Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.04994
Michal Dory, M. Ghaffari
The minimum-cost $k$-edge-connected spanning subgraph ($k$-ECSS) problem is a generalization and strengthening of the well-studied minimum-cost spanning tree (MST) problem. While the round complexity of distributedly computing the latter has been well-understood, the former remains mostly open, especially as soon as $kgeq 3$. In this paper, we present the first distributed algorithm that computes an approximation of $k$-ECSS in sublinear time for general $k$. Concretely, we describe a randomized distributed algorithm that, in $tilde{O}(k(D+ksqrt{n}))$ rounds, computes a $k$-edge-connected spanning subgraph whose cost is within an $O(log nlog k)$ factor of optimal. Here, $n$ and $D$ denote the number of vertices and diameter of the graph, respectively. This time complexity is nearly optimal for any $k=poly(log n)$, almost matching an $tilde{Omega}(D+sqrt{n/k})$ lower bound. Our algorithm is the first to achieve a sublinear round complexity for $kgeq 3$. We note that this case is considerably more challenging than the well-studied and well-understood $k=1$ case -- better known as MST -- and the closely related $k=2$ case. Our algorithm is based on reducing the $k$-ECSS problem to $k$ set cover instances, in which we gradually augment the connectivity of the spanning subgraph. To solve each set cover instance, we combine new structural observations on minimum cuts with graph sketching ideas. One key ingredient in our algorithm is a novel structural lemma that allows us to compress the information about all minimum cuts in a graph into a succinct representation, which is computed in a decentralized fashion. We hope that this succinct representation may find applications in other computational settings or for other problems.
{"title":"A Nearly Time-Optimal Distributed Approximation of Minimum Cost k-Edge-Connected Spanning Subgraph","authors":"Michal Dory, M. Ghaffari","doi":"10.48550/arXiv.2211.04994","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04994","url":null,"abstract":"The minimum-cost $k$-edge-connected spanning subgraph ($k$-ECSS) problem is a generalization and strengthening of the well-studied minimum-cost spanning tree (MST) problem. While the round complexity of distributedly computing the latter has been well-understood, the former remains mostly open, especially as soon as $kgeq 3$. In this paper, we present the first distributed algorithm that computes an approximation of $k$-ECSS in sublinear time for general $k$. Concretely, we describe a randomized distributed algorithm that, in $tilde{O}(k(D+ksqrt{n}))$ rounds, computes a $k$-edge-connected spanning subgraph whose cost is within an $O(log nlog k)$ factor of optimal. Here, $n$ and $D$ denote the number of vertices and diameter of the graph, respectively. This time complexity is nearly optimal for any $k=poly(log n)$, almost matching an $tilde{Omega}(D+sqrt{n/k})$ lower bound. Our algorithm is the first to achieve a sublinear round complexity for $kgeq 3$. We note that this case is considerably more challenging than the well-studied and well-understood $k=1$ case -- better known as MST -- and the closely related $k=2$ case. Our algorithm is based on reducing the $k$-ECSS problem to $k$ set cover instances, in which we gradually augment the connectivity of the spanning subgraph. To solve each set cover instance, we combine new structural observations on minimum cuts with graph sketching ideas. One key ingredient in our algorithm is a novel structural lemma that allows us to compress the information about all minimum cuts in a graph into a succinct representation, which is computed in a decentralized fashion. We hope that this succinct representation may find applications in other computational settings or for other problems.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"6 1","pages":"4296-4334"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86790365","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}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.05170
Manuel Stoeckl
Many problems on data streams have been studied at two extremes of difficulty: either allowing randomized algorithms, in the static setting (where they should err with bounded probability on the worst case stream); or when only deterministic and infallible algorithms are required. Some recent works have considered the adversarial setting, in which a randomized streaming algorithm must succeed even on data streams provided by an adaptive adversary that can see the intermediate outputs of the algorithm. In order to better understand the differences between these models, we study a streaming task called"Missing Item Finding". In this problem, for $r
{"title":"Streaming algorithms for the missing item finding problem","authors":"Manuel Stoeckl","doi":"10.48550/arXiv.2211.05170","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05170","url":null,"abstract":"Many problems on data streams have been studied at two extremes of difficulty: either allowing randomized algorithms, in the static setting (where they should err with bounded probability on the worst case stream); or when only deterministic and infallible algorithms are required. Some recent works have considered the adversarial setting, in which a randomized streaming algorithm must succeed even on data streams provided by an adaptive adversary that can see the intermediate outputs of the algorithm. In order to better understand the differences between these models, we study a streaming task called\"Missing Item Finding\". In this problem, for $r<n$, one is given a data stream $a_1,ldots,a_r$ of elements in $[n]$, (possibly with repetitions), and must output some $x in [n]$ which does not equal any of the $a_i$. We prove that, for $r = n^{Theta(1)}$ and $delta = 1/mathrm{poly}(n)$, the space required for randomized algorithms that solve this problem in the static setting with error $delta$ is $Theta(mathrm{polylog}(n))$; for algorithms in the adversarial setting with error $delta$, $Theta((1 + r^2 / n) mathrm{polylog}(n))$; and for deterministic algorithms, $Theta(r / mathrm{polylog}(n))$. Because our adversarially robust algorithm relies on free access to a string of $O(r log n)$ random bits, we investigate a\"random start\"model of streaming algorithms where all random bits used are included in the space cost. Here we find a conditional lower bound on the space usage, which depends on the space that would be needed for a pseudo-deterministic algorithm to solve the problem. We also prove an $Omega(r / mathrm{polylog}(n))$ lower bound for the space needed by a streaming algorithm with $<1/2^{mathrm{polylog}(n)}$ error against\"white-box\"adversaries that can see the internal state of the algorithm, but not predict its future random decisions.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"118 1","pages":"793-818"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77411029","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}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.05006
M. Henzinger, Jalaj Upadhyay, Sarvagya Upadhyay
The first large-scale deployment of private federated learning uses differentially private counting in the continual release model as a subroutine (Google AI blog titled"Federated Learning with Formal Differential Privacy Guarantees"). In this case, a concrete bound on the error is very relevant to reduce the privacy parameter. The standard mechanism for continual counting is the binary mechanism. We present a novel mechanism and show that its mean squared error is both asymptotically optimal and a factor 10 smaller than the error of the binary mechanism. We also show that the constants in our analysis are almost tight by giving non-asymptotic lower and upper bounds that differ only in the constants of lower-order terms. Our algorithm is a matrix mechanism for the counting matrix and takes constant time per release. We also use our explicit factorization of the counting matrix to give an upper bound on the excess risk of the private learning algorithm of Denisov et al. (NeurIPS 2022). Our lower bound for any continual counting mechanism is the first tight lower bound on continual counting under approximate differential privacy. It is achieved using a new lower bound on a certain factorization norm, denoted by $gamma_F(cdot)$, in terms of the singular values of the matrix. In particular, we show that for any complex matrix, $A in mathbb{C}^{m times n}$, [ gamma_F(A) geq frac{1}{sqrt{m}}|A|_1, ] where $|cdot |$ denotes the Schatten-1 norm. We believe this technique will be useful in proving lower bounds for a larger class of linear queries. To illustrate the power of this technique, we show the first lower bound on the mean squared error for answering parity queries.
私有联邦学习的第一次大规模部署使用持续发布模型中的差分私有计数作为子例程(Google AI博客标题为“具有正式差分隐私保证的联邦学习”)。在这种情况下，错误的具体界限与减少隐私参数非常相关。连续计数的标准机制是二进制机制。我们提出了一种新的机构，并证明了它的均方误差是渐近最优的，并且比二元机构的误差小10倍。我们还通过给出仅在低阶项的常数不同的非渐近下界和上界，证明了我们分析中的常数几乎是紧的。我们的算法是计数矩阵的矩阵机制，每次释放花费恒定的时间。我们还使用计数矩阵的显式分解来给出Denisov等人(NeurIPS 2022)的私有学习算法的超额风险的上限。本文给出的连续计数机制的下界是近似微分隐私下连续计数的第一个紧下界。它是使用一个新的下界在一个特定的分解范数，表示为$gamma_F(cdot)$，在矩阵的奇异值。特别地，我们证明了对于任何复矩阵$A in mathbb{C}^{m times n}$, [ gamma_F(A) geq frac{1}{sqrt{m}}|A|_1, ]，其中$|cdot |$表示schatten1范数。我们相信这种技术在证明更大类线性查询的下界时会很有用。为了说明这种技术的强大功能，我们展示了回答奇偶查询的均方误差的第一个下界。
{"title":"Almost Tight Error Bounds on Differentially Private Continual Counting","authors":"M. Henzinger, Jalaj Upadhyay, Sarvagya Upadhyay","doi":"10.48550/arXiv.2211.05006","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05006","url":null,"abstract":"The first large-scale deployment of private federated learning uses differentially private counting in the continual release model as a subroutine (Google AI blog titled\"Federated Learning with Formal Differential Privacy Guarantees\"). In this case, a concrete bound on the error is very relevant to reduce the privacy parameter. The standard mechanism for continual counting is the binary mechanism. We present a novel mechanism and show that its mean squared error is both asymptotically optimal and a factor 10 smaller than the error of the binary mechanism. We also show that the constants in our analysis are almost tight by giving non-asymptotic lower and upper bounds that differ only in the constants of lower-order terms. Our algorithm is a matrix mechanism for the counting matrix and takes constant time per release. We also use our explicit factorization of the counting matrix to give an upper bound on the excess risk of the private learning algorithm of Denisov et al. (NeurIPS 2022). Our lower bound for any continual counting mechanism is the first tight lower bound on continual counting under approximate differential privacy. It is achieved using a new lower bound on a certain factorization norm, denoted by $gamma_F(cdot)$, in terms of the singular values of the matrix. In particular, we show that for any complex matrix, $A in mathbb{C}^{m times n}$, [ gamma_F(A) geq frac{1}{sqrt{m}}|A|_1, ] where $|cdot |$ denotes the Schatten-1 norm. We believe this technique will be useful in proving lower bounds for a larger class of linear queries. To illustrate the power of this technique, we show the first lower bound on the mean squared error for answering parity queries.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"17 16 1","pages":"5003-5039"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82560020","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}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.05053
Kim-Manuel Klein, Adam Polak, Lars Rohwedder
The starting point of this paper is the problem of scheduling $n$ jobs with processing times and due dates on a single machine so as to minimize the total processing time of tardy jobs, i.e., $1||sum p_j U_j$. This problem was identified by Bringmann et al. (Algorithmica 2022) as a natural subquadratic-time special case of the classic $1||sum w_j U_j$ problem, which likely requires time quadratic in the total processing time $P$, because of a fine-grained lower bound. Bringmann et al.~obtain their $tilde{O}(P^{7/4})$ time scheduling algorithm through a new variant of convolution, dubbed Max-Min Skewed Convolution, which they solve in $tilde{O}(n^{7/4})$ time. Our main technical contribution is a faster and simpler convolution algorithm running in $tilde{O}(n^{5/3})$ time. It implies an $tilde{O}(P^{5/3})$ time algorithm for $1||sum p_j U_j$, but may also be of independent interest. Inspired by recent developments for the Subset Sum and Knapsack problems, we study $1||sum p_j U_j$ parameterized by the maximum job processing time $p_{max}$. With proximity techniques borrowed from integer linear programming (ILP), we show structural properties of the problem that, coupled with a new dynamic programming formulation, lead to an $tilde{O}(n+p_{max}^3)$ time algorithm. Moreover, in the setting with multiple machines, we use similar techniques to get an $n cdot p_{max}^{O(m)}$ time algorithm for $Pm||sum p_j U_j$. Finally, we point out that the considered problems exhibit a particular triangular block structure in the constraint matrices of their ILP formulations. In light of recent ILP research, a question that arises is whether one can devise a generic algorithm for such a class of ILPs. We give a negative answer to this question: we show that already a slight generalization of the structure of the scheduling ILP leads to a strongly NP-hard problem.
{"title":"On Minimizing Tardy Processing Time, Max-Min Skewed Convolution, and Triangular Structured ILPs","authors":"Kim-Manuel Klein, Adam Polak, Lars Rohwedder","doi":"10.48550/arXiv.2211.05053","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05053","url":null,"abstract":"The starting point of this paper is the problem of scheduling $n$ jobs with processing times and due dates on a single machine so as to minimize the total processing time of tardy jobs, i.e., $1||sum p_j U_j$. This problem was identified by Bringmann et al. (Algorithmica 2022) as a natural subquadratic-time special case of the classic $1||sum w_j U_j$ problem, which likely requires time quadratic in the total processing time $P$, because of a fine-grained lower bound. Bringmann et al.~obtain their $tilde{O}(P^{7/4})$ time scheduling algorithm through a new variant of convolution, dubbed Max-Min Skewed Convolution, which they solve in $tilde{O}(n^{7/4})$ time. Our main technical contribution is a faster and simpler convolution algorithm running in $tilde{O}(n^{5/3})$ time. It implies an $tilde{O}(P^{5/3})$ time algorithm for $1||sum p_j U_j$, but may also be of independent interest. Inspired by recent developments for the Subset Sum and Knapsack problems, we study $1||sum p_j U_j$ parameterized by the maximum job processing time $p_{max}$. With proximity techniques borrowed from integer linear programming (ILP), we show structural properties of the problem that, coupled with a new dynamic programming formulation, lead to an $tilde{O}(n+p_{max}^3)$ time algorithm. Moreover, in the setting with multiple machines, we use similar techniques to get an $n cdot p_{max}^{O(m)}$ time algorithm for $Pm||sum p_j U_j$. Finally, we point out that the considered problems exhibit a particular triangular block structure in the constraint matrices of their ILP formulations. In light of recent ILP research, a question that arises is whether one can devise a generic algorithm for such a class of ILPs. We give a negative answer to this question: we show that already a slight generalization of the structure of the scheduling ILP leads to a strongly NP-hard problem.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"3 4","pages":"2947-2960"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72591168","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}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.05217
Josh Alman, Yunfeng Guan, Ashwin Padaki
We give new, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix. A standard argument (e.g., the mixed product property of Kronecker products, or a generalization of the Fast Walsh-Hadamard transform) shows that any such $N times N$ matrix has a depth-2 circuit of size $O(N^{1.5})$. We improve on this for all such matrices, and especially for some such matrices of particular interest: - For any integer $q>1$ and any matrix which is the Kronecker power of a fixed $q times q$ matrix, we construct a depth-2 circuit of size $O(N^{1.5 - a_q})$, where $a_q>0$ is a positive constant depending only on $q$. No bound beating size $O(N^{1.5})$ was previously known for any $q>2$. - For the case $q=2$, i.e., for any matrix which is the Kronecker power of a fixed $2 times 2$ matrix, we construct a depth-2 circuit of size $O(N^{1.446})$, improving the prior best size $O(N^{1.493})$ [Alman, 2021]. - For the Walsh-Hadamard transform, we construct a depth-2 circuit of size $O(N^{1.443})$, improving the prior best size $O(N^{1.476})$ [Alman, 2021]. - For the disjointness matrix (the communication matrix of set disjointness, or equivalently, the matrix for the linear transform that evaluates a multilinear polynomial on all $0/1$ inputs), we construct a depth-2 circuit of size $O(N^{1.258})$, improving the prior best size $O(N^{1.272})$ [Jukna and Sergeev, 2013]. Our constructions also generalize to improving the standard construction for any depth $leq O(log N)$. Our main technical tool is an improved way to convert a nontrivial circuit for any matrix into a circuit for its Kronecker powers. Our new bounds provably could not be achieved using the approaches of prior work.
我们给出了新的、更小的等深度线性电路结构，用于计算任何固定矩阵的克罗内克幂矩阵。一个标准的论证(例如，Kronecker积的混合积性质，或快速Walsh-Hadamard变换的推广)表明，任何这样的$N times N$矩阵都有一个大小为$O(N^{1.5})$的深度2电路。对于所有这样的矩阵，我们改进了这一点，特别是对于一些特别感兴趣的矩阵:—对于任何整数$q>1$和任何固定$q times q$矩阵的Kronecker幂矩阵，我们构建了一个大小为$O(N^{1.5 - a_q})$的深度2电路，其中$a_q>0$是一个仅依赖于$q$的正常数。以前不知道任何$q>2$的固定跳动大小$O(N^{1.5})$。-对于$q=2$的情况，即对于固定$2 times 2$矩阵的Kronecker幂的任何矩阵，我们构建了一个大小为$O(N^{1.446})$的深度2电路，改进了先前的最佳大小$O(N^{1.493})$ [Alman, 2021]。-对于Walsh-Hadamard变换，我们构建了一个大小为$O(N^{1.443})$的深度-2电路，改进了先验最佳大小$O(N^{1.476})$ [Alman, 2021]。-对于脱节矩阵(集合脱节的通信矩阵，或等效地，用于评估所有$0/1$输入上的多线性多项式的线性变换的矩阵)，我们构建了大小为$O(N^{1.258})$的深度2电路，改进了先验最佳大小$O(N^{1.272})$ [Jukna和Sergeev, 2013]。我们的结构也推广到改善任何深度的标准结构$leq O(log N)$。我们的主要技术工具是一种改进的方法，可以将任意矩阵的非平凡电路转换为其克罗内克幂的电路。我们的新边界可以证明不能用先前工作的方法得到。
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Greedy BST (or simply Greedy) is an online self-adjusting binary search tree defined in the geometric view ([Lucas, 1988; Munro, 2000; Demaine, Harmon, Iacono, Kane, Patrascu, SODA 2009). Along with Splay trees (Sleator, Tarjan 1985), Greedy is considered the most promising candidate for being dynamically optimal, i.e., starting with any initial tree, their access costs on any sequence is conjectured to be within $O(1)$ factor of the offline optimal. However, in the past four decades, the question has remained elusive even for highly restricted input. In this paper, we prove new bounds on the cost of Greedy in the ''pattern avoidance'' regime. Our new results include: The (preorder) traversal conjecture for Greedy holds up to a factor of $O(2^{alpha(n)})$, improving upon the bound of $2^{alpha(n)^{O(1)}}$ in (Chalermsook et al., FOCS 2015). This is the best known bound obtained by any online BSTs. We settle the postorder traversal conjecture for Greedy. The deque conjecture for Greedy holds up to a factor of $O(alpha(n))$, improving upon the bound $2^{O(alpha(n))}$ in (Chalermsook, et al., WADS 2015). The split conjecture holds for Greedy up to a factor of $O(2^{alpha(n)})$. Key to all these results is to partition (based on the input structures) the execution log of Greedy into several simpler-to-analyze subsets for which classical forbidden submatrix bounds can be leveraged. Finally, we show the applicability of this technique to handle a class of increasingly complex pattern-avoiding input sequences, called $k$-increasing sequences. As a bonus, we discover a new class of permutation matrices whose extremal bounds are polynomially bounded. This gives a partial progress on an open question by Jacob Fox (2013).
贪心BST(或简称为贪心)是一种定义在几何视图中的在线自调整二叉搜索树(Lucas, 1988;Munro, 2000;Demaine, Harmon, Iacono, Kane, Patrascu, SODA, 2009)。与Splay树(Sleator, Tarjan 1985)一样，贪心被认为是动态最优的最有希望的候选，即从任何初始树开始，它们在任何序列上的访问成本都被推测在离线最优的$O(1)$因子之内。然而，在过去的四十年里，即使对高度限制的投入，这个问题仍然难以捉摸。在本文中，我们证明了“模式回避”状态下贪心代价的新界。我们的新结果包括:贪心的(预序)遍历猜想符合$O(2^{alpha(n)})$的因子，改进了$2^{alpha(n)^{O(1)}}$的边界(Chalermsook et al.， FOCS 2015)。这是任何联机BSTs得到的最知名的界。我们解决了贪心的后置遍历猜想。贪心的deque猜想符合$O(alpha(n))$的因子，改进了(Chalermsook, et al.， WADS 2015)中$2^{O(alpha(n))}$的边界。对于贪心来说，分裂猜想一直成立到一个因子$O(2^{alpha(n)})$。所有这些结果的关键是(基于输入结构)将Greedy的执行日志划分为几个易于分析的子集，这些子集可以利用经典的禁止子矩阵边界。最后，我们展示了这种技术在处理一类越来越复杂的模式避免输入序列(称为$k$递增序列)方面的适用性。作为奖励，我们发现了一类新的排列矩阵，其极界是多项式有界的。这对Jacob Fox(2013)提出的一个开放性问题给出了部分进展。
{"title":"Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition","authors":"Parinya Chalermsook, Manoj Gupta, W. Jiamjitrak, Nidia Obscura Acosta, Akash Pareek, Sorrachai Yingchareonthawornchai","doi":"10.48550/arXiv.2211.04112","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04112","url":null,"abstract":"Greedy BST (or simply Greedy) is an online self-adjusting binary search tree defined in the geometric view ([Lucas, 1988; Munro, 2000; Demaine, Harmon, Iacono, Kane, Patrascu, SODA 2009). Along with Splay trees (Sleator, Tarjan 1985), Greedy is considered the most promising candidate for being dynamically optimal, i.e., starting with any initial tree, their access costs on any sequence is conjectured to be within $O(1)$ factor of the offline optimal. However, in the past four decades, the question has remained elusive even for highly restricted input. In this paper, we prove new bounds on the cost of Greedy in the ''pattern avoidance'' regime. Our new results include: The (preorder) traversal conjecture for Greedy holds up to a factor of $O(2^{alpha(n)})$, improving upon the bound of $2^{alpha(n)^{O(1)}}$ in (Chalermsook et al., FOCS 2015). This is the best known bound obtained by any online BSTs. We settle the postorder traversal conjecture for Greedy. The deque conjecture for Greedy holds up to a factor of $O(alpha(n))$, improving upon the bound $2^{O(alpha(n))}$ in (Chalermsook, et al., WADS 2015). The split conjecture holds for Greedy up to a factor of $O(2^{alpha(n)})$. Key to all these results is to partition (based on the input structures) the execution log of Greedy into several simpler-to-analyze subsets for which classical forbidden submatrix bounds can be leveraged. Finally, we show the applicability of this technique to handle a class of increasingly complex pattern-avoiding input sequences, called $k$-increasing sequences. As a bonus, we discover a new class of permutation matrices whose extremal bounds are polynomially bounded. This gives a partial progress on an open question by Jacob Fox (2013).","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"69 1","pages":"509-534"},"PeriodicalIF":0.0,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74877465","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}
Pub Date : 2022-11-07DOI: 10.48550/arXiv.2211.03292
Xi Chen, Anindya De, Chin Ho Lee, R. Servedio, S. Sinha
The well-known trace reconstruction problem is the problem of inferring an unknown source string $x in {0,1}^n$ from independent"traces", i.e. copies of $x$ that have been corrupted by a $delta$-deletion channel which independently deletes each bit of $x$ with probability $delta$ and concatenates the surviving bits. The current paper considers the extreme data-limited regime in which only a single trace is provided to the reconstruction algorithm. In this setting exact reconstruction is of course impossible, and the question is to what accuracy the source string $x$ can be approximately reconstructed. We give a detailed study of this question, providing algorithms and lower bounds for the high, intermediate, and low deletion rate regimes in both the worst-case ($x$ is arbitrary) and average-case ($x$ is drawn uniformly from ${0,1}^n$) models. In several cases the lower bounds we establish are matched by computationally efficient algorithms that we provide. We highlight our results for the high deletion rate regime: roughly speaking, they show that - Having access to a single trace is already quite useful for worst-case trace reconstruction: an efficient algorithm can perform much more accurate reconstruction, given one trace that is even only a few bits long, than it could given no traces at all. But in contrast, - in the average-case setting, having access to a single trace is provably not very useful: no algorithm, computationally efficient or otherwise, can achieve significantly higher accuracy given one trace that is $o(n)$ bits long than it could with no traces.
众所周知的跟踪重建问题是从独立的“跟踪”中推断出未知源字符串$x in {0,1}^n$的问题，即被$delta$删除通道损坏的$x$的副本，该通道以$delta$的概率独立删除$x$的每个比特并连接幸存的比特。本文考虑了只向重构算法提供单一轨迹的极端数据限制情况。在这种情况下，精确的重建当然是不可能的，问题是源字符串$x$可以近似地重建到什么精度。我们对这个问题进行了详细的研究，提供了在最坏情况($x$是任意的)和平均情况($x$是从${0,1}^n$中均匀抽取的)模型中高、中、低删除率的算法和下界。在一些情况下，我们建立的下界与我们提供的计算效率高的算法相匹配。我们强调了高删除率制度的结果:粗略地说，它们表明-访问单个跟踪对于最坏情况下的跟踪重建已经非常有用:一个有效的算法可以执行更准确的重建，给定一个甚至只有几个比特长的跟踪，而不是根本没有跟踪。但相比之下，在一般情况下，访问单个跟踪可证明不是很有用:给定一个跟踪长度为$o(n)$比特的算法，无论计算效率如何，都无法获得比没有跟踪高得多的精度。
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Pub Date : 2022-11-07DOI: 10.48550/arXiv.2211.03877
Andrew Alseth, Matthew J. Patitz
In the abstract Tile Assembly Model (aTAM) square tiles self-assemble, autonomously binding via glues on their edges, to form structures. Algorithmic aTAM systems can be designed in which the patterns of tile attachments are forced to follow the execution of targeted algorithms. Such systems have been proven to be computationally universal as well as intrinsically universal (IU), a notion borrowed and adapted from cellular automata showing that a single tile set exists which is capable of simulating all aTAM systems (FOCS 2012). The input to an algorithmic aTAM system can be provided in a variety of ways, with a common method being via the"seed"assembly, which is a pre-formed assembly from which all growth propagates. In this paper we present a series of results which investigate the the trade-offs of using seeds consisting of a single tile, versus those containing multiple tiles. We show that arbitrary systems with multi-tile seeds cannot be converted to functionally equivalent systems with single-tile seeds without using a scale factor>1. We prove tight bounds on the scale factor required, and also present a construction which uses a large scale factor but an optimal number of unique tile types. That construction is then used to develop a construction that performs simultaneous simulation of all aTAM systems in parallel, as well as to display a connection to other tile-based self-assembly models via the notion of intrinsic universality.
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Pub Date : 2022-11-07DOI: 10.48550/arXiv.2211.03530
A. Balliu, R. Latypov, Yannic Maus, D. Olivetti, Jara Uitto
We show fast deterministic algorithms for fundamental problems on forests in the challenging low-space regime of the well-known Massive Parallel Computation (MPC) model. A recent breakthrough result by Coy and Czumaj [STOC'22] shows that, in this setting, it is possible to deterministically identify connected components on graphs in $O(log D + loglog n)$ rounds, where $D$ is the diameter of the graph and $n$ the number of nodes. The authors left open a major question: is it possible to get rid of the additive $loglog n$ factor and deterministically identify connected components in a runtime that is completely independent of $n$? We answer the above question in the affirmative in the case of forests. We give an algorithm that identifies connected components in $O(log D)$ deterministic rounds. The total memory required is $O(n+m)$ words, where $m$ is the number of edges in the input graph, which is optimal as it is only enough to store the input graph. We complement our upper bound results by showing that $Omega(log D)$ time is necessary even for component-unstable algorithms, conditioned on the widely believed 1 vs. 2 cycles conjecture. Our techniques also yield a deterministic forest-rooting algorithm with the same runtime and memory bounds. Furthermore, we consider Locally Checkable Labeling problems (LCLs), whose solution can be verified by checking the $O(1)$-radius neighborhood of each node. We show that any LCL problem on forests can be solved in $O(log D)$ rounds with a canonical deterministic algorithm, improving over the $O(log n)$ runtime of Brandt, Latypov and Uitto [DISC'21]. We also show that there is no algorithm that solves all LCL problems on trees asymptotically faster.
我们展示了在众所周知的大规模并行计算(MPC)模型具有挑战性的低空间状态下森林基本问题的快速确定性算法。Coy和Czumaj [STOC'22]最近的一项突破性成果表明，在这种情况下，可以在$O(log D + loglog n)$轮中确定地识别图上的连接组件，其中$D$为图的直径，$n$为节点数。作者留下了一个主要的问题:是否有可能摆脱附加的$loglog n$因素并确定地识别完全独立于$n$的运行时中的连接组件?就森林而言，我们对上述问题的回答是肯定的。我们给出了在$O(log D)$确定性轮中识别连接组件的算法。所需的总内存为$O(n+m)$ words，其中$m$是输入图中的边数，这是最优的，因为它只足够存储输入图。我们通过证明$Omega(log D)$时间是必要的来补充我们的上界结果，即使对于组件不稳定算法，条件是广泛相信的1 vs. 2循环猜想。我们的技术还产生了具有相同运行时和内存边界的确定性森林生根算法。此外，我们考虑了局部可检查标记问题(LCLs)，其解可以通过检查每个节点的$O(1)$ -半径邻域来验证。我们证明了森林上的任何LCL问题都可以用规范确定性算法在$O(log D)$轮内解决，比Brandt, Latypov和Uitto [DISC'21]的$O(log n)$运行时间有所改进。我们还证明了没有一种算法可以更快地解决树上的所有LCL问题。
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Pub Date : 2022-11-07DOI: 10.48550/arXiv.2211.03341
Kyungjin Cho, Eunjin Oh, Seunghyuk Oh
In this paper, we study the textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $iin{1,ldots,k}$. We present a $2^{O(k^2)}n$-time algorithm for the textsf{Planar Disjoint Paths} problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].
本文研究了平面不相交路径问题:给定一个有n个顶点的无向平面图$G$和一个由$k$对$(s_i,t_i) $ {i=1}^k$组成的集合$T$，目标是找到$k$对连接$s_i$和$t_i$的顶点不相交路径的集合$ P$，对于所有索引$i In {1，ldots,k}$。我们提出了一个$2^{O(k^2)}n$时间的算法来求解平面不相交路径问题。这改进了之前最著名的两种算法:$2^{2^{O(k)}}n$ time算法[离散应用数学1995]和$2^{O(k^2)}n^6$ time算法[STOC 2020]。
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