We study algorithmic questions wrt algebraic path properties in concurrent systems, where the transitions of the system are labeled from a complete, closed semiring. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural problems that arise in program analysis. We consider that each component of the concurrent system is a graph with constant treewidth, a property satisfied by the controlflow graphs of most programs. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis. The study of multiple queries allows us to consider the tradeoff between the resource usage of the one-time preprocessing and for each individual query. The traditional approach constructs the product graph of all components and applies the best-known graph algorithm on the product. In this approach, even the answer to a single query requires the transitive closure (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time. Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time, each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results showing that the worst-case running time of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (i.e., improving the worst-case bound for the shortest path problem in general graphs). Preliminary experimental results show that our algorithms perform favorably on several benchmarks.

中文翻译：

---

等树宽分量并发系统中代数路径性质的算法

我们研究并发系统中的代数路径属性的算法问题，其中系统的转换是从一个完整的封闭半环标记的。代数路径属性可以对数据流分析问题、最短路径问题以及程序分析中出现的许多其他自然问题进行建模。我们认为并发系统的每个组件都是一个具有恒定树宽的图，这是大多数程序的控制流图都满足的属性。我们允许多种可能的查询，这些查询在需求驱动的数据流分析中自然出现。对多个查询的研究使我们可以考虑在资源使用情况之间的权衡一度预处理和用于每个人询问。传统方法构建所有组件的乘积图，并将最著名的图算法应用于产品。在这种方法中，即使是单个查询的答案也需要传递闭包（即所有可能查询的结果），这没有为预处理和查询时间之间的权衡提供空间。我们的主要贡献是显着改善传统方法的最坏情况运行时间的算法，并根据查询数量提供各种权衡。例如，在两个组件的并发系统中，传统方法在最坏的情况下需要十六进制时间来回答一个查询以及计算传递闭包，而我们表明，通过几乎立方时间的一次性预处理，每个后续查询最多可以在线性时间内回答，甚至传递闭包也可以在几乎四次的时间内计算出来。此外，我们建立了条件最优结果，表明如果不在图算法上取得重大突破（即改善一般图中最短路径问题的最坏情况界限），我们的算法的最坏情况运行时间就无法得到改善。初步实验结果表明，我们的算法在多个基准测试中表现良好。