Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms最新文献_第6页

A Nearly Time-Optimal Distributed Approximation of Minimum Cost k-Edge-Connected Spanning Subgraph 最小代价k边连通生成子图的近时间最优分布逼近

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-09 DOI: 10.48550/arXiv.2211.04994

Michal Dory, M. Ghaffari

{"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}

引用次数: 0

Streaming algorithms for the missing item finding problem 缺失物品查找问题的流算法

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-09 DOI: 10.48550/arXiv.2211.05170

Manuel Stoeckl

{"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}

引用次数: 3

Almost Tight Error Bounds on Differentially Private Continual Counting 微分私有连续计数的几乎紧误差界

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-09 DOI: 10.48550/arXiv.2211.05006

M. Henzinger, Jalaj Upadhyay, Sarvagya Upadhyay

{"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}

引用次数: 11

On Minimizing Tardy Processing Time, Max-Min Skewed Convolution, and Triangular Structured ILPs 最小化延迟处理时间、最大-最小偏斜卷积和三角结构ilp

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-09 DOI: 10.48550/arXiv.2211.05053

Kim-Manuel Klein, Adam Polak, Lars Rohwedder

{"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}

引用次数: 3

Smaller Low-Depth Circuits for Kronecker Powers 克罗内克功率的小低深度电路

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-09 DOI: 10.48550/arXiv.2211.05217

Josh Alman, Yunfeng Guan, Ashwin Padaki

{"title":"Smaller Low-Depth Circuits for Kronecker Powers","authors":"Josh Alman, Yunfeng Guan, Ashwin Padaki","doi":"10.48550/arXiv.2211.05217","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05217","url":null,"abstract":"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.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"1 1","pages":"4159-4187"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86458723","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}

引用次数: 1

Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition 基于矩阵分解的贪心bst改进模式回避边界

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-08 DOI: 10.48550/arXiv.2211.04112

Parinya Chalermsook, Manoj Gupta, W. Jiamjitrak, Nidia Obscura Acosta, Akash Pareek, Sorrachai Yingchareonthawornchai

{"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}

引用次数: 4

Approximate Trace Reconstruction from a Single Trace 从单个轨迹近似轨迹重建

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-07 DOI: 10.48550/arXiv.2211.03292

Xi Chen, Anindya De, Chin Ho Lee, R. Servedio, S. Sinha

{"title":"Approximate Trace Reconstruction from a Single Trace","authors":"Xi Chen, Anindya De, Chin Ho Lee, R. Servedio, S. Sinha","doi":"10.48550/arXiv.2211.03292","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03292","url":null,"abstract":"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.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"84 1","pages":"605-637"},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85901096","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}

引用次数: 0

The Need for Seed (in the abstract Tile Assembly Model) 对种子的需求(在抽象的瓷砖装配模型中)

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-07 DOI: 10.48550/arXiv.2211.03877

Andrew Alseth, Matthew J. Patitz

{"title":"The Need for Seed (in the abstract Tile Assembly Model)","authors":"Andrew Alseth, Matthew J. Patitz","doi":"10.48550/arXiv.2211.03877","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03877","url":null,"abstract":"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.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"72 1","pages":"4540-4589"},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78648819","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}

引用次数: 1

Optimal Deterministic Massively Parallel Connectivity on Forests 森林的最优确定性大规模并行连通性

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-07 DOI: 10.48550/arXiv.2211.03530

A. Balliu, R. Latypov, Yannic Maus, D. Olivetti, Jara Uitto

{"title":"Optimal Deterministic Massively Parallel Connectivity on Forests","authors":"A. Balliu, R. Latypov, Yannic Maus, D. Olivetti, Jara Uitto","doi":"10.48550/arXiv.2211.03530","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03530","url":null,"abstract":"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.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"19 1","pages":"2589-2631"},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74616216","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}

引用次数: 4

Parameterized Algorithm for the Disjoint Path Problem on Planar Graphs: Exponential in k2 and Linear in n 平面图上不相交路径问题的参数化算法:k2是指数的，n是线性的

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-11-07 DOI: 10.48550/arXiv.2211.03341

Kyungjin Cho, Eunjin Oh, Seunghyuk Oh

引用次数: 0