AF: Small: Collaborative Research: Dynamic Data Structures for Vectors and Graphs in Sublinear Memory

子线性空间数据结构最近引起了人们的极大兴趣，其应用例如在流算法、分布式计算、随机线性代数和压缩传感中。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。该提案旨在为两个看似不同但密切相关的对象（向量和图）开发设计和分析子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，其主题与本项目的重点重叠。此外，这两个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学生的外联活动，并扩大对计算的参与。研究本身也可能产生工业影响，与数据库，网络流量分析和数据挖掘有关。PI将专注于一系列可以通过旋转栅门流模型捕获的问题：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是顶点的集合），并且边插入/删除则对应于从x中的某个条目的加法/减法。本项目旨在进一步理解与向量更新的小空间动态数据结构相关的基本问题，特别是与图形问题相关的问题。例如：* 小空间向量更新数据结构：仅插入情况：在仅插入情况下，向量更新x的增量坐标，因此x是数据流中各种项目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，例如范数估计、重打击者和统计数据的连续监控。完全动态流和图的应用：许多最著名的用于图问题的小空间动态数据结构通过减少到向量更新问题来操作。例如，唯一已知的近似线性空间算法的频谱稀疏化操作减少到l_2重打击，和算法的连接，k边连接，最小生成树，和其他几个减少到矢量坐标采样问题。然而，仍然存在许多开放的问题，例如，什么是最佳的空间复杂度的连接在动态流？该奖项反映了NSF的法定使命，并被认为值得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的支持。

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.