AF: Small: Collaborative Research: Dynamic data structures for vectors and graphs in sublinear memory

亚线性空间数据结构最近引起了人们的极大兴趣，应用于流算法、分布式计算、随机线性代数和压缩感知等领域。小空间解决方案适合快速缓存，因此也提供了时间加速，并且在分布式设置中需要更少的存储和更少的带宽。本提案旨在为两个看似不同但密切相关的对象：向量和图，开发设计和分析亚线性空间数据结构的新方法。pi还计划教授高级研究生课程，其主题与该项目的重点重叠。此外，两个pi计划培训和指导研究生和本科生的研究，组织研讨会，并撰写调查文章。pi还计划参加面向非计算机科学家和K-12学生的外展活动，并扩大对计算机的参与。研究本身也可能有工业影响，涉及到数据库、网络流量分析和数据挖掘。pi将专注于一系列问题，这些问题可以由旋转门流模型捕获：R^n中的一些高维向量x接收坐标更新，在图的情况下可能有n = (|V|选择2)（其中V是顶点集），然后边缘插入/删除对应于x中的某些条目的加法/减法。该项目旨在进一步理解与矢量更新的小空间动态数据结构相关的基本问题，特别是与图问题相关的问题。*小空间向量更新数据结构：只插入的情况：在只插入的情况下，向量更新x的增量坐标，因此x是数据流中各种项出现次数的频率计数向量。pi计划解决该模型中的一些最基本的问题，例如规范估计、重磅打击和统计数据的持续监控。*全动态流和图的应用：许多著名的图问题的小空间动态数据结构通过简化为矢量更新问题来操作。例如，唯一已知的谱稀疏器的近线性空间算法是通过约简到l_2个重打击来操作的，而连通性、k边连通性、最小生成树和其他一些算法则归结为向量坐标采样问题。许多悬而未决的问题仍然存在，例如，动态流中连接的最佳空间复杂度是多少？pi还计划研究其他几个动态图和超图问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Sublinear-space data structures have been of major recent interest, with applications for example in streaming algorithms, distributed computation, randomized linear algebra, and compressed sensing. Small-space solutions fit in fast cache, thus providing time speedups as well, and also require less storage and less bandwidth in distributed settings. This proposal aims to develop novel methods for designing and analyzing sublinear-space data structures for two seemingly different but closely related objects: vectors and graphs. The PIs also plan to teach advanced graduate courses whose topics overlap with the focus of this project. Furthermore, both PIs plan to train and mentor graduate and undergraduate students students in research, organize workshops, and write survey articles. The PIs plan to also participate in outreach activities to non-computer scientists and to K-12 students, and to broaden participation in computing. The research itself may also have industrial impact, being related to databases, network traffic analysis, and data mining.The PIs will focus on a set of problems that can be captured by the turnstile streaming model: some high-dimensional vector x in R^n receives coordinate-wise updates, which in the case of graphs could have n = (|V| choose 2) (where V is the set of vertices), and edge insert/deletion then corresponds to addition/subtraction from some entry in x. This project aims to further the understanding of fundamental questions related to small-space dynamic data structures for vector updates, and especially as they relate to graph problems.For example:* Small-space vector update data structures: the insertion-only case: In the insertion-only case, vector updates increment coordinates of x, so that x is a frequency-count vector of the number of occurrences of various items in a data stream. The PIs plan to attack some of the most fundamental problems in this model, such as norm estimation, heavy hitters, and continuous monitoring of statistics.* Fully dynamic streams and applications to graphs: Many of the best-known small-space dynamic data structures for graph problems operate by reducing to vector-update problems. For example, the only known nearly linear-space algorithm for spectral sparsifiers operates by reduction to l_2 heavy hitters, and algorithms for connectivity, k-edge connectivity, minimum spanning trees, and several others reduce to vector coordinate-sampling problems. Many open problems though still remain, e.g. what is the optimal space complexity for connectivity in dynamic streams? The PIs also plan to investigate several other dynamic graph and hypergraph problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.