Pub Date : 2022-06-29DOI: 10.1007/s00446-022-00431-z
Michael Elkin, Ofer Neiman
Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with (Omega (n log n)) edges, or with a hopbound (n^{Omega (1)}). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on (epsilon )) ((log log n)^{log log n + O(1)}). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [19] for computing hopsets with a constant (i.e., independent of n) hopbound requires (n^{Omega (1)}) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [19]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.
{"title":"Linear-Size hopsets with small hopbound, and constant-hopbound hopsets in RNC","authors":"Michael Elkin, Ofer Neiman","doi":"10.1007/s00446-022-00431-z","DOIUrl":"https://doi.org/10.1007/s00446-022-00431-z","url":null,"abstract":"<p>Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with <span>(Omega (n log n))</span> edges, or with a hopbound <span>(n^{Omega (1)})</span>. In this paper we devise a construction of <i>linear-size</i> hopsets with hopbound (ignoring the dependence on <span>(epsilon )</span>) <span>((log log n)^{log log n + O(1)})</span>. This improves the previous hopbound for linear-size hopsets almost <i>exponentially</i>. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [19] for computing hopsets with a constant (i.e., independent of <i>n</i>) hopbound requires <span>(n^{Omega (1)})</span> time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is <i>exponentially</i> better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [19]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex <i>v</i>, report all approximate shortest paths from <i>v</i> in <i>constant time</i>. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.</p>","PeriodicalId":50569,"journal":{"name":"Distributed Computing","volume":"20 10","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe a simple deterministic $$O( varepsilon ^{-1} log Delta )$$ O ( ε - 1 log Δ ) round distributed algorithm for $$(2alpha +1)(1 + varepsilon )$$ ( 2 α + 1 ) ( 1 + ε ) approximation of minimum weighted dominating set on graphs with arboricity at most $$alpha $$ α . Here $$Delta $$ Δ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized $$O(alpha ^2)$$ O ( α 2 ) approximation in $$O(log n)$$ O ( log n ) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic $$O(alpha log Delta )$$ O ( α log Δ ) approximation in $$O(log Delta )$$ O ( log Δ ) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic $$O(alpha )$$ O ( α ) approximation in $$O(log ^2 Delta )$$ O ( log 2 Δ ) rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized $$O(alpha )$$ O ( α ) approximation in $$O(alpha log n)$$ O ( α log n ) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized $$O(alpha log Delta )$$ O ( α log Δ ) round distributed algorithm that sharpens the approximation factor to $$alpha (1+o(1))$$ α ( 1 + o ( 1 ) ) . If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve $$alpha - 1 - varepsilon $$ α - 1 - ε approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).
我们描述了一个简单的确定性$$O( varepsilon ^{-1} log Delta )$$ O (ε - 1 log Δ)轮分布算法，用于$$(2alpha +1)(1 + varepsilon )$$ (2 α + 1) (1 + ε)最小加权支配集在最大限为$$alpha $$ α的图上的逼近。其中$$Delta $$ Δ表示最大度。我们还展示了一个下界，通过减少分布式顶点覆盖近似上著名的KMW下界(Kuhn等人在JACM 63:116, 2016)，证明即使在未加权的情况下，这种轮复杂度也几乎是最优的。我们的算法改进了之前的所有结果(仅适用于未加权的图)，包括$$O(log n)$$ O (log n)轮的随机$$O(alpha ^2)$$ O (α 2)近似(Lenzen等人在分布式计算国际研讨会上，施普林格，2010)，$$O(log Delta )$$ O (log Δ)轮的确定性$$O(alpha log Delta )$$ O (α log Δ)近似(Lenzen等人在分布式计算国际研讨会上，施普林格，2010)，在$$O(log ^2 Delta )$$ O (log 2 Δ)轮中的确定性$$O(alpha )$$ O (α)近似(隐含在Bansal等人的Inform Process Lett 122:21 - 24,2017中);进行第17届离散算法研讨会(SODA)， 2006年)，以及$$O(alpha log n)$$ O (α log n)轮的随机$$O(alpha )$$ O (α)近似(Morgan等人在第35届国际分布式计算研讨会上，2021年)。我们还提供了一个随机的$$O(alpha log Delta )$$ O (α log Δ)轮分布算法，该算法将近似因子提高到$$alpha (1+o(1))$$ α (1 + O(1))。如果每个节点被限制进行多项式时间计算，我们的近似因子在一阶上是紧密的，因为它是NP-hard实现$$alpha - 1 - varepsilon $$ α - 1- ε近似(Bansal et al. in Inform Process Lett 122:21- 24,2017)。
{"title":"Near-optimal distributed dominating set in bounded arboricity graphs","authors":"Michal Dory, M. Ghaffari, S. Ilchi","doi":"10.1145/3519270.3538437","DOIUrl":"https://doi.org/10.1145/3519270.3538437","url":null,"abstract":"We describe a simple deterministic $$O( varepsilon ^{-1} log Delta )$$ O ( ε - 1 log Δ ) round distributed algorithm for $$(2alpha +1)(1 + varepsilon )$$ ( 2 α + 1 ) ( 1 + ε ) approximation of minimum weighted dominating set on graphs with arboricity at most $$alpha $$ α . Here $$Delta $$ Δ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized $$O(alpha ^2)$$ O ( α 2 ) approximation in $$O(log n)$$ O ( log n ) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic $$O(alpha log Delta )$$ O ( α log Δ ) approximation in $$O(log Delta )$$ O ( log Δ ) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic $$O(alpha )$$ O ( α ) approximation in $$O(log ^2 Delta )$$ O ( log 2 Δ ) rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized $$O(alpha )$$ O ( α ) approximation in $$O(alpha log n)$$ O ( α log n ) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized $$O(alpha log Delta )$$ O ( α log Δ ) round distributed algorithm that sharpens the approximation factor to $$alpha (1+o(1))$$ α ( 1 + o ( 1 ) ) . If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve $$alpha - 1 - varepsilon $$ α - 1 - ε approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).","PeriodicalId":50569,"journal":{"name":"Distributed Computing","volume":"1 1","pages":"1-12"},"PeriodicalIF":1.3,"publicationDate":"2022-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47763316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Background: To assess the multicenter outcomes of posterior chamber phakic intraocular lens implantation with a central hole (EVO-ICL, STAAR Surgical) for patients undergoing previous laser in situ keratomileusis (LASIK).
Methods: This case series enrolled 31 eyes of 21 consecutive patients undergoing EVO-ICL implantation to correct residual refractive errors after LASIK at 7 nationwide major surgical sites. We investigated safety, efficacy, predictability, stability, and adverse events at 1 week, 1, 3, and 6 months postoperatively, and at the final visit.
Results: The mean observation period was 1.6 ± 1.8 years. Uncorrected and corrected visual acuities were - 0.14 ± 0.11 and - 0.22 ± 0.09 logMAR at 6 months postoperatively. At 6 months postoperatively, 81% and 100% of eyes were within ± 0.5 D and ± 1.0 D, respectively, of the targeted correction. We found neither significant manifest refraction changes of 0.05 ± 0.38 D from 1 week to 6 months nor apparent intraoperative or postoperative complications in any case.
Conclusions: Our multicenter study confirmed that the EVO-ICL provided good outcomes in safety, efficacy, predictability, and stability, even in post-LASIK eyes. Therefore, EVO-ICL implantation may be a viable surgical option, even for correcting residual refractive errors after LASIK. Trial registration University Hospital Medical Information Network Clinical Trial Registry (000045295).
{"title":"Posterior chamber phakic intraocular lens implantation after laser in situ keratomileusis.","authors":"Kazutaka Kamiya, Kimiya Shimizu, Akihito Igarashi, Yoshihiro Kitazawa, Takashi Kojima, Tomoaki Nakamura, Kazuo Ichikawa, Sachiko Fukuoka, Kahoko Fujimoto","doi":"10.1186/s40662-022-00282-6","DOIUrl":"10.1186/s40662-022-00282-6","url":null,"abstract":"<p><strong>Background: </strong>To assess the multicenter outcomes of posterior chamber phakic intraocular lens implantation with a central hole (EVO-ICL, STAAR Surgical) for patients undergoing previous laser in situ keratomileusis (LASIK).</p><p><strong>Methods: </strong>This case series enrolled 31 eyes of 21 consecutive patients undergoing EVO-ICL implantation to correct residual refractive errors after LASIK at 7 nationwide major surgical sites. We investigated safety, efficacy, predictability, stability, and adverse events at 1 week, 1, 3, and 6 months postoperatively, and at the final visit.</p><p><strong>Results: </strong>The mean observation period was 1.6 ± 1.8 years. Uncorrected and corrected visual acuities were - 0.14 ± 0.11 and - 0.22 ± 0.09 logMAR at 6 months postoperatively. At 6 months postoperatively, 81% and 100% of eyes were within ± 0.5 D and ± 1.0 D, respectively, of the targeted correction. We found neither significant manifest refraction changes of 0.05 ± 0.38 D from 1 week to 6 months nor apparent intraoperative or postoperative complications in any case.</p><p><strong>Conclusions: </strong>Our multicenter study confirmed that the EVO-ICL provided good outcomes in safety, efficacy, predictability, and stability, even in post-LASIK eyes. Therefore, EVO-ICL implantation may be a viable surgical option, even for correcting residual refractive errors after LASIK. Trial registration University Hospital Medical Information Network Clinical Trial Registry (000045295).</p>","PeriodicalId":50569,"journal":{"name":"Distributed Computing","volume":"1 1","pages":"15"},"PeriodicalIF":4.1,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9008970/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89832009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-03DOI: 10.1007/s00446-021-00416-4
Ittai Abraham, Danny Dolev, Gilad Stern
The celebrated result of Fischer, Lynch and Paterson is the fundamental lower bound for asynchronous fault tolerant computation: any 1-crash resilient asynchronous agreement protocol must have some (possibly measure zero) probability of not terminating. In 1994, Ben-Or, Kelmer and Rabin published a proof-sketch of a lesser known lower bound for asynchronous fault tolerant computation with optimal resilience in face of a Byzantine adversary: if (nle 4t) then any t-resilient asynchronous verifiable secret sharing protocol must have some non-zero probability of not terminating. Our main contribution is to revisit this lower bound and provide a rigorous and more general proof. Our second contribution is to show how to avoid this lower bound. We provide a protocol with optimal resilience that is almost surely terminating for a strong common coin functionality. Using this new primitive we provide an almost surely terminating protocol with optimal resilience for asynchronous Byzantine agreement that has a new fair validity property. To the best of our knowledge this is the first asynchronous Byzantine agreement with fair validity in the information theoretic setting.
{"title":"Revisiting asynchronous fault tolerant computation with optimal resilience","authors":"Ittai Abraham, Danny Dolev, Gilad Stern","doi":"10.1007/s00446-021-00416-4","DOIUrl":"https://doi.org/10.1007/s00446-021-00416-4","url":null,"abstract":"<p>The celebrated result of Fischer, Lynch and Paterson is the fundamental lower bound for asynchronous fault tolerant computation: any 1-crash resilient asynchronous agreement protocol must have some (possibly measure zero) probability of not terminating. In 1994, Ben-Or, Kelmer and Rabin published a <i>proof-sketch</i> of a lesser known lower bound for asynchronous fault tolerant computation with optimal resilience in face of a Byzantine adversary: if <span>(nle 4t)</span> then any t-resilient asynchronous verifiable secret sharing protocol must have some <b>non-zero</b> probability of not terminating. Our main contribution is to revisit this lower bound and provide a rigorous and more general proof. Our second contribution is to show how to avoid this lower bound. We provide a protocol with optimal resilience that is almost surely terminating for a <i>strong common coin</i> functionality. Using this new primitive we provide an almost surely terminating protocol with optimal resilience for asynchronous Byzantine agreement that has a new <i>fair validity</i> property. To the best of our knowledge this is the first asynchronous Byzantine agreement with fair validity in the information theoretic setting.\u0000</p>","PeriodicalId":50569,"journal":{"name":"Distributed Computing","volume":"34 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.1007/s00446-022-00422-0
R. Guerraoui, P. Kuznetsov, M. Monti, M. Pavlovic, Dragos-Adrian Seredinschi
{"title":"Correction to: The consensus number of a cryptocurrency","authors":"R. Guerraoui, P. Kuznetsov, M. Monti, M. Pavlovic, Dragos-Adrian Seredinschi","doi":"10.1007/s00446-022-00422-0","DOIUrl":"https://doi.org/10.1007/s00446-022-00422-0","url":null,"abstract":"","PeriodicalId":50569,"journal":{"name":"Distributed Computing","volume":"35 1","pages":"17"},"PeriodicalIF":1.3,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41830087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01Epub Date: 2022-01-20DOI: 10.1007/s00446-021-00418-2
Danupon Nanongkai, Michele Scquizzato
<p><p>The <i>Massively Parallel Computation</i> (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the <i>one cycle versus two cycles</i> problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., <math><mrow><mi>P</mi> <mo>≠</mo> <mi>NP</mi></mrow> </math> ), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by <math><mrow><mi>MPC</mi> <mo>(</mo> <mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo> <mo>)</mo></mrow> </math> , and the standard space complexity classes <math><mi>L</mi></math> and <math><mi>NL</mi></math> , and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model. Specifically, our main results are as follows.Lower bounds conditioned on the one cycle versus two cycles conjecture can be instead argued under the <math><mrow><mi>L</mi> <mo>⊈</mo> <mi>MPC</mi> <mo>(</mo> <mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo> <mo>)</mo></mrow> </math> conjecture: these two assumptions are equivalent, and refuting either of them would lead to <math><mrow><mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo></mrow> </math> -round MPC algorithms for a large number of challenging problems, including list ranking, minimum cut, and planarity testing. In fact, we show that these problems and many others require asymptotically the same number of rounds as the seemingly much easier problem of distinguishing between a graph being one cycle or two cycles.Many lower bounds previously argued under the one cycle versus two cycles conjecture can be argued under an even more robust (thus harder to refute) conjecture, namely <math><mrow><mi>NL</mi> <mo>⊈</mo> <mi>MPC</mi> <mo>(</mo> <mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo> <mo>)</mo></mrow> </math> . Refuting this conjecture would lead to <math><mrow><mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo></mrow> </math> -round MPC algorithms for an even larger set of problems, including all-pairs shortest paths, betweenness centrality, and all aforementioned ones. Lower bounds under this conjecture hold for probl
大规模并行计算（MPC）模型是许多现代大规模数据处理框架的通用抽象，在过去几年中受到越来越多的关注，尤其是在经典图问题方面。迄今为止，论证该模型下限的唯一方法是以某些特定问题的难易度猜想为条件，例如单循环或双循环许诺图上的图连通性，通常称为单循环与双循环问题。这与传统的基于复杂度类猜想（如 P ≠ NP）的论证不同，后者通常更稳健，因为反驳它们会为一大堆问题带来突破性的算法。在本文中，我们将介绍允许后一类论证的问题和问题类之间的联系。这些联系涉及在 MPC 模型中可在亚对数回合数内求解的一类问题，用 MPC ( o ( log N ) ) 表示。 以及标准空间复杂度类 L 和 NL，并提出了一些稳健的猜想，即反驳这些猜想会在 MPC 模型中产生许多速度惊人的新算法。我们还获得了新的条件下界，并证明了 MPC 模型中问题间的新还原和等价性。以一个循环与两个循环猜想为条件的下界可以在 L ⊈ MPC ( o ( log N ) ) 猜想下进行论证：这两个假设是等价的，反驳其中任何一个假设都会为大量具有挑战性的问题带来 o ( log N ) 轮 MPC 算法，包括列表排序、最小切割和平面性检验。事实上，我们证明了这些问题和许多其他问题所需的回合数，与区分一个图是一个循环还是两个循环这个看似简单得多的问题所需的回合数渐近相同。以前根据一个循环与两个循环猜想提出的许多下界，可以根据一个更稳健（因此更难反驳）的猜想提出，即 NL ⊈ MPC ( o ( log N ) ) 。反驳这一猜想将为更多问题带来 o ( log N ) 轮 MPC 算法，包括全对最短路径、间度中心性以及上述所有问题。这一猜想下的下界在完全匹配和网络流等问题上也是成立的。
{"title":"Equivalence classes and conditional hardness in massively parallel computations.","authors":"Danupon Nanongkai, Michele Scquizzato","doi":"10.1007/s00446-021-00418-2","DOIUrl":"10.1007/s00446-021-00418-2","url":null,"abstract":"<p><p>The <i>Massively Parallel Computation</i> (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the <i>one cycle versus two cycles</i> problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., <math><mrow><mi>P</mi> <mo>≠</mo> <mi>NP</mi></mrow> </math> ), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by <math><mrow><mi>MPC</mi> <mo>(</mo> <mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo> <mo>)</mo></mrow> </math> , and the standard space complexity classes <math><mi>L</mi></math> and <math><mi>NL</mi></math> , and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model. Specifically, our main results are as follows.Lower bounds conditioned on the one cycle versus two cycles conjecture can be instead argued under the <math><mrow><mi>L</mi> <mo>⊈</mo> <mi>MPC</mi> <mo>(</mo> <mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo> <mo>)</mo></mrow> </math> conjecture: these two assumptions are equivalent, and refuting either of them would lead to <math><mrow><mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo></mrow> </math> -round MPC algorithms for a large number of challenging problems, including list ranking, minimum cut, and planarity testing. In fact, we show that these problems and many others require asymptotically the same number of rounds as the seemingly much easier problem of distinguishing between a graph being one cycle or two cycles.Many lower bounds previously argued under the one cycle versus two cycles conjecture can be argued under an even more robust (thus harder to refute) conjecture, namely <math><mrow><mi>NL</mi> <mo>⊈</mo> <mi>MPC</mi> <mo>(</mo> <mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo> <mo>)</mo></mrow> </math> . Refuting this conjecture would lead to <math><mrow><mi>o</mi> <mo>(</mo> <mo>log</mo> <mi>N</mi> <mo>)</mo></mrow> </math> -round MPC algorithms for an even larger set of problems, including all-pairs shortest paths, betweenness centrality, and all aforementioned ones. Lower bounds under this conjecture hold for probl","PeriodicalId":50569,"journal":{"name":"Distributed Computing","volume":"35 1","pages":"165-183"},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8907129/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44815276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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