Pub Date : 2024-12-10DOI: 10.1016/j.ipl.2024.106551
David Flores-Peñaloza , Mario A. Lopez , Nestaly Marín , David Orden
Let P be a k-colored set of n points in the plane, . We study the problem of deciding if P contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point c in the plane such that each of the open quadrants defined by c contains a point of P, each of them having a different color. We provide an -time algorithm for this problem, where the hidden constant does not depend on k; then, we prove that this problem has time complexity in the algebraic computation tree model. No general position assumptions for P are required.
{"title":"An efficient algorithm for identifying rainbow ortho-convex 4-sets in k-colored point sets","authors":"David Flores-Peñaloza , Mario A. Lopez , Nestaly Marín , David Orden","doi":"10.1016/j.ipl.2024.106551","DOIUrl":"10.1016/j.ipl.2024.106551","url":null,"abstract":"<div><div>Let <em>P</em> be a <em>k</em>-colored set of <em>n</em> points in the plane, <span><math><mn>4</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>. We study the problem of deciding if <em>P</em> contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point <em>c</em> in the plane such that each of the open quadrants defined by <em>c</em> contains a point of <em>P</em>, each of them having a different color. We provide an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>-time algorithm for this problem, where the hidden constant does not depend on <em>k</em>; then, we prove that this problem has time complexity <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> in the algebraic computation tree model. No general position assumptions for <em>P</em> are required.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106551"},"PeriodicalIF":0.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 : 2024-12-09DOI: 10.1016/j.ipl.2024.106550
Adam J. Przeździecki
We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.
Its complexity, for a matrix, is , where is the cost of matrix multiplication. In case of a sparse matrix, is the cost of the dense-sparse matrix multiplication.
The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.
{"title":"A simple division-free algorithm for computing Pfaffians","authors":"Adam J. Przeździecki","doi":"10.1016/j.ipl.2024.106550","DOIUrl":"10.1016/j.ipl.2024.106550","url":null,"abstract":"<div><div>We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.</div><div>Its complexity, for a <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> matrix, is <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of matrix multiplication. In case of a sparse matrix, <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of the dense-sparse matrix multiplication.</div><div>The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106550"},"PeriodicalIF":0.7,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 : 2024-11-27DOI: 10.1016/j.ipl.2024.106549
Naoyuki Kamiyama
The topic of this paper is the stable matching problem in a bipartite graph. Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the stable matching problem with ties by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we consider the setting where we are allowed to delete agents on only one side. We prove that, in this setting, our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. Next, we consider the setting where we are given an upper bound on the number of deleted agents for each side, and we are allowed to delete agents on both sides. We prove that, in this setting, our problem is NP-complete.
{"title":"Modifying an instance of the super-stable matching problem","authors":"Naoyuki Kamiyama","doi":"10.1016/j.ipl.2024.106549","DOIUrl":"10.1016/j.ipl.2024.106549","url":null,"abstract":"<div><div>The topic of this paper is the stable matching problem in a bipartite graph. Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the stable matching problem with ties by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we consider the setting where we are allowed to delete agents on only one side. We prove that, in this setting, our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. Next, we consider the setting where we are given an upper bound on the number of deleted agents for each side, and we are allowed to delete agents on both sides. We prove that, in this setting, our problem is NP-complete.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106549"},"PeriodicalIF":0.7,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 : 2024-11-22DOI: 10.1016/j.ipl.2024.106542
Chris R. Giannella
Because many dissimilarity functions behave differently in low versus high-dimensional spaces, the behavior of high-dimensional nearest neighbor search has been studied extensively. One line of research involves the characterization of nearest neighbor queries as unstable if their query points have nearly identical dissimilarity with most points in the dataset. This research has shown that, for various data distributions and dissimilarity functions, the probability of query instability approaches one. Previous work in Information Processing Letters by C. Giannella in 2021 explicated this phenomenon for centered Gaussian data and Euclidean distance. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data and a fundamentally different dissimilarity function, cosine dissimilarity. Conditions are provided on the covariance matrices and dataset size function guaranteeing that the probability of query instability goes to one. Furthermore, conditions are provided under which the instability probability is bounded away from one.
{"title":"Instability results for cosine-dissimilarity-based nearest neighbor search on high dimensional Gaussian data","authors":"Chris R. Giannella","doi":"10.1016/j.ipl.2024.106542","DOIUrl":"10.1016/j.ipl.2024.106542","url":null,"abstract":"<div><div>Because many dissimilarity functions behave differently in low versus high-dimensional spaces, the behavior of high-dimensional nearest neighbor search has been studied extensively. One line of research involves the characterization of nearest neighbor queries as unstable if their query points have nearly identical dissimilarity with most points in the dataset. This research has shown that, for various data distributions and dissimilarity functions, the probability of query instability approaches one. Previous work in <em>Information Processing Letters</em> by C. Giannella in 2021 explicated this phenomenon for centered Gaussian data and Euclidean distance. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data and a fundamentally different dissimilarity function, cosine dissimilarity. Conditions are provided on the covariance matrices and dataset size function guaranteeing that the probability of query instability goes to one. Furthermore, conditions are provided under which the instability probability is bounded away from one.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106542"},"PeriodicalIF":0.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 : 2024-11-12DOI: 10.1016/j.ipl.2024.106541
Peter Kostolányi
The class of all finitely generated semigroups with a deterministic context-free word problem is shown to be closed under free products, answering a question of T. Brough, A. J. Cain, and M. Pfeiffer. On the other hand, it is proved that the class of all finitely generated monoids with a deterministic context-free word problem is not closed under monoid free products.
证明了具有确定性无上下文词问题的所有有限生成半群的类在自由积下是封闭的，回答了 T. Brough、A. J. Cain 和 M. Pfeiffer 的一个问题。另一方面，证明了具有确定性无上下文词问题的所有有限生成单元的类在单元自由积下不是封闭的。
{"title":"Free products of semigroups and monoids with a deterministic context-free word problem","authors":"Peter Kostolányi","doi":"10.1016/j.ipl.2024.106541","DOIUrl":"10.1016/j.ipl.2024.106541","url":null,"abstract":"<div><div>The class of all finitely generated semigroups with a deterministic context-free word problem is shown to be closed under free products, answering a question of T. Brough, A. J. Cain, and M. Pfeiffer. On the other hand, it is proved that the class of all finitely generated monoids with a deterministic context-free word problem is not closed under monoid free products.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106541"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Covering a set of segments in a plane with vehicles of limited autonomy is a problem of practical interest. The limited battery endurance imposes periodical visits to a static base station. Typically, two optimization problems are considered: minimize the number of tours, and minimize the total traveled distance. In a general setting, the problems are NP-hard and in this letter, we study the one-dimensional version. For covering segments on a line, we design efficient solutions for both optimization problems. First, we design a greedy algorithm that is optimal for the first task, and for both tasks when only one segment is considered. Being n and m the number of segments and tours of an optimal solution, respectively, our algorithm runs in time. For the second criterion, our solution is based on Dynamic Programming and runs in time.
用自主能力有限的车辆在平面上覆盖一组区段是一个具有实际意义的问题。由于电池续航时间有限，因此需要定期访问静态基站。通常，需要考虑两个优化问题：最小化巡视次数和最小化总行程。在一般情况下，这两个问题都很难解决，在这封信中，我们研究的是一维问题。对于线路上的覆盖线段，我们为这两个优化问题设计了高效的解决方案。首先，我们设计了一种贪婪算法，该算法对第一项任务和只考虑一个线段时的两项任务都是最优的。由于 n 和 m 分别为最优解的线段数和游程数，我们的算法运行时间为 O(m+n)。对于第二个标准，我们的解决方案基于动态编程，运行时间为 O(n2m)。
{"title":"Covering segments on a line with drones","authors":"Sergey Bereg , José-Miguel Díaz-Báñez , Alina Kasiuk , Miguel-Angel Pérez-Cutiño , Fabio Rodríguez","doi":"10.1016/j.ipl.2024.106540","DOIUrl":"10.1016/j.ipl.2024.106540","url":null,"abstract":"<div><div>Covering a set of segments in a plane with vehicles of limited autonomy is a problem of practical interest. The limited battery endurance imposes periodical visits to a static base station. Typically, two optimization problems are considered: minimize the number of tours, and minimize the total traveled distance. In a general setting, the problems are NP-hard and in this letter, we study the one-dimensional version. For covering segments on a line, we design efficient solutions for both optimization problems. First, we design a greedy algorithm that is optimal for the first task, and for both tasks when only one segment is considered. Being <em>n</em> and <em>m</em> the number of segments and tours of an optimal solution, respectively, our algorithm runs in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span> time. For the second criterion, our solution is based on Dynamic Programming and runs in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>m</mi><mo>)</mo></math></span> time.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106540"},"PeriodicalIF":0.7,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 : 2024-10-31DOI: 10.1016/j.ipl.2024.106539
Roberto Colomboni , Emmanuel Esposito , Andrea Paudice
The fat-shattering dimension characterizes the uniform convergence property of real-valued function classes. The state-of-the-art upper bounds in [6] feature a multiplicative squared logarithmic factor on the sample complexity, leaving an open gap with the existing lower bound. By relying on a refined packing number bound given in [20], we provide an improved uniform convergence bound that closes this gap.
{"title":"An improved uniform convergence bound with fat-shattering dimension","authors":"Roberto Colomboni , Emmanuel Esposito , Andrea Paudice","doi":"10.1016/j.ipl.2024.106539","DOIUrl":"10.1016/j.ipl.2024.106539","url":null,"abstract":"<div><div>The fat-shattering dimension characterizes the uniform convergence property of real-valued function classes. The state-of-the-art upper bounds in <span><span>[6]</span></span> feature a multiplicative squared logarithmic factor on the sample complexity, leaving an open gap with the existing lower bound. By relying on a refined packing number bound given in <span><span>[20]</span></span>, we provide an improved uniform convergence bound that closes this gap.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106539"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 : 2024-10-28DOI: 10.1016/j.ipl.2024.106538
Hossein Jowhari , Shamisa Nematollahi
In the problem of airports and railways with unsplittable demand (ARUD), we are given a complete graph with weights on the vertices , and the length of the edges . Additionally, a positive integer k serves as the capacity parameter. We are also provided with a function that defines a non-zero demand for each city. The goal is to compute a spanning forest R of G and a subset of minimum cost such that each component in R has one open facility and the total demand in each component is at most k (the capacity constraint). The cost of the solution is defined as . This problem is a generalization of the Airport and Railways (AR) problem introduced by Adamaszek et al. (STACS 2016). In Adamaszek et al. version, each vertex has a unit demand.
This paper presents a bi-criteria approximation algorithm for the metric ARUD problem in the sense that the algorithm is allowed to exceed the capacity constraints by while the cost of the solution is compared with the cost of an optimal solution that does not violate the capacity constraint. Our approach builds upon an existing approximation algorithm for the metric AR problem, developed by Adamaszek et al. (STACS 2018), and further leverages the well-known rounding algorithm of Shmoys and Tardos for the Generalized Assignment Problem (GAP). Assuming the total demand is polynomially bounded in the number of vertices, our algorithm runs in polynomial time. We also show that it is NP-hard to find an approximate solution for ARUD within any factor without violating the capacity constraints. This is the case even when each demand is polynomially bounded in the number of vertices. Furthermore, we determine the complexity of ARUD for some fixed values of k.
在需求不可分割的机场和铁路（ARUD）问题中，我们给定了一个完整的图 G=（V,E），其顶点的权重为 a:V→R+，边的长度为 ℓ:V×V→R+。此外，还有一个正整数 k 作为容量参数。我们还得到了一个函数 b:V→N，它定义了每个城市的非零需求。我们的目标是计算 G 的生成林 R 和成本最小的子集 A⊆V，使得 R 中的每个部分都有一个开放设施，且每个部分的总需求最多为 k（容量约束）。解 (A,R) 的成本定义为：∑v∈Aa(v)+∑e∈E(R)ℓ(e)。该问题是 Adamaszek 等人（STACS 2016）提出的机场与铁路（AR）问题的一般化。在 Adamaszek 等人的版本中，每个顶点都有一个单位需求。本文针对公制 ARUD 问题提出了一种双标准近似算法，即允许算法超出容量约束 O(k)，同时将解的成本与不违反容量约束的最优解的成本进行比较。我们的方法借鉴了 Adamaszek 等人（STACS 2018）针对度量 AR 问题开发的现有近似算法，并进一步利用了 Shmoys 和 Tardos 针对广义分配问题（GAP）的著名舍入算法。假设总需求在顶点数量上是多项式有界的，我们的算法将在多项式时间内运行。我们还证明，在不违反容量约束的情况下，在任何系数内找到 ARUD 的近似解都是 NP 难的。即使每个需求的顶点数都是多项式有界的，情况也是如此。此外，我们还确定了一些固定 k 值的 ARUD 复杂性。
{"title":"Airports and railways with unsplittable demand","authors":"Hossein Jowhari , Shamisa Nematollahi","doi":"10.1016/j.ipl.2024.106538","DOIUrl":"10.1016/j.ipl.2024.106538","url":null,"abstract":"<div><div>In the problem of airports and railways with unsplittable demand (ARUD), we are given a complete graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> with weights on the vertices <span><math><mi>a</mi><mo>:</mo><mi>V</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, and the length of the edges <span><math><mi>ℓ</mi><mo>:</mo><mi>V</mi><mo>×</mo><mi>V</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>. Additionally, a positive integer <em>k</em> serves as the capacity parameter. We are also provided with a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>N</mi></math></span> that defines a non-zero demand for each city. The goal is to compute a spanning forest <em>R</em> of <em>G</em> and a subset <span><math><mi>A</mi><mo>⊆</mo><mi>V</mi></math></span> of minimum cost such that each component in <em>R</em> has one open facility and the total demand in each component is at most <em>k</em> (the capacity constraint). The cost of the solution <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> is defined as <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>A</mi></mrow></msub><mi>a</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>+</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></msub><mi>ℓ</mi><mo>(</mo><mi>e</mi><mo>)</mo></math></span>. This problem is a generalization of the Airport and Railways (AR) problem introduced by Adamaszek et al. (STACS 2016). In Adamaszek et al. version, each vertex has a unit demand.</div><div>This paper presents a bi-criteria approximation algorithm for the metric ARUD problem in the sense that the algorithm is allowed to exceed the capacity constraints by <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> while the cost of the solution is compared with the cost of an optimal solution that does not violate the capacity constraint. Our approach builds upon an existing approximation algorithm for the metric AR problem, developed by Adamaszek et al. (STACS 2018), and further leverages the well-known rounding algorithm of Shmoys and Tardos for the Generalized Assignment Problem (GAP). Assuming the total demand is polynomially bounded in the number of vertices, our algorithm runs in polynomial time. We also show that it is NP-hard to find an approximate solution for ARUD within any factor without violating the capacity constraints. This is the case even when each demand is polynomially bounded in the number of vertices. Furthermore, we determine the complexity of ARUD for some fixed values of <em>k</em>.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106538"},"PeriodicalIF":0.7,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 : 2024-10-24DOI: 10.1016/j.ipl.2024.106537
Samuel Hartman, Jonathan P. Sorenson
We present a version of the sieve of Eratosthenes that can factor all integers ≤x in arithmetic operations using at most bits of space. Among algorithms that take the optimal time, this new space bound is an improvement of a factor proportional to over the implied previous bound of . We also show our algorithm performs well in practice.
{"title":"Reducing the space used by the sieve of Eratosthenes when factoring","authors":"Samuel Hartman, Jonathan P. Sorenson","doi":"10.1016/j.ipl.2024.106537","DOIUrl":"10.1016/j.ipl.2024.106537","url":null,"abstract":"<div><div>We present a version of the sieve of Eratosthenes that can factor all integers ≤<em>x</em> in <span><math><mi>O</mi><mo>(</mo><mi>x</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></math></span> arithmetic operations using at most <span><math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo>/</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></math></span> bits of space. Among algorithms that take the optimal <span><math><mi>O</mi><mo>(</mo><mi>x</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></math></span> time, this new space bound is an improvement of a factor proportional to <span><math><mi>log</mi><mo></mo><mi>x</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>x</mi></math></span> over the implied previous bound of <span><math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></math></span>. We also show our algorithm performs well in practice.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106537"},"PeriodicalIF":0.7,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 show that for every n-vertex graph with at least one edge, its treewidth is greater than or equal to , where Δ and are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [Inf. Process. Lett., 2003] and the subsequent one by the authors of the present paper [IEICE Trans. Inf. Syst., 2024]. The new lower bound is almost tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.
{"title":"An improved spectral lower bound of treewidth","authors":"Tatsuya Gima , Tesshu Hanaka , Kohei Noro , Hirotaka Ono , Yota Otachi","doi":"10.1016/j.ipl.2024.106536","DOIUrl":"10.1016/j.ipl.2024.106536","url":null,"abstract":"<div><div>We show that for every <em>n</em>-vertex graph with at least one edge, its treewidth is greater than or equal to <span><math><mi>n</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><mo>(</mo><mi>Δ</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where Δ and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [<em>Inf. Process. Lett.</em>, 2003] and the subsequent one by the authors of the present paper [<em>IEICE Trans. Inf. Syst.</em>, 2024]. The new lower bound is <em>almost</em> tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106536"},"PeriodicalIF":0.7,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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