Self-improving Algorithms for Coordinate-wise Maxima

Computing the coordinate-wise maxima of a planar point set is a classic and well-studied problem in computational geometry. We give an algorithm for this problem in the \emph{self-improving setting}. We have $n$ (unknown) independent distributions \cD_1, \cD_2, ..., \cD_n of planar points. An input pointset $(p_1, p_2, ..., p_n)$ is generated by taking an independent sample $p_i$ from each \cD_i, so the input distribution \cD is the product \prod_i \cD_i. A self-improving algorithm repeatedly gets input sets from the distribution \cD (which is \emph{a priori} unknown) and tries to optimize its running time for \cD. Our algorithm uses the first few inputs to learn salient features of the distribution, and then becomes an optimal algorithm for distribution \cD. Let \OPT_\cD denote the expected depth of an \emph{optimal} linear comparison tree computing the maxima for distribution \cD. Our algorithm eventually has an expected running time of O(\text{OPT}_\cD + n), even though it did not know \cD to begin with. Our result requires new tools to understand linear comparison trees for computing maxima. We show how to convert general linear comparison trees to very restricted versions, which can then be related to the running time of our algorithm. An interesting feature of our algorithm is an interleaved search, where the algorithm tries to determine the likeliest point to be maximal with minimal computation. This allows the running time to be truly optimal for the distribution \cD.Comment: To appear in Symposium of Computational Geometry 2012 (17 pages, 2 figures

Length limited coding and optimal height-limited binary trees

A new O(nL)-time algorithm is given for finding an optimal prefix-free binary code for a weighted alphabet of size n, with the restriction that no code string be longer than L. An O(nL logn)-time algorithm is given for the corresponding alphabetic problem problem, which is equivalent to optimizing a dictionary of n words, implemented as a binary tree of height

Optimal Binary Search Trees with Near Minimal Height

Suppose we have n keys, n access probabilities for the keys, and n+1 access probabilities for the gaps between the keys. Let h_min(n) be the minimal height of a binary search tree for n keys. We consider the problem to construct an optimal binary search tree with near minimal height, i.e.\ with height h <= h_min(n) + Delta for some fixed Delta. It is shown, that for any fixed Delta optimal binary search trees with near minimal height can be constructed in time O(n^2). This is as fast as in the unrestricted case. So far, the best known algorithms for the construction of height-restricted optimal binary search trees have running time O(L n^2), whereby L is the maximal permitted height. Compared to these algorithms our algorithm is at least faster by a factor of log n, because L is lower bounded by log n

Dynamic Ordered Sets with Exponential Search Trees

We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully-dynamic linear space data structures. This leads to an optimal bound of O(sqrt(log n/loglog n)) for searching and updating a dynamic set of n integer keys in linear space. Here searching an integer y means finding the maximum key in the set which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein: Introduction to Algorithms, 2nd ed., MIT Press, 2001). The best previous deterministic linear space bound was O(log n/loglog n) due Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space. We also get the following worst-case linear space trade-offs between the number n, the word length w, and the maximal key U < 2^w: O(min{loglog n+log n/log w, (loglog n)(loglog U)/(logloglog U)}). These trade-offs are, however, not likely to be optimal. Our results are generalized to finger searching and string searching, providing optimal results for both in terms of n.Comment: Revision corrects some typoes and state things better for applications in subsequent paper

Efficient Monitoring of ??-languages

We present a technique for generating efficient monitors for Omega-regular-languages. We show how Buchi automata can be reduced in size and transformed into special, statistically optimal nondeterministic finite state machines, called binary transition tree finite state machines (BTT-FSMs), which recognize precisely the minimal bad prefixes of the original omega-regular-language. The presented technique is implemented as part of a larger monitoring framework and is available for download

Reinforcement Learning via AIXI Approximation

This paper introduces a principled approach for the design of a scalable general reinforcement learning agent. This approach is based on a direct approximation of AIXI, a Bayesian optimality notion for general reinforcement learning agents. Previously, it has been unclear whether the theory of AIXI could motivate the design of practical algorithms. We answer this hitherto open question in the affirmative, by providing the first computationally feasible approximation to the AIXI agent. To develop our approximation, we introduce a Monte Carlo Tree Search algorithm along with an agent-specific extension of the Context Tree Weighting algorithm. Empirically, we present a set of encouraging results on a number of stochastic, unknown, and partially observable domains.Comment: 8 LaTeX pages, 1 figur

Longest Common Pattern between two Permutations

In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations

A subquadratic algorithm for constructing approximately optimal binary search trees

An algorithm is presented which constructs an optimal binary search tree for an ordered list of n items, and which requires subquadratic time if there is no long sublist of very low frequency items. For example, time = O(n^1.6) if the frequency of each item is at least ε/ n for some constant ε &gt; 0.A second algorithm is presented which constructs an approximately optimal binary search tree. This algorithm has one parameter, and exhibits a tradeoff between speed and accuracy. It is possible to choose the parameter such that time = O(n^1.6) and error = o(l)