Pub Date : 2022-10-10DOI: https://dl.acm.org/doi/10.1145/3546074
Saladi Rahul, Yufei Tao
A reporting query returns the objects satisfying a predicate q from an input set. In prioritized reporting, each object carries a real-valued weight (which can be query dependent), and a query returns the objects that satisfy q and have weights at least a threshold τ. A top-k query finds, among all the objects satisfying q, the k ones of the largest weights; a max query is a special instance with k = 1. We want to design data structures of small space to support queries (and possibly updates) efficiently.
Previous work has shown that a top-k structure can also support max and prioritized queries with no performance deterioration. This article explores the opposite direction: do prioritized queries, possibly combined with max queries, imply top-k search? Subject to mild conditions, we provide affirmative answers with two reduction techniques. The first converts a prioritized structure into a static top-k structure with the same space complexity and only a logarithmic blowup in query time. If a max structure is available in addition, our second reduction yields a top-k structure with no degradation in expected performance (this holds for the space, query, and update complexities). Our techniques significantly simplify the design of top-k structures because structures for max and prioritized queries are often easier to obtain. We demonstrate this by developing top-k structures for interval stabbing, 3D dominance, halfspace reporting, linear ranking, and L∞ nearest neighbor search in the RAM and the external memory computation models.
{"title":"Generic Techniques for Building Top-k Structures","authors":"Saladi Rahul, Yufei Tao","doi":"https://dl.acm.org/doi/10.1145/3546074","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3546074","url":null,"abstract":"<p>A <i>reporting query</i> returns the objects satisfying a predicate <i>q</i> from an input set. In <i>prioritized reporting</i>, each object carries a real-valued weight (which can be query dependent), and a query returns the objects that satisfy <i>q</i> and have weights at least a threshold τ. A <i>top-<i>k</i> query</i> finds, among all the objects satisfying <i>q</i>, the <i>k</i> ones of the largest weights; a <i>max query</i> is a special instance with <i>k</i> = 1. We want to design data structures of small space to support queries (and possibly updates) efficiently.</p><p>Previous work has shown that a top-<i>k</i> structure can also support max and prioritized queries with no performance deterioration. This article explores the opposite direction: do prioritized queries, possibly combined with max queries, imply top-<i>k</i> search? Subject to mild conditions, we provide affirmative answers with two reduction techniques. The first converts a prioritized structure into a static top-<i>k</i> structure with the same space complexity and only a logarithmic blowup in query time. If a max structure is available in addition, our second reduction yields a top-<i>k</i> structure with no degradation in expected performance (this holds for the space, query, and update complexities). Our techniques significantly simplify the design of top-<i>k</i> structures because structures for max and prioritized queries are often easier to obtain. We demonstrate this by developing top-<i>k</i> structures for interval stabbing, 3D dominance, halfspace reporting, linear ranking, and <i>L</i><sub>∞</sub> nearest neighbor search in the RAM and the external memory computation models.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"27 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-10DOI: https://dl.acm.org/doi/10.1145/3459096
Sepehr Abbasi-Zadeh, Nikhil Bansal, Guru Guruganesh, Aleksandar Nikolov, Roy Schwartz, Mohit Singh
Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [31] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known.
In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.
{"title":"Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems","authors":"Sepehr Abbasi-Zadeh, Nikhil Bansal, Guru Guruganesh, Aleksandar Nikolov, Roy Schwartz, Mohit Singh","doi":"https://dl.acm.org/doi/10.1145/3459096","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3459096","url":null,"abstract":"<p>Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [31] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., <span>Max-Cut</span>, for many others, e.g., <span>Max-SAT</span> and <span>Max-DiCut</span>, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. </p><p>In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including <span>Max-Cut</span>, <span>Max-2SAT</span>, and <span>Max-DiCut</span>, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the <span>Max-Cut</span> problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"2 4","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-10DOI: https://dl.acm.org/doi/10.1145/3531527
Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova
We initiate a systematic study of tolerant testers of image properties or, equivalently, algorithms that approximate the distance from a given image to the desired property. Image processing is a particularly compelling area of applications for sublinear-time algorithms and, specifically, property testing. However, for testing algorithms to reach their full potential in image processing, they have to be tolerant, which allows them to be resilient to noise.
We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so it becomes a half-plane? A representation of a convex object? A representation of a connected object? More precisely, our algorithms approximate the distance to three basic properties (being a half-plane, convexity, and connectedness) within a small additive error ε, after reading poly(1/ε) pixels, independent of the image size. We also design an efficient agnostic proper PAC learner of convex sets (continuous and discrete) in two dimensions under the uniform distribution.
Our algorithms require very simple access to the input: uniform random samples for the half-plane property and convexity, and samples from uniformly random blocks for connectedness. However, the analysis of the algorithms, especially for convexity, requires many geometric and combinatorial insights. For example, in the analysis of the algorithm for convexity, we define a set of reference polygons Pε such that (1) every convex image has a nearby polygon in Pε and (2) one can use dynamic programming to quickly compute the smallest empirical distance to a polygon in Pε.
{"title":"Tolerant Testers of Image Properties","authors":"Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova","doi":"https://dl.acm.org/doi/10.1145/3531527","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3531527","url":null,"abstract":"<p>We initiate a systematic study of tolerant testers of image properties or, equivalently, algorithms that approximate the distance from a given image to the desired property. Image processing is a particularly compelling area of applications for sublinear-time algorithms and, specifically, property testing. However, for testing algorithms to reach their full potential in image processing, they have to be tolerant, which allows them to be resilient to noise.</p><p>We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so it becomes a half-plane? A representation of a convex object? A representation of a connected object? More precisely, our algorithms approximate the distance to three basic properties (being a half-plane, convexity, and connectedness) within a small additive error ε, after reading <i>poly</i>(1/ε) pixels, independent of the image size. We also design an efficient agnostic proper PAC learner of convex sets (continuous and discrete) in two dimensions under the uniform distribution.</p><p>Our algorithms require very simple access to the input: uniform random samples for the half-plane property and convexity, and samples from uniformly random blocks for connectedness. However, the analysis of the algorithms, especially for convexity, requires many geometric and combinatorial insights. For example, in the analysis of the algorithm for convexity, we define a set of reference polygons <i>P</i><sub>ε</sub> such that (1) every convex image has a nearby polygon in <i>P</i><sub>ε</sub> and (2) one can use dynamic programming to quickly compute the smallest empirical distance to a polygon in <i>P</i><sub>ε</sub>.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"2 6‐7","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-10DOI: https://dl.acm.org/doi/10.1145/3459095
Matthew Joseph, Jieming Mao, Aaron Roth
We prove a general connection between the communication complexity of two-player games and the sample complexity of their multi-player locally private analogues. We use this connection to prove sample complexity lower bounds for locally differentially private protocols as straightforward corollaries of results from communication complexity. In particular, we (1) use a communication lower bound for the hidden layers problem to prove an exponential sample complexity separation between sequentially and fully interactive locally private protocols, and (2) use a communication lower bound for the pointer chasing problem to prove an exponential sample complexity separation between k-round and (k+1)-round sequentially interactive locally private protocols, for every k.
{"title":"Exponential Separations in Local Privacy","authors":"Matthew Joseph, Jieming Mao, Aaron Roth","doi":"https://dl.acm.org/doi/10.1145/3459095","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3459095","url":null,"abstract":"<p>We prove a general connection between the <i>communication</i> complexity of two-player games and the <i>sample</i> complexity of their multi-player locally private analogues. We use this connection to prove sample complexity lower bounds for locally differentially private protocols as straightforward corollaries of results from communication complexity. In particular, we (1) use a communication lower bound for the hidden layers problem to prove an exponential sample complexity separation between sequentially and fully interactive locally private protocols, and (2) use a communication lower bound for the pointer chasing problem to prove an exponential sample complexity separation between <i>k</i>-round and (<i>k+1</i>)-round sequentially interactive locally private protocols, for every <i>k</i>.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"2 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Shah, Cheng Sheng, Sharma V. Thankachan, J. Vitter
The ranked (or top-k) document retrieval problem is defined as follows: preprocess a collection {T1,T2,… ,Td} of d strings (called documents) of total length n into a data structure, such that for any given query (P,k), where P is a string (called pattern) of length p ≥ 1 and k ∈ [1,d] is an integer, the identifiers of those k documents that are most relevant to P can be reported, ideally in the sorted order of their relevance. The seminal work by Hon et al. [FOCS 2009 and Journal of the ACM 2014] presented an O(n)-space (in words) data structure with O(p+k log k) query time. The query time was later improved to O(p+k) [SODA 2012] and further to O(p/ log σn+k) [SIAM Journal on Computing 2017] by Navarro and Nekrich, where σ is the alphabet size. We revisit this problem in the external memory model and present three data structures. The first one takes O(n)-space and answer queries in O(p/B + log B n + k/B+ log * (n/B)) I/Os, where B is the block size. The second one takes O(n log * (n/B)) space and answer queries in optimal O(p/B + log B n + k/B) I/Os. In both cases, the answers are reported in the unsorted order of relevance. To handle sorted top-k document retrieval, we present an O(n log (d/B)) space data structure with optimal query cost.
{"title":"Ranked Document Retrieval in External Memory","authors":"R. Shah, Cheng Sheng, Sharma V. Thankachan, J. Vitter","doi":"10.1145/3559763","DOIUrl":"https://doi.org/10.1145/3559763","url":null,"abstract":"The ranked (or top-k) document retrieval problem is defined as follows: preprocess a collection {T1,T2,… ,Td} of d strings (called documents) of total length n into a data structure, such that for any given query (P,k), where P is a string (called pattern) of length p ≥ 1 and k ∈ [1,d] is an integer, the identifiers of those k documents that are most relevant to P can be reported, ideally in the sorted order of their relevance. The seminal work by Hon et al. [FOCS 2009 and Journal of the ACM 2014] presented an O(n)-space (in words) data structure with O(p+k log k) query time. The query time was later improved to O(p+k) [SODA 2012] and further to O(p/ log σn+k) [SIAM Journal on Computing 2017] by Navarro and Nekrich, where σ is the alphabet size. We revisit this problem in the external memory model and present three data structures. The first one takes O(n)-space and answer queries in O(p/B + log B n + k/B+ log * (n/B)) I/Os, where B is the block size. The second one takes O(n log * (n/B)) space and answer queries in optimal O(p/B + log B n + k/B) I/Os. In both cases, the answers are reported in the unsorted order of relevance. To handle sorted top-k document retrieval, we present an O(n log (d/B)) space data structure with optimal query cost.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 12"},"PeriodicalIF":1.3,"publicationDate":"2022-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43217564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix A of dimension n. The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by O(n3(log n + log ||A||)2(log n)2) bit operations, where ||A|| = max ij|Aij| denotes the largest entry of A in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time (n3log ||A||)1 + o(1) bit operations, where the exponent `` + o(1)′′ captures additional factors (c_1 (log n)^{c_2} (rm {loglog} ||A||)^{c_3} ) for positive real constants c1, c2, c3.
{"title":"A Cubic Algorithm for Computing the Hermite Normal Form of a Nonsingular Integer Matrix","authors":"Stavros Birmpilis, G. Labahn, A. Storjohann","doi":"10.1145/3617996","DOIUrl":"https://doi.org/10.1145/3617996","url":null,"abstract":"A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix A of dimension n. The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by O(n3(log n + log ||A||)2(log n)2) bit operations, where ||A|| = max ij|Aij| denotes the largest entry of A in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time (n3log ||A||)1 + o(1) bit operations, where the exponent `` + o(1)′′ captures additional factors (c_1 (log n)^{c_2} (rm {loglog} ||A||)^{c_3} ) for positive real constants c1, c2, c3.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47916917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [38], with an approximation guarantee of (0.3393 + δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a ((frac{1}{3}+delta) ) -Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [38], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.
{"title":"A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games","authors":"Argyrios Deligkas, M. Fasoulakis, E. Markakis","doi":"10.1145/3606697","DOIUrl":"https://doi.org/10.1145/3606697","url":null,"abstract":"Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [38], with an approximation guarantee of (0.3393 + δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a ((frac{1}{3}+delta) ) -Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [38], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49225586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity. The result extends to the splittable CVRP.
{"title":"A PTAS for Capacitated Vehicle Routing on Trees","authors":"Claire Mathieu, Hang Zhou","doi":"10.1145/3575799","DOIUrl":"https://doi.org/10.1145/3575799","url":null,"abstract":"We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity. The result extends to the splittable CVRP.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 28"},"PeriodicalIF":1.3,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46329433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-05DOI: 10.1137/1.9781611977073.10
Timothy M. Chan, D. Zheng
We revisit Hopcroft’s problem and related fundamental problems about geometric range searching. Given n points and n lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs in O(n4/3) time, which matches the conjectured lower bound and improves the best previous time bound of (n^{4/3}2^{O(log ^*n)} ) obtained almost 30 years ago by Matoušek. We describe two interesting and different ways to achieve the result: the first is randomized and uses a new 2D version of fractional cascading for arrangements of lines; the second is deterministic and uses decision trees in a manner inspired by the sorting technique of Fredman (1976). The second approach extends to any constant dimension. Many consequences follow from these new ideas: for example, we obtain an O(n4/3)-time algorithm for line segment intersection counting in the plane, O(n4/3)-time randomized algorithms for distance selection in the plane and bichromatic closest pair and Euclidean minimum spanning tree in three or four dimensions, and a randomized data structure for halfplane range counting in the plane with O(n4/3) preprocessing time and space and O(n1/3) query time.
{"title":"Hopcroft’s Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees","authors":"Timothy M. Chan, D. Zheng","doi":"10.1137/1.9781611977073.10","DOIUrl":"https://doi.org/10.1137/1.9781611977073.10","url":null,"abstract":"We revisit Hopcroft’s problem and related fundamental problems about geometric range searching. Given n points and n lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs in O(n4/3) time, which matches the conjectured lower bound and improves the best previous time bound of (n^{4/3}2^{O(log ^*n)} ) obtained almost 30 years ago by Matoušek. We describe two interesting and different ways to achieve the result: the first is randomized and uses a new 2D version of fractional cascading for arrangements of lines; the second is deterministic and uses decision trees in a manner inspired by the sorting technique of Fredman (1976). The second approach extends to any constant dimension. Many consequences follow from these new ideas: for example, we obtain an O(n4/3)-time algorithm for line segment intersection counting in the plane, O(n4/3)-time randomized algorithms for distance selection in the plane and bichromatic closest pair and Euclidean minimum spanning tree in three or four dimensions, and a randomized data structure for halfplane range counting in the plane with O(n4/3) preprocessing time and space and O(n1/3) query time.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49116051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-03DOI: 10.1137/1.9781611977073.144
Dvir Fried, Shay Golan, T. Kociumaka, T. Kopelowitz, E. Porat, Tatiana Starikovskaya
A Dyck sequence is a sequence of opening and closing parentheses (of various types) that is balanced. The Dyck edit distance of a given sequence of parentheses S is the smallest number of edit operations (insertions, deletions, and substitutions) needed to transform S into a Dyck sequence. We consider the threshold Dyck edit distance problem, where the input is a sequence of parentheses S and a positive integer k, and the goal is to compute the Dyck edit distance of S only if the distance is at most k, and otherwise report that the distance is larger than k. Backurs and Onak [PODS’16] showed that the threshold Dyck edit distance problem can be solved in O(n + k16) time. In this work, we design new algorithms for the threshold Dyck edit distance problem which costs O(n + k4.544184) time with high probability or O(n + k4.853059) deterministically. Our algorithms combine several new structural properties of the Dyck edit distance problem, a refined algorithm for fast (min , +) matrix product, and a careful modification of ideas used in Valiant’s parsing algorithm.
{"title":"An Improved Algorithm for The k-Dyck Edit Distance Problem","authors":"Dvir Fried, Shay Golan, T. Kociumaka, T. Kopelowitz, E. Porat, Tatiana Starikovskaya","doi":"10.1137/1.9781611977073.144","DOIUrl":"https://doi.org/10.1137/1.9781611977073.144","url":null,"abstract":"A Dyck sequence is a sequence of opening and closing parentheses (of various types) that is balanced. The Dyck edit distance of a given sequence of parentheses S is the smallest number of edit operations (insertions, deletions, and substitutions) needed to transform S into a Dyck sequence. We consider the threshold Dyck edit distance problem, where the input is a sequence of parentheses S and a positive integer k, and the goal is to compute the Dyck edit distance of S only if the distance is at most k, and otherwise report that the distance is larger than k. Backurs and Onak [PODS’16] showed that the threshold Dyck edit distance problem can be solved in O(n + k16) time. In this work, we design new algorithms for the threshold Dyck edit distance problem which costs O(n + k4.544184) time with high probability or O(n + k4.853059) deterministically. Our algorithms combine several new structural properties of the Dyck edit distance problem, a refined algorithm for fast (min , +) matrix product, and a careful modification of ideas used in Valiant’s parsing algorithm.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"230 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75908560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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