We prove a query complexity lower bound for approximating the top r dimensional eigenspace of a matrix. We consider an oracle model where, given a symmetric matrix M ∈ ℝd × d, an algorithm Alg is allowed to make T exact queries of the form w(i) = M v(i) for i in {1,...,T}, where v(i) is drawn from a distribution which depends arbitrarily on the past queries and measurements {v(j),w(i)}1 ≤ j ≤ i−1. We show that for every gap ∈ (0,1/2], there exists a distribution over matrices M for which 1) gapr(M) = Ω(gap) (where gapr(M) is the normalized gap between the r and r+1-st largest-magnitude eigenvector of M), and 2) any Alg which takes fewer than const × r logd/√gap queries fails (with overwhelming probability) to identity a matrix V ∈ ℝd × r with orthonormal columns for which ⟨ V, M V⟩ ≥ (1 − const × gap)∑i=1r λi(M). Our bound requires only that d is a small polynomial in 1/gap and r, and matches the upper bounds of Musco and Musco ’15. Moreover, it establishes a strict separation between convex optimization and “strict-saddle” non-convex optimization of which PCA is a canonical example: in the former, first-order methods can have dimension-free iteration complexity, whereas in PCA, the iteration complexity of gradient-based methods must necessarily grow with the dimension. Our argument proceeds via a reduction to estimating a rank-r spike in a deformed Wigner model M =W + λ U U⊤, where W is from the Gaussian Orthogonal Ensemble, U is uniform on the d × r-Stieffel manifold and λ > 1 governs the size of the perturbation. Surprisingly, this ubiquitous random matrix model witnesses the worst-case rate for eigenspace approximation, and the ‘accelerated’ gap−1/2 in the rate follows as a consequence of the correspendence between the asymptotic eigengap and the size of the perturbation λ, when λ is near the “phase transition” λ = 1. To verify that d need only be polynomial in gap−1 and r, we prove a finite sample convergence theorem for top eigenvalues of a deformed Wigner matrix, which may be of independent interest. We then lower bound the above estimation problem with a novel technique based on Fano-style data-processing inequalities with truncated likelihoods; the technique generalizes the Bayes-risk lower bound of Chen et al. ’16, and we believe it is particularly suited to lower bounds in adaptive settings like the one considered in this paper.
我们证明了近似矩阵的上r维特征空间的查询复杂度下界。我们考虑一个oracle模型，其中，给定一个对称矩阵M∈v x d，允许算法Alg对i在{1，…，T}，其中v(i)是从任意依赖于过去查询和测量的分布中得出的{v(j)，w(i)}1≤j≤i−1。我们表明，对于每个间隙∈(0,1/2)，存在矩阵M上的分布，其中1)gapr(M) = Ω(gap)(其中gapr(M)是r和r+1-st最大特征向量之间的归一化间隙)，并且2)任何小于const × r logd/√gap查询的Alg都无法(以压倒性的概率)识别矩阵V∈V x x r，其标准正交列为⟨V, M V⟩≥(1 - const × gap)∑i=1r λi(M)。我们的边界只要求d是1/gap和r中的一个小多项式，并且匹配Musco和Musco ' 15的上界。此外，它将凸优化与“严格鞍形”非凸优化严格区分开来，其中PCA是一个典型的例子:在前者中，一阶方法可以具有无维迭代复杂度，而在PCA中，基于梯度的方法的迭代复杂度必然随着维数的增加而增加。我们的论证通过简化到估计变形Wigner模型M =W + λ U U，其中W来自高斯正交系综，U在d × r-Stieffel流形上是均匀的，λ >1控制扰动的大小。令人惊讶的是，这个无处不在的随机矩阵模型见证了特征空间近似的最坏情况速率，并且当λ接近“相变”λ = 1时，由于渐近特征与扰动λ的大小之间的对应关系，速率中的“加速”间隙- 1/2。为了证明d只需要是gap - 1和r中的多项式，我们证明了变形Wigner矩阵的上特征值的有限样本收敛定理，这可能是一个独立的兴趣。然后，我们使用一种基于截断似然的fano式数据处理不等式的新技术对上述估计问题下界;该技术推广了Chen等人的贝叶斯风险下界[16]，我们认为它特别适合于本文所考虑的自适应设置中的下界。
{"title":"Tight query complexity lower bounds for PCA via finite sample deformed wigner law","authors":"Max Simchowitz, A. Alaoui, B. Recht","doi":"10.1145/3188745.3188796","DOIUrl":"https://doi.org/10.1145/3188745.3188796","url":null,"abstract":"We prove a query complexity lower bound for approximating the top r dimensional eigenspace of a matrix. We consider an oracle model where, given a symmetric matrix M ∈ ℝd × d, an algorithm Alg is allowed to make T exact queries of the form w(i) = M v(i) for i in {1,...,T}, where v(i) is drawn from a distribution which depends arbitrarily on the past queries and measurements {v(j),w(i)}1 ≤ j ≤ i−1. We show that for every gap ∈ (0,1/2], there exists a distribution over matrices M for which 1) gapr(M) = Ω(gap) (where gapr(M) is the normalized gap between the r and r+1-st largest-magnitude eigenvector of M), and 2) any Alg which takes fewer than const × r logd/√gap queries fails (with overwhelming probability) to identity a matrix V ∈ ℝd × r with orthonormal columns for which ⟨ V, M V⟩ ≥ (1 − const × gap)∑i=1r λi(M). Our bound requires only that d is a small polynomial in 1/gap and r, and matches the upper bounds of Musco and Musco ’15. Moreover, it establishes a strict separation between convex optimization and “strict-saddle” non-convex optimization of which PCA is a canonical example: in the former, first-order methods can have dimension-free iteration complexity, whereas in PCA, the iteration complexity of gradient-based methods must necessarily grow with the dimension. Our argument proceeds via a reduction to estimating a rank-r spike in a deformed Wigner model M =W + λ U U⊤, where W is from the Gaussian Orthogonal Ensemble, U is uniform on the d × r-Stieffel manifold and λ > 1 governs the size of the perturbation. Surprisingly, this ubiquitous random matrix model witnesses the worst-case rate for eigenspace approximation, and the ‘accelerated’ gap−1/2 in the rate follows as a consequence of the correspendence between the asymptotic eigengap and the size of the perturbation λ, when λ is near the “phase transition” λ = 1. To verify that d need only be polynomial in gap−1 and r, we prove a finite sample convergence theorem for top eigenvalues of a deformed Wigner matrix, which may be of independent interest. We then lower bound the above estimation problem with a novel technique based on Fano-style data-processing inequalities with truncated likelihoods; the technique generalizes the Bayes-risk lower bound of Chen et al. ’16, and we believe it is particularly suited to lower bounds in adaptive settings like the one considered in this paper.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81587496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zeyuan Allen-Zhu, A. Garg, Yuanzhi Li, R. Oliveira, A. Wigderson
We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, "commutative" metric (for which the above problem is not convex). Our method is general and applicable to other settings. As a consequence, we solve the equivalence problem for the left-right group action underlying the operator scaling problem. This yields a deterministic polynomial-time algorithm for a new class of Polynomial Identity Testing (PIT) problems, which was the original motivation for studying operator scaling.
{"title":"Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing","authors":"Zeyuan Allen-Zhu, A. Garg, Yuanzhi Li, R. Oliveira, A. Wigderson","doi":"10.1145/3188745.3188942","DOIUrl":"https://doi.org/10.1145/3188745.3188942","url":null,"abstract":"We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, \"commutative\" metric (for which the above problem is not convex). Our method is general and applicable to other settings. As a consequence, we solve the equivalence problem for the left-right group action underlying the operator scaling problem. This yields a deterministic polynomial-time algorithm for a new class of Polynomial Identity Testing (PIT) problems, which was the original motivation for studying operator scaling.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74415294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be an edge-weighted directed graph with n vertices embedded on an orientable surface of genus g. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest paths in G. The perturbations take O(gn) time to compute. We use our perturbation scheme in a black box manner to derive a deterministic O(n loglogn) time algorithm for minimum cut in directed edge-weighted planar graphs and a deterministic O(g2 n logn) time proprocessing scheme for the multiple-source shortest paths problem of computing a shortest path oracle for all vertices lying on a common face of a surface embedded graph. The latter result yields faster deterministic near-linear time algorithms for a variety of problems in constant genus surface embedded graphs. Finally, we open the black box in order to generalize a recent linear-time algorithm for multiple-source shortest paths in unweighted undirected planar graphs to work in arbitrary orientable surfaces. Our algorithm runs in O(g2 n logg) time in this setting, and it can be used to give improved linear time algorithms for several problems in unweighted undirected surface embedded graphs of constant genus including the computation of minimum cuts, shortest topologically non-trivial cycles, and minimum homology bases.
设G是一个边加权有向图，其中n个顶点嵌入在G属的可定向表面上。我们描述了一个简单的确定性字典摄动方案，它保证了G中最小代价流和最短路径的唯一性。摄动需要O(gn)时间来计算。我们以黑盒方式使用我们的摄动格式导出了一个确定的O(n logn)时间算法，用于有向边加权平面图中的最小切割，以及一个确定的O(g2 n logn)时间处理方案，用于计算位于表面嵌入图的公共面上的所有顶点的最短路径oracle的多源最短路径问题。后一种结果为常属曲面嵌入图的各种问题提供了更快的确定性近线性时间算法。最后，我们打开黑盒子，以推广最近的一种线性时间算法，该算法适用于无加权无向平面图中的多源最短路径，适用于任意可定向曲面。在这种情况下，我们的算法在O(g2 n log)时间内运行，并且它可以用于在常数属的无加权无向曲面嵌入图中提供改进的线性时间算法，包括最小切割，最短拓扑非平凡循环和最小同调基的计算。
{"title":"Holiest minimum-cost paths and flows in surface graphs","authors":"Jeff Erickson, K. Fox, Luvsandondov Lkhamsuren","doi":"10.1145/3188745.3188904","DOIUrl":"https://doi.org/10.1145/3188745.3188904","url":null,"abstract":"Let G be an edge-weighted directed graph with n vertices embedded on an orientable surface of genus g. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest paths in G. The perturbations take O(gn) time to compute. We use our perturbation scheme in a black box manner to derive a deterministic O(n loglogn) time algorithm for minimum cut in directed edge-weighted planar graphs and a deterministic O(g2 n logn) time proprocessing scheme for the multiple-source shortest paths problem of computing a shortest path oracle for all vertices lying on a common face of a surface embedded graph. The latter result yields faster deterministic near-linear time algorithms for a variety of problems in constant genus surface embedded graphs. Finally, we open the black box in order to generalize a recent linear-time algorithm for multiple-source shortest paths in unweighted undirected planar graphs to work in arbitrary orientable surfaces. Our algorithm runs in O(g2 n logg) time in this setting, and it can be used to give improved linear time algorithms for several problems in unweighted undirected surface embedded graphs of constant genus including the computation of minimum cuts, shortest topologically non-trivial cycles, and minimum homology bases.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80981796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mikkel Abrahamsen, Anna Adamaszek, K. Bringmann, Vincent Cohen-Addad, M. Mehr, E. Rotenberg, A. Roytman, M. Thorup
We consider very natural ”fence enclosure” problems studied by Capoyleas, Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a set S of n points in the plane, we aim at finding a set of closed curves such that (1) each point is enclosed by a curve and (2) the total length of the curves is minimized. We consider two main variants. In the first variant, we pay a unit cost per curve in addition to the total length of the curves. An equivalent formulation of this version is that we have to enclose n unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most k closed curves and pay no cost per curve. For the variant with at most k closed curves,we present an algorithm that is polynomialin bothn andk. For the variant with unit cost per curve, or unit disks, we presenta near-linear time algorithm. Capoyleas, Rote, and Woeginger solved the problem with at most k curves in nO(k) time. Arkin, Khuller, and Mitchell used this to solve the unit cost per curve version in exponential time. At the time, they conjectured that the problem with k curves is NP-hard for general k. Our polynomial time algorithm refutes this unless P equals NP.
{"title":"Fast fencing","authors":"Mikkel Abrahamsen, Anna Adamaszek, K. Bringmann, Vincent Cohen-Addad, M. Mehr, E. Rotenberg, A. Roytman, M. Thorup","doi":"10.1145/3188745.3188878","DOIUrl":"https://doi.org/10.1145/3188745.3188878","url":null,"abstract":"We consider very natural ”fence enclosure” problems studied by Capoyleas, Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a set S of n points in the plane, we aim at finding a set of closed curves such that (1) each point is enclosed by a curve and (2) the total length of the curves is minimized. We consider two main variants. In the first variant, we pay a unit cost per curve in addition to the total length of the curves. An equivalent formulation of this version is that we have to enclose n unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most k closed curves and pay no cost per curve. For the variant with at most k closed curves,we present an algorithm that is polynomialin bothn andk. For the variant with unit cost per curve, or unit disks, we presenta near-linear time algorithm. Capoyleas, Rote, and Woeginger solved the problem with at most k curves in nO(k) time. Arkin, Khuller, and Mitchell used this to solve the unit cost per curve version in exponential time. At the time, they conjectured that the problem with k curves is NP-hard for general k. Our polynomial time algorithm refutes this unless P equals NP.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76843957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new model of stochastic bandits with adversarial corruptions which aims to capture settings where most of the input follows a stochastic pattern but some fraction of it can be adversarially changed to trick the algorithm, e.g., click fraud, fake reviews and email spam. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In our model, the rewards for all arms are initially drawn from a distribution and are then altered by an adaptive adversary. We provide a simple algorithm whose performance gracefully degrades with the total corruption the adversary injected in the data, measured by the sum across rounds of the biggest alteration the adversary made in the data in that round; this total corruption is denoted by C. Our algorithm provides a guarantee that retains the optimal guarantee (up to a logarithmic term) if the input is stochastic and whose performance degrades linearly to the amount of corruption C, while crucially being agnostic to it. We also provide a lower bound showing that this linear degradation is necessary if the algorithm achieves optimal performance in the stochastic setting (the lower bound works even for a known amount of corruption, a special case in which our algorithm achieves optimal performance without the extra logarithm).
{"title":"Stochastic bandits robust to adversarial corruptions","authors":"Thodoris Lykouris, V. Mirrokni, R. Leme","doi":"10.1145/3188745.3188918","DOIUrl":"https://doi.org/10.1145/3188745.3188918","url":null,"abstract":"We introduce a new model of stochastic bandits with adversarial corruptions which aims to capture settings where most of the input follows a stochastic pattern but some fraction of it can be adversarially changed to trick the algorithm, e.g., click fraud, fake reviews and email spam. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In our model, the rewards for all arms are initially drawn from a distribution and are then altered by an adaptive adversary. We provide a simple algorithm whose performance gracefully degrades with the total corruption the adversary injected in the data, measured by the sum across rounds of the biggest alteration the adversary made in the data in that round; this total corruption is denoted by C. Our algorithm provides a guarantee that retains the optimal guarantee (up to a logarithmic term) if the input is stochastic and whose performance degrades linearly to the amount of corruption C, while crucially being agnostic to it. We also provide a lower bound showing that this linear degradation is necessary if the algorithm achieves optimal performance in the stochastic setting (the lower bound works even for a known amount of corruption, a special case in which our algorithm achieves optimal performance without the extra logarithm).","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73848499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a generalized network design (GND) problem, a set of resources are assigned (non-exclusively) to multiple requests. Each request contributes its weight to the resources it uses and the total load on a resource is then translated to the cost it incurs via a resource specific cost function. Motivated by energy efficiency applications, recently, there is a growing interest in GND using cost functions that exhibit (dis)economies of scale ((D)oS), namely, cost functions that appear subadditive for small loads and superadditive for larger loads. The current paper advances the existing literature on approximation algorithms for GND problems with (D)oS cost functions in various aspects: (1) while the existing results are restricted to routing requests in undirected graphs, identifying the resources with the graph's edges, the current paper presents a generic approximation framework that yields approximation results for a much wider family of requests (including various types of Steiner tree and Steiner forest requests) in both directed and undirected graphs, where the resources can be identified with either the edges or the vertices; (2) while the existing results assume that a request contributes the same weight to each resource it uses, our approximation framework allows for unrelated weights, thus providing the first non-trivial approximation for the problem of scheduling unrelated parallel machines with (D)oS cost functions; (3) while most of the existing approximation algorithms are based on convex programming, our approximation framework is fully combinatorial and runs in strongly polynomial time; (4) the family of (D)oS cost functions considered in the current paper is more general than the one considered in the existing literature, providing a more accurate abstraction for practical energy conservation scenarios; and (5) we obtain the first approximation ratio for GND with (D)oS cost functions that depends only on the parameters of the resources' technology and does not grow with the number of resources, the number of requests, or their weights. The design of our approximation framework relies heavily on Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible usefulness of this toolbox in the area of approximation algorithms.
{"title":"Approximating generalized network design under (dis)economies of scale with applications to energy efficiency","authors":"Y. Emek, S. Kutten, R. Lavi, Yangguang Shi","doi":"10.1145/3188745.3188812","DOIUrl":"https://doi.org/10.1145/3188745.3188812","url":null,"abstract":"In a generalized network design (GND) problem, a set of resources are assigned (non-exclusively) to multiple requests. Each request contributes its weight to the resources it uses and the total load on a resource is then translated to the cost it incurs via a resource specific cost function. Motivated by energy efficiency applications, recently, there is a growing interest in GND using cost functions that exhibit (dis)economies of scale ((D)oS), namely, cost functions that appear subadditive for small loads and superadditive for larger loads. The current paper advances the existing literature on approximation algorithms for GND problems with (D)oS cost functions in various aspects: (1) while the existing results are restricted to routing requests in undirected graphs, identifying the resources with the graph's edges, the current paper presents a generic approximation framework that yields approximation results for a much wider family of requests (including various types of Steiner tree and Steiner forest requests) in both directed and undirected graphs, where the resources can be identified with either the edges or the vertices; (2) while the existing results assume that a request contributes the same weight to each resource it uses, our approximation framework allows for unrelated weights, thus providing the first non-trivial approximation for the problem of scheduling unrelated parallel machines with (D)oS cost functions; (3) while most of the existing approximation algorithms are based on convex programming, our approximation framework is fully combinatorial and runs in strongly polynomial time; (4) the family of (D)oS cost functions considered in the current paper is more general than the one considered in the existing literature, providing a more accurate abstraction for practical energy conservation scenarios; and (5) we obtain the first approximation ratio for GND with (D)oS cost functions that depends only on the parameters of the resources' technology and does not grow with the number of resources, the number of requests, or their weights. The design of our approximation framework relies heavily on Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible usefulness of this toolbox in the area of approximation algorithms.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88620630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove conditional near-quadratic running time lower bounds for approximate Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance. Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false, for every δ>0 there exists a constant ε>0 such that computing a (1+ε)-approximation to the Bichromatic Closest Pair requires Ω(n2−δ) time. In particular, this implies a near-linear query time for Approximate Nearest Neighbor search with polynomial preprocessing time. Our reduction uses the recently introduced Distributed PCP framework, but obtains improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG codes have been constructed in other settings before, but our construction is the first to yield new hardness results.
{"title":"Hardness of approximate nearest neighbor search","authors":"A. Rubinstein","doi":"10.1145/3188745.3188916","DOIUrl":"https://doi.org/10.1145/3188745.3188916","url":null,"abstract":"We prove conditional near-quadratic running time lower bounds for approximate Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance. Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false, for every δ>0 there exists a constant ε>0 such that computing a (1+ε)-approximation to the Bichromatic Closest Pair requires Ω(n2−δ) time. In particular, this implies a near-linear query time for Approximate Nearest Neighbor search with polynomial preprocessing time. Our reduction uses the recently introduced Distributed PCP framework, but obtains improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG codes have been constructed in other settings before, but our construction is the first to yield new hardness results.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77086977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sepehr Assadi, Krzysztof Onak, B. Schieber, Shay Solomon
A maximal independent set (MIS) can be maintained in an evolving m-edge graph by simply recomputing it from scratch in O(m) time after each update. But can it be maintained in time sublinear in m in fully dynamic graphs? We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time O(min{Δ,m3/4}), where Δ is a fixed bound on the maximum degree in the graph and m is the (dynamically changing) number of edges. We further present a distributed implementation of our algorithm with O(min{Δ,m3/4}) amortized message complexity, and O(1) amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC’16) that required an assumption of a non-adaptive oblivious adversary.
{"title":"Fully dynamic maximal independent set with sublinear update time","authors":"Sepehr Assadi, Krzysztof Onak, B. Schieber, Shay Solomon","doi":"10.1145/3188745.3188922","DOIUrl":"https://doi.org/10.1145/3188745.3188922","url":null,"abstract":"A maximal independent set (MIS) can be maintained in an evolving m-edge graph by simply recomputing it from scratch in O(m) time after each update. But can it be maintained in time sublinear in m in fully dynamic graphs? We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time O(min{Δ,m3/4}), where Δ is a fixed bound on the maximum degree in the graph and m is the (dynamically changing) number of edges. We further present a distributed implementation of our algorithm with O(min{Δ,m3/4}) amortized message complexity, and O(1) amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC’16) that required an assumption of a non-adaptive oblivious adversary.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85824558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures (Allen,Munro, 1978; Sleator, Tarjan, 1983; Fredman, Sedgewick, Sleator, Tarjan, 1986; Wilber, 1989; Fredman, 1999; Iacono, Özkan, 2014). Roughly speaking, we map an arbitrary heap algorithm within a broad and natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature (e.g. Pettie; 2005, 2008). Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, conjectured to be instance-optimal (Lucas, 1988; Munro, 2000; Demaine et al., 2009). Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a “power-of-two-choices” type of heuristic. For the smooth heap we obtain instance-specific upper bounds, with applications in adaptive sorting, and we see it as a promising candidate for the long-standing question of a simpler alternative to Fibonacci heaps.
{"title":"Smooth heaps and a dual view of self-adjusting data structures","authors":"L. Kozma, Thatchaphol Saranurak","doi":"10.1145/3188745.3188864","DOIUrl":"https://doi.org/10.1145/3188745.3188864","url":null,"abstract":"We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures (Allen,Munro, 1978; Sleator, Tarjan, 1983; Fredman, Sedgewick, Sleator, Tarjan, 1986; Wilber, 1989; Fredman, 1999; Iacono, Özkan, 2014). Roughly speaking, we map an arbitrary heap algorithm within a broad and natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature (e.g. Pettie; 2005, 2008). Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, conjectured to be instance-optimal (Lucas, 1988; Munro, 2000; Demaine et al., 2009). Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a “power-of-two-choices” type of heuristic. For the smooth heap we obtain instance-specific upper bounds, with applications in adaptive sorting, and we see it as a promising candidate for the long-standing question of a simpler alternative to Fibonacci heaps.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86221123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1−1/e-competitive in our model even for bipartite graphs.
{"title":"How to match when all vertices arrive online","authors":"Zhiyi Huang, N. Kang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang, Xue Zhu","doi":"10.1145/3188745.3188858","DOIUrl":"https://doi.org/10.1145/3188745.3188858","url":null,"abstract":"We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1−1/e-competitive in our model even for bipartite graphs.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76548172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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