AF: Small: Efficient Approximations for Dynamic Programs and Other Topics in Algorithms

该项目支持在算法和计算复杂性方面的几个主题的新的和正在进行的研究。该项目的一个主要焦点将是某些组合优化问题，例如确定两个数据序列的最长公共子序列，这些问题可以表示为特殊网络中的最短路径问题。我们的目标是开发可以证明与正确答案非常接近的算法，并且比现有算法快得多。该项目的另一个目标是为网络构建稀疏网络，稀疏网络是具有较少边的子网络，保持(部分或近似)原始网络的连通性或距离属性。该项目的第三部分将寻求在MapReduce范式中建立对并行程序效率的内在限制，这是一种日益流行的并行编程范式，其中的计算是在一系列精确定义的循环中进行的。其目的是通过证明某些基本计算任务所需的计算轮数的下界来建立该模型的一些固有限制。该项目的另一部分将开发新的算法并确定文件维护问题的效率限度，即数字以在线方式显示，并加载到线性数组中(可能在项目之间有间隙)，以便项目从左到右的顺序与自然顺序相匹配。成本是通过在装载过程中移动任何物品的总次数来衡量的。这里的目的是获得比使用随机化的现有算法更好的算法，或者证明随机化不能显著改进现有的最佳算法。通过推进算法和复杂性的理论，该奖项将增加可用于高效算法设计的工具集。为有效估计动态程序而开发的算法技术对于为诸如串匹配之类的问题开发算法的实践者可能是有用的，串匹配是在诸如数据检索和生物数据分析等各种领域中出现的基本问题。建立对解决各种问题的计算资源的内在要求可以指导为相关问题寻找改进的算法。该项目的一个重要部分是培养研究生在该领域进行研究。

This project supports new and ongoing research on several topics in algorithms and computational complexity. A major focus of the project will be certain combinatorical optimization problems, such as determining the longest common subsequence of two data sequences, that can be formulated as shortest path problems in special networks. The goal is to develop algorithms that provably give close approximations to the correct answer and are significantly faster than existing algorithms. Another goal of the project is to construct sparse spanners for networks, which are subnetworks with few edges that preserve (partially or approximately) the connectivity or distance properties of the original network. A third part of the project will seek to establish inherent limitations on the efficiency of parallel programs in the MapReduce paradigm, which is an increasingly popular paradigm for parallel programming in which computation occurs in a sequence of precisely defined rounds. The aim is to establish some inherent limitations on this model by proving lower bounds on the number of computation rounds needed for certain basic computational tasks. Another part of the project will develop new algorithms and determine limits to efficiency for the file maintenance problem, in which numbers are presented in an online manner and are loaded into a linear array (possibly with gaps between items) so that the left-to-right order of the items matches the natural order. The cost is measured by the total number of times any item is moved during the loading process. The aim here is to obtain better algorithms than the existing ones using randomization, or to establish that randomization can not significantly improve on the best existing algorithms.By advancing the theory of algorithms and complexity, this award will increase the set of tools available for efficient design of algorithms. The algorithmic techniques developed for efficient estimation of dynamic programs may be useful for practitioners developing algorithms for problems such as string matching, which is a fundamental problem that arises in varied areas such as data retrieval and analysis of biological data. Establishing inherent requirements on computational resources for solving various problems can guide the search for improved algorithms for related problems. An important part of the project is the training of graduate students to do research in the field.