},
  URN =		{urn:nbn:de:0030-drops-178877},
  doi =		{10.4230/LIPIcs.SoCG.2023.37},
  annote =	{Keywords: combinatorial Laplacian, persistent Laplacian, Schur complement, persistent homology, persistent Betti number}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.51
Ephemeral Persistence Features and the Stability of Filtered Chain Complexes

Authors: Facundo Mémoli and Ling Zhou

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes ephemeral points, i.e. points with zero persistence, which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics which are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes thus confirming that these ephemeral points strengthen the discriminating power of the standard VR barcode.
Cite as

Facundo Mémoli and Ling Zhou. Ephemeral Persistence Features and the Stability of Filtered Chain Complexes. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{memoli_et_al:LIPIcs.SoCG.2023.51,
  author =	{M\'{e}moli, Facundo and Zhou, Ling},
  title =	{{Ephemeral Persistence Features and the Stability of Filtered Chain Complexes}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{51:1--51:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.51},
  URN =		{urn:nbn:de:0030-drops-179014},
  doi =		{10.4230/LIPIcs.SoCG.2023.51},
  annote =	{Keywords: filtered chain complexes, Vietoris-Rips complexes, barcode, bottleneck distance, matching distance, Gromov-Hausdorff distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.31
Persistent Cup-Length

Authors: Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces. In this paper, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.
Cite as

Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent Cup-Length. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{contessoto_et_al:LIPIcs.SoCG.2022.31,
  author =	{Contessoto, Marco and M\'{e}moli, Facundo and Stefanou, Anastasios and Zhou, Ling},
  title =	{{Persistent Cup-Length}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{31:1--31:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.31},
  URN =		{urn:nbn:de:0030-drops-160398},
  doi =		{10.4230/LIPIcs.SoCG.2022.31},
  annote =	{Keywords: cohomology, cup product, persistence, cup length, Gromov-Hausdorff distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.34
Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications

Authors: Tamal K. Dey, Woojin Kim, and Facundo Mémoli

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval I of a 𝐙²-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank over I by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing M to that path. If the bifiltration and I have at most t simplices and points respectively, this computation takes O(t^ω) time where ω ∈ [2,2.373) is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its summands.
Cite as

Tamal K. Dey, Woojin Kim, and Facundo Mémoli. Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2022.34,
  author =	{Dey, Tamal K. and Kim, Woojin and M\'{e}moli, Facundo},
  title =	{{Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{34:1--34:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.34},
  URN =		{urn:nbn:de:0030-drops-160420},
  doi =		{10.4230/LIPIcs.SoCG.2022.34},
  annote =	{Keywords: Multiparameter persistent homology, Zigzag persistent homology, Generalized Persistence Diagrams, M\"{o}bius inversion}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.15
The Reeb Graph Edit Distance Is Universal

Authors: Ulrich Bauer, Claudia Landi, and Facundo Mémoli

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.
Cite as

Ulrich Bauer, Claudia Landi, and Facundo Mémoli. The Reeb Graph Edit Distance Is Universal. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2020.15,
  author =	{Bauer, Ulrich and Landi, Claudia and M\'{e}moli, Facundo},
  title =	{{The Reeb Graph Edit Distance Is Universal}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.15},
  URN =		{urn:nbn:de:0030-drops-121730},
  doi =		{10.4230/LIPIcs.SoCG.2020.15},
  annote =	{Keywords: Reeb graphs, topological descriptors, edit distance, interleaving distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.26
Elder-Rule-Staircodes for Augmented Metric Spaces

Authors: Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.
Cite as

Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang. Elder-Rule-Staircodes for Augmented Metric Spaces. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cai_et_al:LIPIcs.SoCG.2020.26,
  author =	{Cai, Chen and Kim, Woojin and M\'{e}moli, Facundo and Wang, Yusu},
  title =	{{Elder-Rule-Staircodes for Augmented Metric Spaces}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.26},
  URN =		{urn:nbn:de:0030-drops-121848},
  doi =		{10.4230/LIPIcs.SoCG.2020.26},
  annote =	{Keywords: Persistent homology, Multiparameter persistence, Barcodes, Elder rule, Hierarchical clustering, Graded Betti numbers}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.36
Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers

Authors: Tamal K. Dey, Facundo Mémoli, and Yusu Wang

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called nerve of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex N(U) of a path-connected cover U of a domain X cannot be richer than that of the domain X itself. Intuitively, this result means that no new H_1-homology class can be "created" under a natural map from X to the nerve complex N(U). Equipping X with a pseudometric d, we further refine this result and characterize the classes of H_1(X) that may survive in the nerve complex using the notion of size of the covering elements in U. These fundamental results about nerve complexes then lead to an analysis of the H_1-homology of Reeb spaces, mappers and multiscale mappers. The analysis of H_1-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.
Cite as

Tamal K. Dey, Facundo Mémoli, and Yusu Wang. Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2017.36,
  author =	{Dey, Tamal K. and M\'{e}moli, Facundo and Wang, Yusu},
  title =	{{Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.36},
  URN =		{urn:nbn:de:0030-drops-72220},
  doi =		{10.4230/LIPIcs.SoCG.2017.36},
  annote =	{Keywords: Topology, Nerves, Mapper, Multiscale Mapper, Reeb Spaces}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.85
Quasimetric Embeddings and Their Applications

Authors: Facundo Mémoli, Anastasios Sidiropoulos, and Vijay Sridhar

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces. Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t*log^2(n)), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t*log^{O(1)}(n)-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.
Cite as

Facundo Mémoli, Anastasios Sidiropoulos, and Vijay Sridhar. Quasimetric Embeddings and Their Applications. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 85:1-85:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{memoli_et_al:LIPIcs.ICALP.2016.85,
  author =	{M\'{e}moli, Facundo and Sidiropoulos, Anastasios and Sridhar, Vijay},
  title =	{{Quasimetric Embeddings and Their Applications}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{85:1--85:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.85},
  URN =		{urn:nbn:de:0030-drops-62007},
  doi =		{10.4230/LIPIcs.ICALP.2016.85},
  annote =	{Keywords: metric embeddings, quasimetrics, outliers, random embeddings, treewidth, Directed Sparsest-Cut, Directed Multicut}
}
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