Figure 1: A high dynamic range 1024×512 environment map [Debevec 98] sampled with 3000 point lights. In this image, importance density is represented by the lightness of the background. It took 0.064 seconds on a 2.6 GHz P4 to generate this point set. Similar results using a hardware accelerated Lloyd relaxation [Hoff et al. 1999] required 1 second, while Structured Importance Sampling [Agarwal et al. 2003] took 1393 seconds. This paper presents a novel method for efficiently generating a good sampling pattern given an importance density over a 2D domain. A Penrose tiling is hierarchically subdivided creating a sufficiently large number of sample points. These points are numbered using the Fibonacci number system, and these numbers are used to threshold the samples against the local value of the importance density. Pre-computed correction vectors, obtained using relaxation, are used to improve the spectral characteristics of the sampling pattern. The technique is deterministic and very fast; the sampling time grows linearly with the required number of samples. We illustrate our technique with importance-based environment mapping, but the technique is versatile enough to be used in a large variety of computer graphics applications, such as light transport calculations, digital halftoning, geometry processing, and various rendering techniques.

This paper is self-contained, no difference field knowledge but only basic facts from algebra are required. In the following we briefly review its sections. Section 2 presents the basic GFF notions, in particular the Fundamental Lemma and an algorithm for computing the GFF-form of a polynomial. In Section 3 we investigate the relation to the dispersion function (Abramov, 1971) and discuss "shift-saturated" polynomials which are polynomials with sufficiently nice GFF-form. Due to lattice properties of K[x] with respect to gcd, a minimal shift-saturated polynomial sat(p) can be assigned to each p 2 K[x]. The canonical S-form of a rational function is introduced as the quotient of two polynomials with denominator of type sat(p). In Section 4 rational telescoping is treated; based on S-form representation, Theorem 4.1 explains why factorials rather than powers play the essential role in summation. Section 5 presents a new and algebraically motivated approach to Gosper's algorithm; together with the basic notions of GFF and Symbolic Summation 3 Section 2 this section can be read independently from the rest of the paper. In Section 6 we consider the general rational summation problem from GFF point of view. Two new algorithms are given. The first one works iteratively similar to the approach sketched by Moenck (1977). His approach is implemented in the computer algebra system Maple to sum rational functions, but due to several gaps in Moenck's original description the Maple algorithm fails on certain rational function inputs as observed by the author of this paper; see Example 6.6. The second algorithm provides an analogue to what is called "Horowitz' Method" or "Hermite-Ostrogradsky Formula" for rational function integration. In addition, discussing minimal-degree answers to...

This paper studies compact routing schemes for networks with low doubling dimension. Two variants are explored, name-independent routing and labeled routing. The key results obtained for this model are the following. First, we provide the first name-independent solution. Specifically, we achieve constant stretch and polylogarithmic storage. Second, we obtain the first truly scale-free solutions, namely, the network’s aspect ratio is not a factor in the stretch. Scale-free schemes are given for three problem models: name-independent routing on graphs, labeled routing on metric spaces, and labeled routing on graphs. Third, we prove a lower bound requiring linear storage for stretch < 3 schemes. This has the important ramification of separating for the first time the name-independent problem model from the labeled model for these networks, since compact stretch-1+ε labeled schemes are known to be possible.

In this paper, we examine how the complexity of domain-independent planning with strips-style operators depends on the nature of the planning operators. We show how the time complexity varies depending on a wide variety of conditions: • whether or not delete lists are allowed; • whether or not negative preconditions are allowed; • whether or not the predicates are restricted to be propositions (i.e., 0-ary); • whether the planning operators are given as part of the input to the planning problem, or instead are fixed in advance.

We derive general bounds on the complexity of learning in the Statistical Query model and in the PAC model with classification noise. We do so by considering the problem of boosting the accuracy of weak learning algorithms which fall within the Statistical Query model. This new model was introduced by Kearns [12] to provide a general framework for efficient PAC learning in the presence of classification noise. We first show a general scheme for boosting the accuracy of weak SQ learning algorithms, proving that weak SQ learning is equivalent to strong SQ learning. The boosting is efficient and is used to show our main result of the first general upper bounds on the complexity of strong SQ learning. Specifically, we derive simultaneous upper bounds with respect to 6 on the number of queries, O(log2:), the Vapnik-Chervonenkis dimension of the query space, O(1og log log +), and the inverse of the minimum tolerance, O(+ log 3). In addition, we show that these general upper bounds are nearly optimal by describing a class of learning problems for which we simultaneously lower bound the number of queries by R(1og f) and the inverse of the minimum tolerance by a(:). We further apply our boosting results in the SQ model to learning in the PAC model with classification noise. Since nearly all PAC learning algorithms can be cast in the SQ model, we can apply our boosting techniques to convert these PAC algorithms into highly efficient SQ algorithms. By simulating these efficient SQ algorithms in the PAC model with classification noise, we show that nearly all PAC algorithms can be converted into highly efficient PAC algorithms which *Author was supported by DARPA Contract N00014-87-K-825 and by NSF Grant CCR-89-14428. Author’s net address: jaaQtheory.lca.rit.edu +.Author was supported by an NDSEG Fellowship and

Given two permutations of n elements, a pair of intervals of these permutations consisting of the same set of elements is called a common interval. Some genetic algorithms based on such common intervals have been proposed for sequencing problems and have exhibited good prospects. In this paper, we propose three types of fast algorithms to enumerate all common intervals: i) a simple O(n 2 ) time algorithm (LHP), whose expected running time becomes O(n) for two randomly generated permutations, ii) a practically fast O(n 2 ) time algorithm (MNG) using the reverse Monge property, and iii) an O(n + K) time algorithm (RC), where K ( 0 n 2 ) is the number of common intervals. It will be also shown that the expected number of common intervals for two random permutations is O(1). This result gives a reason for the phenomenon that the expected time complexity O(n) of the algorithm LHP is independent of K. Among the proposed algorithms, RC is most desirable from the theoretical point ...

We study the parameterized complexity of three NP-hard graph completion problems. The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the big-O notation are small and do not depend on k. In particular, this implies that the problem is fixed-parameter tractable (FPT). The PROPER

Evolutionary computation (EC) consists of the design and analysis of probabilistic algorithms inspired by the principles of natural selection and variation. Genetic Programming (GP) is one subfield of EC that emphasizes desirable features such as the use of procedural representations, the capability to discover and exploit intrinsic characteristics of the application domain, and the flexibility to adapt the shape and complexity of learned models. Approaches that learn monolithic representations are considerably less likely to be effective for complex problems, and standard GP is no exception. The main goal of this dissertation is to extend GP capabilities with automatic mechanisms to cope with problems of increasing complexity. Humans succeed here by skillfully using hierarchical decomposition and abstraction mechanisms. The translation of such mechanisms into a general computer implementation is a tremendous challenge, which requires a firm understanding of the interplay between repr...

. We present and analyze a probabilistic method for verification by explicit state enumeration, which improves on the "hashcompact" method of Wolper and Leroy. The hashcompact method maintains a hash table in which compressed values for states instead of full state descriptors are stored. This method saves space but allows a non-zero probability of omitting states during verification, which may cause verification to miss design errors (i.e. verification may produce "false positives"). Our method improves on Wolper and Leroy's by calculating the hash and compressed values independently, and by using a specific hashing scheme that requires a low number of probes in the hash table. The result is a large reduction in the probability of omitting a state. Hence, we can achieve a given upper bound on the probability of omitting a state using fewer bits per compressed state. For example, we can reduce the number of bytes stored for each state from the eight recommended by Wolper and Leroy to o...

Existing array region representation techniques are sensitive to the complexityofarray subscripts. In general, these techniques are very accurate and efficient for simple subscript expressions, but lose accuracy or require potentially expensive algorithms for complex subscripts. We found that in scientific applications, many access patterns are simple even when the subscript expressions are complex. In this work, we present a new, general array access representation and define operations for it. This allows us to aggregate and simplify the representation enough that precise region operations may be applied to enable compiler optimizations. Our experiments show that these techniques hold promise for speeding up applications.