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

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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.