14049212
The Incompressibility Method
Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source. It was proposed by A.N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner. This is impossible for classical probability theory. Kolmogorov complexity is known variously as 'algorithmic information', 'algorithmic entropy', 'Kolmogorov-Chaitin complexity', 'descriptional complexity', 'shortest program length', 'algorithmic randomness', and others. Using it, we developed a new mathematical proof technique, now known as the 'incompressibility method'. The incompressibility method is a basic general technique such as the 'pigeon hole' argument, 'the counting method' or the 'probabilistic method'. The new method has been quite successful and we present recent examples. The first example concerns a "static" problem in combinatorial geometry. From among (n 3)triangles with vertices chosen from among n points in the unit square, U, let T be the one with the smallest area, and let A be the area of T. Heilbronn's triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the average-case: If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c/n3 < µn < C/n3 for all large enough n, where µn is the expectation of A. Moreover, c/n3 < A < C/n3 for almost all A, that is, almost all A are close to the expectation value so that we determine the area of the smallest triangle for an arrangement in "general position". Our second example concerns a "dynamic" problem in average-case running time of algorithms. The question of a nontrivial general lower bound (or upper bound) on the average-case complexity of Shellsort has been open for about forty years. We obtain the first such lower bound.
871253
Kolmogorov Complexity Arguments in Combinatorics
Abstract The utility of a Kolmogorov complexity method in combinatorial theory is demonstrated by several examples.
14168735
Inductive Reasoning and Kolmogorov Complexity
Abstract Reasoning to obtain the “truth” about reality, from external data, is an important, controversial, and complicated issue in man's effort to understand nature. (Yet, today, we try to make machines do this.) There have been old useful principles, new exciting models, and intricate theories scattered in vastly different areas including philosophy of science, statistics, computer science, and psychology. We focus on inductive reasoning in correspondence with ideas of R. J. Solomonoff. While his proposals result in perfect procedures, they involve the noncomputable notion of Kolmogorov complexity. In this paper we develop the thesis that Solomonoff's method is fundamental in the sense that many other induction principles can be viewed as particular ways to obtain computable approximations to it. We demonstrate this explicitly in the cases of Gold's paradigm for inductive inference, Rissanen's minimum description length (MDL) principle, Fisher's maximum likelihood principle, and Jaynes' maximum entropy principle. We present several new theorems and derivations to this effect. We also delimit what can be learned and what cannot be learned in terms of Kolmogorov complexity, and we describe an experiment in machine learning of handwritten characters. We also give an application of Kolmogorov complexity in Valiant style learning, where we want to learn a concept probably approximately correct in feasible time and examples.
9397376
Inferring a DNA Sequence from Erroneous Copies
Abstract We suggest a novel approach for efficiently reconstructing an original DNA sequence from erroneous copies.
43310555
Twist-Rotation Transformations of Binary Trees and Arithmetic Expressions
The paper studies the computational complexity and efficient algorithms for the twist?rotation transformations of binary trees, which is equivalent to the transformation of arithmetic expressions over an associative and commutative binary operation. The main results are (1) a full binary tree with n labeled leaves can be transformed into any other in at most 3nlogn+2n twist and rotation operations, (2) deciding the twist?rotation distance between two binary trees is NP-complete, and (3) the twist?rotation transformation can be approximated with ratio 6logn+4 in polynomial time for full binary trees with n uniquely labeled leaves.
553776
Linear approximation of shortest superstrings
We consider the following problem: given a collection of strings s1, ••• ,sm, find the shortest string s such that each s; appears as a substring (a consecutive block) of s. Although this problem is known to be NP-hard, a simple greedy procedure appears to do quite well and is routinely used in DNA sequencing and data compression practice, namely: repeatedly merge the pair of (distinct) strings with maximum overlap until only one string remains. Let n denote the A. Blum was supported by a National Science Foundation (NSF) Graduate Fellowship. Part of this work was done while A. Blum was visiting AT & T Bell Labs. T. Jiang was supported in part by a grant from SERB, McMaster University and NSERC Operating Grant OGP0046613. M. Li was supported in part by the NSERC Operating Grants OGP0036747 and OGP0046506. J. Tromp was supported in part by NSERC Grant OGP0036747 while the author was visiting at the University of Waterloo. Authors' present addresses: A. Blum, School of Computer Science, Carnegie-Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213. E-mail: avrim@theory.cs.cmu.edu; T. Jiang, Department of Computer Science, McMaster University, Hamilton, Ont., Canada L8S 4Kl. E-mail: jiang<Q•maccs. mcmaster.ca; M. Li, Department of Computer Science, University of Waterloo, Waterloo, Ont., Canada N3L 30 I. E-mail: mli@watmath.waterloo.edu; J. Tromp, CWI, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands. E-mail: tromp@cwi.nl; M. Yannakakis, Room 2C-319, AT&T Bell Labs, 600 Mountain Avenue, Murray Hill, NJ 07974. E-mail: mihalis@research.att.com. Permission to copy without fee all or part of this material is granted provided that the copies arc not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1994 ACM 0004-5411/94/0700-0630$03.50 Journal of the Association for Computing Machinery, Vol. 41, No. 4, July 1994, pages 630-647. Linear Approximation of Shortest Superstrings 631 length of the optimal superstring. A common conjecture states that the above greedy procedure produces a superstring of length O(n) (in fact, 2n), yet the only previous nontrivial bound known for any polynomial-time algorithm is a recent O(n log n) result. We show that the greedy algorithm does in fact achieve a constant factor approximation, proving an upper bound of 4n. Furthermore, we present a simple modified version of the greedy algorithm that we show produces a superstring of length at most 3n. We also show the superstring problem to be MAX SNP-hard, which implies that a polynomial-time approximation scheme for this problem is unlikely.
32860736
Nonrigid motion recovery for 3D surfaces
We present a spline-based nonrigid motion and point correspondence recovery method for 3D surfaces. This method is based on differential geometry. Shape information is used to recover the point correspondences. In contrast to the majority of shape-based methods, which assume that shape (unit normal, curvature) changes are minimum after motion, our method focuses on the nonrigid relationship between before-motion and after-motion shapes. This nonrigid shape relationship is described by modeling the underlying nonrigid motion; we model it as a spline transformation, which has global control over the entire motion field along with the local deformation integrated within. This provides our method certain advantages over some pure differential geometric methods, which also utilize the nonrigid shape relationship but only work on local areas without a global control. For example, motion regularity is hard to implement in these pure differential geometric methods but is not a problem when the motion field is controlled by a spline transformation. The orthogonal parameterization requirement of the nonrigid shape relationship has to be approximated in these previous methods but is easy to meet in our method. Furthermore, the small deformation constraint introduced by the previous works is relaxed in our method. Experiments on both synthetic and real motions have been conducted. The quantitative and qualitative evaluations of our method are presented. The application of our method to the human tongue motion analysis is also presented in this paper.
44742126
A New Association Rules Mining Algorithms Based on Directed Itemsets Graph
In this paper, we introduced a new data structure called DISG (Directed itemsets graph) in which the information of frequent itemsets was stored. Based on it, a new algorithm called DBDG (DFS Based -DISG) was developed by using depth first searching strategy. At last we performed a experiment on a real dataset to test the run time of DBDG. The experiment showd that it was efficient for mining dense datasets.
14969576
Semi-Supervised Regression with Co-Training
In many practical machine learning and data mining applications, unlabeled training examples are readily available but labeled ones are fairly expensive to obtain. Therefore, semi-supervised learning algorithms such as co-training have attracted much attention. Previous research mainly focuses on semi-supervised classification. In this paper, a co-training style semi-supervised regression algorithm, i.e. COREG, is proposed. This algorithm uses two k-nearest neighbor regressors with different distance metrics, each of which labels the unlabeled data for the other regressor where the labeling confidence is estimated through consulting the influence of the labeling of unlabeled examples on the labeled ones. Experiments show that COREG can effectively exploit unlabeled data to improve regression estimates.
3009418
The hamiltonian index of a graph and its branch-bonds
Let $G$ be an undirected and loopless finite graph that is not a path. The minimum $m$ such that the iterated line graph $L^m(G)$ is hamiltonian is called the hamiltonian index of $G,$ denoted by $h(G).$ A reduction method to determine the hamiltonian index of a graph $G$ with $h(G)\geq 2$ is given here. With it we will establish a sharp lower bound and a sharp upper bound for $h(G)$, respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that $h(G)$ may reach all integers between the lower bound and the upper bound.
5412677
Middleware for streaming 3D progressive meshes over lossy networks
Streaming 3D graphics have been widely used in multimedia applications such as online gaming and virtual reality. However, a gap exists between the zero-loss-tolerance of the existing compression schemes and the lossy network transmissions. In this article, we propose a generic 3D middleware between the 3D application layer and the transport layer for the transmission of triangle-based progressively compressed 3D models. Significant features of the proposed middleware include. 1) handling 3D compressed data streams from multiple progressive compression techniques. 2) considering end user hardware capabilities for effectively saving the data size for network delivery. 3) a minimum cost dynamic reliable set selector to choose the transport protocol for each sublayer based on the real-time network traffic. Extensive simulations with TCP/UDP and SCTP show that the proposed 3D middleware can achieve the dual objectives of maintaining low transmission delay and small distortion, and thus supporting high quality 3D streaming with high flexibility.