参数化线性时间传递闭包
Parameterized Linear Time Transitive Closure
AI总结：
本文提出基于链分解的可达性索引方案，实现常数时间查询与参数化线性时间构建，并通过加速Fulkerson求DAG宽度验证其有效性。
摘要：
诸如任务A是否依赖于任务B、作者A是否受到论文B的影响、某种蛋白质是否与特定生物过程或分子功能相关、或者类A是否继承自类B等查询，只是可以建模为网络（有向图）上可达性查询的几个例子。数字系统每天都要回答无数此类查询。本文讨论传递闭包问题。我们专注于可应用的解决方案，使我们能够快速、在常数时间内回答查询，并可服务于实际应用。与围绕二维邻接矩阵构建的大多数研究不同，我们提出了一种构建可达性索引方案的方法。该方案使我们能够在常数时间内回答查询，并且可以在参数化线性时间内构建。此外，它包含一种压缩数据结构。我们的方法和算法通过大量实验得到验证，这些实验揭示了在该问题中起关键作用的因素。为了强调该解决方案的效率并展示将我们的方法应用于重要问题的潜力，我们使用它来加速Fulkerson求解DAG宽度的方法。我们的结果挑战了过去三十年来反复重申的关于该方法效率的普遍信念。我们的方法基于链分解的概念。在深入描述之前，我们引入、分析并利用了一种链分解算法。此外，我们探索了链分解如何促进传递闭包解决方案，引入了一种通用的线性时间归约技术，该技术在给定任何链分解的情况下移除大量传递边。
英文摘要：
Inquiries such as whether a task A depends on a task B, whether an author A has been influenced by a paper B, whether a certain protein is associated with a specific biological process or molecular function, or whether class A inherits from class B, are just a few examples of inquiries that can be modeled as reachability queries on a network (Directed Graph). Digital systems answer myriad such inquiries every day. In this paper, we discuss the transitive closure problem. We focus on applicable solutions that enable us to answer queries fast, in constant time, and can serve in real-world applications. In contrast to the majority of research on this topic that revolves around the construction of a two-dimensional adjacency matrix, we present an approach that builds a reachability indexing scheme. This scheme enables us to answer queries in constant time and can be built in parameterized linear time. In addition, it captures a compressed data structure. Our approach and algorithms are validated by extensive experiments that shed light on the factors that play a key role in this problem. To stress the efficiency of this solution and demonstrate the potential to apply our approach to important problems, we use it to speed up Fulkerson's method for finding the width of a DAG. Our results challenge the prevailing belief, reiterated over the last thirty years, regarding the efficiency of this method. Our approach is based on the concept of chain decomposition. Before we delve into its description, we introduce, analyze, and utilize a chain decomposition algorithm. Furthermore, we explore how chain decomposition can facilitate transitive closure solutions introducing a general purpose linear time reduction technique that removes a large subset of transitive edges given any chain decomposition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
