Computational Complexity of the Interleaving Distance

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.Comment: Discussion related to the characteristic of the field added. Paper accepted to the 34th International Symposium on Computational Geometr

Computing the interleaving distance is NP-hard

We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are $1$-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement of the above reduction, showing that also the approximation of the interleaving distance is NP-hard for any approximation factor smaller than $3$. Additionally, we obtain corresponding hardness results for the case that the modules are indecomposable, and in the setting of one-sided stability. Furthermore, we show that checking for injections (resp. surjections) between persistence modules is NP-hard. In conjunction with earlier results from computational algebra this gives a complete characterization of the computational complexity of one-sided stability. Lastly, we show that it is in general NP-hard to approximate distances induced by noise systems within a factor of 2.Comment: 25 pages. Several expository improvements and minor corrections. Also added a section on noise system

Stabilizing decomposition of multiparameter persistence modules

While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is well known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition, as naive attempts to do this tend to fail. We introduce tools and definitions, in particular $\epsilon$-refinements and the $\epsilon$-erosion neighborhood of a module, to make sense of the question of how to build such a theory. Having built this foundation, we define the $\epsilon$-pruning of a module, which is a new invariant acting like a ``refined barcode'' that shows great promise to extract features from a module by approximately decomposing it. The main theorem of the paper can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of $2r$, where $r$ is the maximal pointwise dimension of one of the modules. Another interpretation is that the pruning of a module $M$ detects all summands appearing in modules in a neighborhood of $M$. Furthermore, we show that the factor $2r$ appearing in the theorem is close to optimal. We also define a distance that we conjecture to be a stable bottleneck-type distance, which has long been a missing piece in multipersistence theory. Finally, we discuss whether the main theorem can be improved for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the complexity of approximating the interleaving distance, and recent applications of homological algebra to multipersistence.Comment: 40 pages, 7 figure

Computing pp-presentation distances is hard

Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced for merge trees and multiparameter persistence modules as more sensitive variations of the respective interleaving distances ($p=\infty$). It is well-known that computing the interleaving distance is NP-hard in both cases. We extend this result by showing that computing the $p$-presentation distance is NP-hard for all $p\in [1,\infty)$ for both merge trees and $t$-parameter persistence modules for any $t\geq 2$. Though the details differ, both proofs follow the same novel strategy, suggesting that our approach can be adapted to proving the NP-hardness of other distances based on sums or $p$-norms.Comment: 28 pages, 7 figure

Quasi-universality of Reeb graph distances

We establish bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.Comment: 17 pages + 6 pages appendix, 5 figures; this version includes the appendix to the conference paper for SoCG 2022 with the same content otherwis

Middleware for transparent TCP connection migration : masking faulty TCP-based services

Masteroppgave i informasjons- og kommunikasjonsteknologi 2004 - Høgskolen i Agder, GrimstadMission critical TCP-based services create a demand for robust and fault tolerant TCP communication. Sense Intellifield monitors drill operations on rig sites offshore. Critical TCP-based services need to be available 24 hours, 7 days a week, and the service providers need to tolerate server failure. How to make TCP robust and fault tolerant without modifying existing infrastructure like existing client/server applications, services, TCP stacks, kernels, or operating systems is the motivation of this thesis. We present a new middleware approach, first of its kind, to allow TCP-based services to survive server failure by migrating TCP connections from failed servers to replicated surviving servers. The approach is based on a proxy technique, which requires modifications to existing infrastructure. Our unique middleware approach is simple, practical, and can be built into existing infrastructure without modifying it. A middleware approach has never been used to implement the proxy based technique. Experiments for validation of functionality and measurement of performance of the middleware prototype are conducted. The results show that our technique adds significant robustness and fault tolerance to TCP, without modifying existing infrastructure. One of the consequences of using a middleware to make TCP communication robust and fault tolerant is added latency. Another consequence is that TCP communication can survive server failure, and mask it. Companies providing robust and fault tolerant TCP, is no longer dependant of third party hardware and/or software. By implementing our solution, they can gain economical advantages. A main focus of this report is to present a prototype that demonstrates our technique and middleware approach. We present relevant background theory which has lead to the design architecture of a middleware approach to make TCP communication fault tolerant. Finally we conduct experiments to uncover the feasibility and performance of the prototype, followed by a discussion and conclusion

Multiple expressed MHC class II loci in salmonids; details of one non-classical region in Atlantic salmon (Salmo salar)

<p>Abstract</p> <p>Background</p> <p>In teleosts, the Major Histocompatibility Complex (MHC) class I and class II molecules reside on different linkage groups as opposed to tetrapods and shark, where the class I and class II genes reside in one genomic region. Several teleost MHC class I regions have been sequenced and show varying number of class I genes. Salmonids have one major expressed MHC class I locus (UBA) in addition to varying numbers of non-classical genes. Two other more distant lineages are also identifyed denoted L and ZE. For class II, only one major expressed class II alpha (DAA) and beta (DAB) gene has been identified in salmonids so far.</p> <p>Results</p> <p>We sequenced a genomic region of 211 kb encompassing divergent MHC class II alpha (<it>Sasa-DBA</it>) and beta (<it>Sasa-DBB</it>) genes in addition to NRGN, TIPRL, TBCEL and TECTA. The region was not linked to the classical class II genes and had some synteny to genomic regions from other teleosts. Two additional divergent and expressed class II sequences denoted DCA and DDA were also identified in both salmon and trout. Expression patterns and lack of polymorphism make these genes non-classical class II analogues. <it>Sasa-DBB</it>, <it>Sasa-DCA </it>and <it>Sasa-DDA </it>had highest expression levels in liver, hindgut and spleen respectively, suggestive of distinctive functions in these tissues. Phylogenetic studies revealed more yet undescribed divergent expressed MHC class II molecules also in other teleosts.</p> <p>Conclusion</p> <p>We have characterised one genomic region containing expressed non-classical MHC class II genes in addition to four other genes not involved in immune function. Salmonids contain at least two expressed MHC class II beta genes and four expressed MHC class II alpha genes with properties suggestive of new functions for MHC class II in vertebrates. Collectively, our data suggest that the class II is worthy of more elaborate studies also in other teleost species.</p

Comprehensive analysis of MHC class I genes from the U-, S-, and Z-lineages in Atlantic salmon

<p>Abstract</p> <p>Background</p> <p>We have previously sequenced more than 500 kb of the duplicated MHC class I regions in Atlantic salmon. In the IA region we identified the loci for the MHC class I gene <it>Sasa-UBA </it>in addition to a soluble MHC class I molecule, <it>Sasa-ULA</it>. A pseudolocus for <it>Sasa-UCA </it>was identified in the nonclassical IB region. Both regions contained genes for antigen presentation, as wells as orthologues to other genes residing in the human MHC region.</p> <p>Results</p> <p>The genomic localisation of two MHC class I lineages (Z and S) has been resolved. 7 BACs were sequenced using a combination of standard Sanger and 454 sequencing. The new sequence data extended the IA region with 150 kb identifying the location of one Z-lineage locus, <it>ZAA</it>. The IB region was extended with 350 kb including three new Z-lineage loci, <it>ZBA</it>, <it>ZCA </it>and <it>ZDA </it>in addition to a <it>UGA </it>locus. An allelic version of the IB region contained a functional <it>UDA </it>locus in addition to the <it>UCA </it>pseudolocus. Additionally a BAC harbouring two MHC class I genes (UHA) was placed on linkage group 14, while a BAC containing the S-lineage locus <it>SAA </it>(previously known as <it>UAA</it>) was placed on LG10. Gene expression studies showed limited expression range for all class I genes with exception of <it>UBA </it>being dominantly expressed in gut, spleen and gills, and <it>ZAA </it>with high expression in blood.</p> <p>Conclusion</p> <p>Here we describe the genomic organization of MHC class I loci from the U-, Z-, and S-lineages in Atlantic salmon. Nine of the described class I genes are located in the extension of the duplicated IA and IB regions, while three class I genes are found on two separate linkage groups. The gene organization of the two regions indicates that the IB region is evolving at a different pace than the IA region. Expression profiling, polymorphic content, peptide binding properties and phylogenetic relationship show that Atlantic salmon has only one MHC class Ia gene (<it>UBA</it>), in addition to a multitude of nonclassical MHC class I genes from the U-, S- and Z-lineages.</p

Genomic Organization of Duplicated Major Histocompatibility Complex Class I Regions in Atlantic Salmon (Salmo Salar)

Background: We have previously identified associations between major histocompatibility complex(MHC) class I and resistance towards bacterial and viral pathogens in Atlantic salmon. To evaluate if onlyMHC or also closely linked genes contributed to the observed resistance we ventured into sequencing ofthe duplicated MHC class I regions of Atlantic salmon.Results: Nine BACs covering more than 500 kb of the two duplicated MHC class I regions of Atlanticsalmon were sequenced and the gene organizations characterized. Both regions contained the proteasomecomponents PSMB8, PSMB9, PSMB9-like and PSMB10 in addition to the transporter for antigen processingTAP2, as well as genes for KIFC1, ZBTB22, DAXX, TAPBP, BRD2, COL11A2, RXRB and SLC39A7. TheIA region contained the recently reported MHC class I Sasa-ULA locus residing approximately 50 kbupstream of the major Sasa-UBA locus. The duplicated class IB region contained an MHC class I locusresembling the rainbow trout UCA locus, but although transcribed it was a pseudogene. No other MHCclass I-like genes were detected in the two duplicated regions. Two allelic BACs spanning the UBA locushad 99.2% identity over 125 kb, while the IA region showed 82.5% identity over 136 kb to the IB region.The Atlantic salmon IB region had an insert of 220 kb in comparison to the IA region containing threechitin synthase genes.Conclusion: We have characterized the gene organization of more than 500 kb of the two duplicatedMHC class I regions in Atlantic salmon. Although Atlantic salmon and rainbow trout are closely related,the gene organization of their IB region has undergone extensive gene rearrangements. The Atlanticsalmon has only one class I UCA pseudogene in the IB region while trout contains the four MHC UCA, UDA,UEA and UFA class I loci. The large differences in gene content and most likely function of the salmon andtrout class IB region clearly argues that sequencing of salmon will not necessarily provide informationrelevant for trout and vice versa

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