Form, scales and optimality in the basin landscape and its channel network

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 1994.Includes bibliographical references (leaves 252-263).by Ede Jorge Ijjász-Vásquez.Ph.D

Merge-and-shrink abstractions for classical planning : theory, strategies, and implementation

Classical planning is the problem of finding a sequence of deterministic actions in a state space that lead from an initial state to a state satisfying some goal condition. The dominant approach to optimally solve planning tasks is heuristic search, in particular A* search combined with an admissible heuristic. While there exist many different admissible heuristics, we focus on abstraction heuristics in this thesis, and in particular, on the well-established merge-and-shrink heuristics. Our main theoretical contribution is to provide a comprehensive description of the merge-and-shrink framework in terms of transformations of transition systems. Unlike previous accounts, our description is fully compositional, i.e. can be understood by understanding each transformation in isolation. In particular, in addition to the name-giving merge and shrink transformations, we also describe pruning and label reduction as such transformations. The latter is based on generalized label reduction, a new theory that removes all of the restrictions of the previous definition of label reduction. We study the four types of transformations in terms of desirable formal properties and explain how these properties transfer to heuristics being admissible and consistent or even perfect. We also describe an optimized implementation of the merge-and-shrink framework that substantially improves the efficiency compared to previous implementations. Furthermore, we investigate the expressive power of merge-and-shrink abstractions by analyzing factored mappings, the data structure they use for representing functions. In particular, we show that there exist certain families of functions that can be compactly represented by so-called non-linear factored mappings but not by linear ones. On the practical side, we contribute several non-linear merge strategies to the merge-and-shrink toolbox. In particular, we adapt a merge strategy from model checking to planning, provide a framework to enhance existing merge strategies based on symmetries, devise a simple score-based merge strategy that minimizes the maximum size of transition systems of the merge-and-shrink computation, and describe another framework to enhance merge strategies based on an analysis of causal dependencies of the planning task. In a large experimental study, we show the evolution of the performance of merge-and-shrink heuristics on planning benchmarks. Starting with the state of the art before the contributions of this thesis, we subsequently evaluate all of our techniques and show that state-of-the-art non-linear merge-and-shrink heuristics improve significantly over the previous state of the art

On the Horton-Strahler number for combinatorial tries

Nebel M. On the Horton-Strahler number for combinatorial tries. ITA. 2000;34(4):279--296

On the Horton-Strahler Number for Combinatorial Tries

In this paper we investigate the average Horton-Strahler number of all possible tree-structures of binary tries. For that purpose we consider a generalization of extended binary trees where leaves are distinguished in order to represent the location of keys within a corresponding trie. Assuming a uniform distribution for those trees we prove that the expected Horton-Strahler number of a tree with α internal nodes and β leaves that correspond to a key is asymptotically given by $$\frac{4^{2\beta-\alpha}\log(\alpha)(2\beta-1)(\alpha+1)(\alpha+2){2\alpha+1\choose \alpha-1}}{8\sqrt{\pi}\alpha^{3/2}\log(2)(\beta-1)\beta{2\beta\choose \beta}^2}$$ provided that α and β grow in some fixed proportion ρ when α → ∞ . A similar result is shown for trees with α internal nodes but with an arbitrary number of keys