Graham Cormode, Zohar Karnin, Edo Liberty, Justin Thaler, Pavel Veselý
Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in n . Given the sketch and a query item y , one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to y . Most works to date focused on additive ε n error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This article investigates multiplicative (1± ε)-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms due to prior work store either O(log (ε 2 n )/ε 2 ) or O (log 3 (ε n )/ε) universe items. We present a randomized sketch storing O (log 1.5 (ε n )/ε) items that can (1± ε)-approximate the rank of each universe item with high constant probability; this space bound is within an (O(sqrt {log (varepsilon n)})) factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments.
{"title":"Relative Error Streaming Quantiles","authors":"Graham Cormode, Zohar Karnin, Edo Liberty, Justin Thaler, Pavel Veselý","doi":"10.1145/3617891","DOIUrl":"https://doi.org/10.1145/3617891","url":null,"abstract":"Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in n . Given the sketch and a query item y , one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to y . Most works to date focused on additive ε n error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This article investigates multiplicative (1± ε)-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms due to prior work store either O(log (ε 2 n )/ε 2 ) or O (log 3 (ε n )/ε) universe items. We present a randomized sketch storing O (log 1.5 (ε n )/ε) items that can (1± ε)-approximate the rank of each universe item with high constant probability; this space bound is within an (O(sqrt {log (varepsilon n)})) factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 prove that the PoA of First Price Auctions is 1-1/ e 2 ≈ 0.8647, closing the gap between the best known bounds [0.7430, 0.8689].
我们证明了首价拍卖的PoA为1-1/ e 2≈0.8647，缩小了已知边界[0.7430,0.8689]之间的差距。
{"title":"First Price Auction is 1 − 1/ <i>e</i> <sup>2</sup> Efficient","authors":"Yaonan Jin, Pinyan Lu","doi":"10.1145/3617902","DOIUrl":"https://doi.org/10.1145/3617902","url":null,"abstract":"We prove that the PoA of First Price Auctions is 1-1/ e 2 ≈ 0.8647, closing the gap between the best known bounds [0.7430, 0.8689].","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135767360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eli Ben-Sasson, Dan Carmon, Yuval Ishai, Swastik Kopparty, Shubhangi Saraf
A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are δ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are δ-close to the property. In particular, no set in the collection has roughly half of its members δ-close to the property and the others δ-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed–Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any δ smaller than the Johnson/Guruswami–Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for δ smaller than the unique decoding radius. Concretely, if δ is smaller than half the minimal distance of an RS code V ⊂ 𝔽 q n , then every affine space is either entirely δ-close to the code or, alternatively, at most an ( n/q )-fraction of it is δ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed–Solomon codes (due to Berlekamp–Welch and Guruswami–Sudan) on a formal element of an affine space. This involves working with Reed–Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests.
{"title":"Proximity Gaps for Reed–Solomon Codes","authors":"Eli Ben-Sasson, Dan Carmon, Yuval Ishai, Swastik Kopparty, Shubhangi Saraf","doi":"10.1145/3614423","DOIUrl":"https://doi.org/10.1145/3614423","url":null,"abstract":"A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are δ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are δ-close to the property. In particular, no set in the collection has roughly half of its members δ-close to the property and the others δ-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed–Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any δ smaller than the Johnson/Guruswami–Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for δ smaller than the unique decoding radius. Concretely, if δ is smaller than half the minimal distance of an RS code V ⊂ 𝔽 q n , then every affine space is either entirely δ-close to the code or, alternatively, at most an ( n/q )-fraction of it is δ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed–Solomon codes (due to Berlekamp–Welch and Guruswami–Sudan) on a formal element of an affine space. This involves working with Reed–Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136058224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Soheil Behnezhad, MohammadTaghi Hajiaghayi, David G. Harris
The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, we still have a limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximal matching either take polylogarithmic time which is considered inefficient, or require a strictly super-linear space of n 1+Ω (1) per machine. In this work, we close this gap by providing a novel analysis of an extremely simple algorithm, which is a variant of an algorithm conjectured to work by Czumaj, Lacki, Madry, Mitrovic, Onak, and Sankowski [ 15 ]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among other results, leads to an O (log log Δ) round algorithm for maximal matching with O(n) space (or even mildly sublinear in n using standard techniques). As an immediate corollary, we get a 2 approximate minimum vertex cover in essentially the same rounds and space, which is the optimal approximation factor under standard assumptions. We also get an improved O (log log Δ) round algorithm for 1 + ε approximate matching. All these results can also be implemented in the congested clique model in the same number of rounds.
{"title":"Exponentially Faster Massively Parallel Maximal Matching","authors":"Soheil Behnezhad, MohammadTaghi Hajiaghayi, David G. Harris","doi":"10.1145/3617360","DOIUrl":"https://doi.org/10.1145/3617360","url":null,"abstract":"The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, we still have a limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximal matching either take polylogarithmic time which is considered inefficient, or require a strictly super-linear space of n 1+Ω (1) per machine. In this work, we close this gap by providing a novel analysis of an extremely simple algorithm, which is a variant of an algorithm conjectured to work by Czumaj, Lacki, Madry, Mitrovic, Onak, and Sankowski [ 15 ]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among other results, leads to an O (log log Δ) round algorithm for maximal matching with O(n) space (or even mildly sublinear in n using standard techniques). As an immediate corollary, we get a 2 approximate minimum vertex cover in essentially the same rounds and space, which is the optimal approximation factor under standard assumptions. We also get an improved O (log log Δ) round algorithm for 1 + ε approximate matching. All these results can also be implemented in the congested clique model in the same number of rounds.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136057754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.
{"title":"Pliability and Approximating Max-CSPs","authors":"Miguel Romero, Marcin Wrochna, Stanislav Živný","doi":"10.1145/3626515","DOIUrl":"https://doi.org/10.1145/3626515","url":null,"abstract":"We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135347694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There has been a long history of studying randomized greedy matching algorithms since the work by Dyer and Frieze (RSA 1991). We follow this trend and consider the problem formulated in the oblivious setting, in which the vertex set of a graph is known to the algorithm, but not the edge set. The algorithm can make queries for the existence of the edge between any pair of vertices but must include the edge into the matching if it exists, i.e., as in the query-commit model by Gamlath et al. (SODA 2019). We revisit the Modified Randomized Greedy (MRG) algorithm by Aronson et al. (RSA 1995) that is proved to achieve a (0.5 + ϵ)-approximation. In each step of the algorithm, an unmatched vertex is chosen uniformly at random and matched to a randomly chosen neighbor (if exists). We study a weaker version of the algorithm named Random Decision Order (RDO) that, in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor (if exists). We prove that the RDO algorithm provides a 0.639-approximation for bipartite graphs and 0.531-approximation for general graphs. As a corollary, we substantially improve the approximation ratio of MRG . Furthermore, we generalize the RDO algorithm to the edge-weighted case and prove that it achieves a 0.501 approximation ratio. This result solves the open question by Chan et al. (SICOMP 2018) and Gamlath et al. (SODA 2019) about the existence of an algorithm that beats greedy in edge-weighted general graphs, where the greedy algorithm probes the edges in descending order of edge-weights. We also present a variant of the algorithm that achieves a (1 − 1/ e )-approximation for edge-weighted bipartite graphs, which generalizes the (1 − 1/ e ) approximation ratio of Gamlath et al. (SODA 2019) for the stochastic setting to the case when the realizations of edges are arbitrarily correlated, where in the stochastic setting, there is a known probability associated with each pair of vertices that indicates the probability that an edge exists between the two vertices, when the pair is probed.
自Dyer和Frieze (RSA 1991)的工作以来，对随机贪婪匹配算法的研究已经有了很长的历史。我们遵循这一趋势，并考虑在遗忘设置下制定的问题，其中图的顶点集是算法已知的，但不知道边集。该算法可以查询任何对顶点之间是否存在边，但如果存在，则必须将边包含到匹配中，即，如Gamlath等人(SODA 2019)的查询提交模型中所示。我们重新审视Aronson等人(RSA 1995)的改进随机贪婪(MRG)算法，该算法已被证明可以实现(0.5 + ε)-近似。在算法的每一步中，均匀随机地选择一个不匹配的顶点，并与随机选择的邻居(如果存在)匹配。我们研究了一种弱版本的算法，称为随机决策顺序(RDO)，在每一步中，随机选择一个不匹配的顶点，并将其与任意邻居(如果存在)匹配。我们证明了RDO算法对二部图提供0.639逼近，对一般图提供0.531逼近。因此，我们大大提高了MRG的近似比。进一步，我们将RDO算法推广到边缘加权的情况，并证明了它达到了0.501的近似比。这一结果解决了Chan等人(SICOMP 2018)和Gamlath等人(SODA 2019)提出的一个开放问题，即在边加权一般图中存在一种击败贪心算法的算法，其中贪心算法按边权重降序探测边。我们也提出一个算法的变体,达到(1−1 / e)光纤edge-weighted由两部分构成的图表,可以推广(1−1 / e)近似Gamlath et al .(2019年苏打水)的比例随机设置的边的实现时,任意相关,在随机环境中,有一个已知的概率与每一对顶点表示的概率之间存在一条边的两个顶点,当对被探测时。
{"title":"Towards a Better Understanding of Randomized Greedy Matching","authors":"Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang","doi":"10.1145/3614318","DOIUrl":"https://doi.org/10.1145/3614318","url":null,"abstract":"There has been a long history of studying randomized greedy matching algorithms since the work by Dyer and Frieze (RSA 1991). We follow this trend and consider the problem formulated in the oblivious setting, in which the vertex set of a graph is known to the algorithm, but not the edge set. The algorithm can make queries for the existence of the edge between any pair of vertices but must include the edge into the matching if it exists, i.e., as in the query-commit model by Gamlath et al. (SODA 2019). We revisit the Modified Randomized Greedy (MRG) algorithm by Aronson et al. (RSA 1995) that is proved to achieve a (0.5 + ϵ)-approximation. In each step of the algorithm, an unmatched vertex is chosen uniformly at random and matched to a randomly chosen neighbor (if exists). We study a weaker version of the algorithm named Random Decision Order (RDO) that, in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor (if exists). We prove that the RDO algorithm provides a 0.639-approximation for bipartite graphs and 0.531-approximation for general graphs. As a corollary, we substantially improve the approximation ratio of MRG . Furthermore, we generalize the RDO algorithm to the edge-weighted case and prove that it achieves a 0.501 approximation ratio. This result solves the open question by Chan et al. (SICOMP 2018) and Gamlath et al. (SODA 2019) about the existence of an algorithm that beats greedy in edge-weighted general graphs, where the greedy algorithm probes the edges in descending order of edge-weights. We also present a variant of the algorithm that achieves a (1 − 1/ e )-approximation for edge-weighted bipartite graphs, which generalizes the (1 − 1/ e ) approximation ratio of Gamlath et al. (SODA 2019) for the stochastic setting to the case when the realizations of edges are arbitrarily correlated, where in the stochastic setting, there is a known probability associated with each pair of vertices that indicates the probability that an edge exists between the two vertices, when the pair is probed.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135347470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael A. Bender, Alex Conway, Martín Farach-Colton, William Kuszmaul, Guido Tagliavini
Despite being one of the oldest data structures in computer science, hash tables continue to be the focus of a great deal of both theoretical and empirical research. A central reason for this is that many of the fundamental properties that one desires from a hash table are difficult to achieve simultaneously; thus many variants offering different trade-offs have been proposed. This paper introduces Iceberg hashing, a hash table that simultaneously offers the strongest known guarantees on a large number of core properties. Iceberg hashing supports constant-time operations while improving on the state of the art for space efficiency, cache efficiency, and low failure probability. Iceberg hashing is also the first hash table to support a load factor of up to 1 − o (1) while being stable, meaning that the position where an element is stored only ever changes when resizes occur. In fact, in the setting where keys are Θ (log n ) bits, the space guarantees that Iceberg hashing offers, namely that it uses at most (log binom{|U|}{n} + O(n log log n) ) bits to store n items from a universe U , matches a lower bound by Demaine et al. that applies to any stable hash table. Iceberg hashing introduces new general-purpose techniques for some of the most basic aspects of hash-table design. Notably, our indirection-free technique for dynamic resizing, which we call waterfall addressing, and our techniques for achieving stability and very-high probability guarantees, can be applied to any hash table that makes use of the front-yard/backyard paradigm for hash table design.
{"title":"Iceberg Hashing: Optimizing Many Hash-Table Criteria at Once","authors":"Michael A. Bender, Alex Conway, Martín Farach-Colton, William Kuszmaul, Guido Tagliavini","doi":"10.1145/3625817","DOIUrl":"https://doi.org/10.1145/3625817","url":null,"abstract":"Despite being one of the oldest data structures in computer science, hash tables continue to be the focus of a great deal of both theoretical and empirical research. A central reason for this is that many of the fundamental properties that one desires from a hash table are difficult to achieve simultaneously; thus many variants offering different trade-offs have been proposed. This paper introduces Iceberg hashing, a hash table that simultaneously offers the strongest known guarantees on a large number of core properties. Iceberg hashing supports constant-time operations while improving on the state of the art for space efficiency, cache efficiency, and low failure probability. Iceberg hashing is also the first hash table to support a load factor of up to 1 − o (1) while being stable, meaning that the position where an element is stored only ever changes when resizes occur. In fact, in the setting where keys are Θ (log n ) bits, the space guarantees that Iceberg hashing offers, namely that it uses at most (log binom{|U|}{n} + O(n log log n) ) bits to store n items from a universe U , matches a lower bound by Demaine et al. that applies to any stable hash table. Iceberg hashing introduces new general-purpose techniques for some of the most basic aspects of hash-table design. Notably, our indirection-free technique for dynamic resizing, which we call waterfall addressing, and our techniques for achieving stability and very-high probability guarantees, can be applied to any hash table that makes use of the front-yard/backyard paradigm for hash table design.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135830528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 consider the allocation of m balls (jobs) into n bins (servers). In the standard Two-Choice process, at each step t = 1, 2, …, m we first sample two randomly chosen bins, compare their two loads and then place a ball in the least loaded bin. It is well-known that for any m ≥ n , this results in a gap (difference between the maximum and average load) of log 2 log n + Θ (1) (with high probability). In this work, we consider Two-Choice in different settings with noisy load comparisons. One key setting involves an adaptive adversary whose power is limited by some threshold (g in mathbb {N} ) . In each step, such adversary can determine the result of any load comparison between two bins whose loads differ by at most g , while if the load difference is greater than g , the comparison is correct. For this adversarial setting, we first prove that for any m ≥ n the gap is (mathcal {O}(g+log n) ) with high probability. Then through a refined analysis we prove that if g ≤ log n , then for any m ≥ n the gap is (mathcal {O}(frac{g}{log g} cdot log log n) ) . For constant values of g , this generalizes the heavily loaded analysis of [19, 61] for the Two-Choice process, and establishes that asymptotically the same gap bound holds even if load comparisons among “similarly loaded” bins are wrong. Finally, we complement these upper bounds with tight lower bounds, which establish an interesting phase transition on how the parameter g impacts the gap. The analysis also applies to settings with outdated and delayed information. For example, for the setting of [18] where balls are allocated in consecutive batches of size b = n , we present an improved and tight gap bound of (Theta big (frac{log n}{log log n}big) ) . This bound also extends for a range of values of b and applies to a relaxed setting where the reported load of a bin can be any load value from the last b steps.
{"title":"Balanced Allocations with the Choice of Noise","authors":"Dimitrios Los, Thomas Sauerwald","doi":"10.1145/3625386","DOIUrl":"https://doi.org/10.1145/3625386","url":null,"abstract":"We consider the allocation of m balls (jobs) into n bins (servers). In the standard Two-Choice process, at each step t = 1, 2, …, m we first sample two randomly chosen bins, compare their two loads and then place a ball in the least loaded bin. It is well-known that for any m ≥ n , this results in a gap (difference between the maximum and average load) of log 2 log n + Θ (1) (with high probability). In this work, we consider Two-Choice in different settings with noisy load comparisons. One key setting involves an adaptive adversary whose power is limited by some threshold (g in mathbb {N} ) . In each step, such adversary can determine the result of any load comparison between two bins whose loads differ by at most g , while if the load difference is greater than g , the comparison is correct. For this adversarial setting, we first prove that for any m ≥ n the gap is (mathcal {O}(g+log n) ) with high probability. Then through a refined analysis we prove that if g ≤ log n , then for any m ≥ n the gap is (mathcal {O}(frac{g}{log g} cdot log log n) ) . For constant values of g , this generalizes the heavily loaded analysis of [19, 61] for the Two-Choice process, and establishes that asymptotically the same gap bound holds even if load comparisons among “similarly loaded” bins are wrong. Finally, we complement these upper bounds with tight lower bounds, which establish an interesting phase transition on how the parameter g impacts the gap. The analysis also applies to settings with outdated and delayed information. For example, for the setting of [18] where balls are allocated in consecutive batches of size b = n , we present an improved and tight gap bound of (Theta big (frac{log n}{log log n}big) ) . This bound also extends for a range of values of b and applies to a relaxed setting where the reported load of a bin can be any load value from the last b steps.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135539141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vishwas Bhargava, Sumanta Ghosh, Mrinal Kumar, Chandra Kanta Mohapatra
Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [7], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans [25] and Kedlaya & Umans [16] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables n is at most d o (1) where the degree of the input polynomial in every variable is less than d . They also stated the question of designing fast algorithms for the large variable case (i.e. n ∉ d o (1) ) as an open problem. use in the preparation of the documentation of their work. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field (mathbb {F}_{q} ) of characteristic p which evaluates an n -variate polynomial of degree less than d in each variable on N inputs in time [ left((N + d^n)^{1 + o(1)}text{poly}(log q, d, n, p)right) ,, ] provided that p is at most d o (1) , and q is at most (exp (exp (exp (⋅⋅⋅(exp ( d ))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. n ∉ d o (1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the following two independently interesting applications. • We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [21] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. • We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program [26]. More precisely, for every fixed prime p , we show that for every constant ϵ > 0, and large enough n , the rank of any n × n Vandermonde matrix V over the field (mathbb {F}_{p^a} ) can be reduced to ( n /exp ( Ω (poly(ϵ)log 0.53 n ))) by changing at most n Θ (ϵ) entries in every row of V , provided a ≤ poly(log n
{"title":"Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications","authors":"Vishwas Bhargava, Sumanta Ghosh, Mrinal Kumar, Chandra Kanta Mohapatra","doi":"10.1145/3625226","DOIUrl":"https://doi.org/10.1145/3625226","url":null,"abstract":"Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [7], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans [25] and Kedlaya & Umans [16] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables n is at most d o (1) where the degree of the input polynomial in every variable is less than d . They also stated the question of designing fast algorithms for the large variable case (i.e. n ∉ d o (1) ) as an open problem. use in the preparation of the documentation of their work. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field (mathbb {F}_{q} ) of characteristic p which evaluates an n -variate polynomial of degree less than d in each variable on N inputs in time [ left((N + d^n)^{1 + o(1)}text{poly}(log q, d, n, p)right) ,, ] provided that p is at most d o (1) , and q is at most (exp (exp (exp (⋅⋅⋅(exp ( d ))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. n ∉ d o (1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the following two independently interesting applications. • We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [21] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. • We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program [26]. More precisely, for every fixed prime p , we show that for every constant ϵ > 0, and large enough n , the rank of any n × n Vandermonde matrix V over the field (mathbb {F}_{p^a} ) can be reduced to ( n /exp ( Ω (poly(ϵ)log 0.53 n ))) by changing at most n Θ (ϵ) entries in every row of V , provided a ≤ poly(log n","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aïna Linn Georges, Armaël Guéneau, Thomas Van Strydonck, Amin Timany, Alix Trieu, Dominique Devriese, Lars Birkedal
A capability machine is a type of CPU allowing fine-grained privilege separation using capabilities , machine words that represent certain kinds of authority. We present a mathematical model and accompanying proof methods that can be used for formal verification of functional correctness of programs running on a capability machine, even when they invoke and are invoked by unknown (and possibly malicious) code. We use a program logic called Cerise for reasoning about known code, and an associated logical relation, for reasoning about unknown code. The logical relation formally captures the capability safety guarantees provided by the capability machine. The Cerise program logic, logical relation, and all the examples considered in the paper have been mechanized using the Iris program logic framework in the Coq proof assistant. The methodology we present underlies recent work of the authors on formal reasoning about capability machines [15, 33, 37], but was left somewhat implicit in those publications. In this paper we present a pedagogical introduction to the methodology, in a simpler setting (no exotic capabilities), and starting from minimal examples. We work our way up to new results about a heap-based calling convention and implementations of sophisticated object-capability patterns of the kind previously studied for high-level languages with object-capabilities, demonstrating that the methodology scales to such reasoning.
{"title":"Cerise: Program Verification on a Capability Machine in the Presence of Untrusted Code","authors":"Aïna Linn Georges, Armaël Guéneau, Thomas Van Strydonck, Amin Timany, Alix Trieu, Dominique Devriese, Lars Birkedal","doi":"10.1145/3623510","DOIUrl":"https://doi.org/10.1145/3623510","url":null,"abstract":"A capability machine is a type of CPU allowing fine-grained privilege separation using capabilities , machine words that represent certain kinds of authority. We present a mathematical model and accompanying proof methods that can be used for formal verification of functional correctness of programs running on a capability machine, even when they invoke and are invoked by unknown (and possibly malicious) code. We use a program logic called Cerise for reasoning about known code, and an associated logical relation, for reasoning about unknown code. The logical relation formally captures the capability safety guarantees provided by the capability machine. The Cerise program logic, logical relation, and all the examples considered in the paper have been mechanized using the Iris program logic framework in the Coq proof assistant. The methodology we present underlies recent work of the authors on formal reasoning about capability machines [15, 33, 37], but was left somewhat implicit in those publications. In this paper we present a pedagogical introduction to the methodology, in a simpler setting (no exotic capabilities), and starting from minimal examples. We work our way up to new results about a heap-based calling convention and implementations of sophisticated object-capability patterns of the kind previously studied for high-level languages with object-capabilities, demonstrating that the methodology scales to such reasoning.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134913118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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