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.
6593224
DNA sequencing and string learning
In laboratories the majority of large-scale DNA sequencing is done following theshotgun strategy, which is to sequence large amount of relatively short fragments randomly and then heuristically find a shortest common superstring of the fragments [26].We study mathematical frameworks, under plausible assumptions, suitable for massive automated DNA sequencing and for analyzing DNA sequencing algorithms. We model the DNA sequencing problem as learning a string from its randomly drawn substrings. Under certain restrictions, this may be viewed as string learning in Valiant's distribution-free learning model and in this case we give an efficient learning algorithm and a quantitative bound on how many examples suffice.One major obstacle to our approach turns out to be a quite well-known open question on how to approximate a shortest common superstring of a set of strings, raised by a number of authors in the last 10 years [9], [29], [30]. We give the firstprovably good algorithm which approximates a shortest superstring of lengthn by a superstring of lengthO(n logn). The algorithm works equally well even in the presence of negative examples, i.e., when merging of some strings is prohibited.
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.
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.
28836653
Statistical properties of finite sequences with high Kolmogorov complexity
We investigate to what extent finite binary sequences with high Kolmogorov complexity are normal (all blocks of equal length occur equally frequently), and the maximal length of all-zero or all-one runs which occur with certainty.
5801265
Reversibility and adiabatic computation: trading time and space for energy
Future miniaturization and mobilization of computing devices requires energy parsimonious ‘adiabatic’ computation. This is contingent on logical reversibility of computation. An example is the idea of quantum computations which are reversible except for the irreversible observation steps. We propose to study quantitatively the exchange of computational resources like time and space for irreversibility in computations. Reversible simulations of irreversible computations are memory intensive. Such (polynomial time) simulations are analysed here in terms of ‘reversible’ pebble games. We show that Bennett’s pebbling strategy uses least additional space for the greatest number of simulated steps. We derive a trade-off for storage space versus irreversible erasure. Next we consider reversible computation itself. An alternative proof is provided for the precise expression of the ultimate irreversibility cost of an otherwise reversible computation without restrictions on time and space use. A time-irreversibility trade-off hierarchy in the exponential time region is exhibited. Finally, extreme time-irreversibility trade-offs for reversible computations in the thoroughly unrealistic range of computable versus non-computable time-bounds are given.
3605809
Performance evaluation for energy efficient topologic control in ad hoc wireless networks
Minimizing total energy to keep an ad hoc wireless network symmetrically connected is an NP-hard problem. Recently, several greedy approximations have been proposed, based on k-restricted decompositions of the network. Their performance ratios are established through estimations of the least upper bound ρk for the ratio between total powers of best possible k-restricted decomposition and the optimal solution. In this paper, we determine the exact value of ρk for all k.
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.
15017510
Min-Energy Voltage Allocation for Tree-Structured Tasks
We study job scheduling on processors capable of running at variable voltage/speed to minimize energy consumption. Each job in a problem instance is specified by its arrival time and deadline, together with required number of CPU cycles. It is known that the minimum energy schedule for n jobs can be computed in O(n3) time, assuming a convex energy function. We investigate more efficient algorithms for computing the optimal schedule when the job sets have certain special structures. When the time intervals are structured as trees, the minimum energy schedule is shown to have a succinct characterization and is computable in time O(P) where P is the tree’s total path length. We also study an on-line average-rate heuristics AVR and prove that its energy consumption achieves a small constant competitive ratio for nested job sets and for job sets with limited overlap. Some simulation results are also given.
16510993
Rational quadratic approximation to real algebraic curves
An algorithm is proposed to give a global approximation to an implicit real plane algebraic curve with rational quadratic B-splines. The algorithm consists of three steps: curve segmentation, segment approximation and curve tracing. The curve is first divided into so-called triangle convex segments. Then each segment is approximated with several rational quadratic Bezier curves. At last, the curve segments are connected into several maximal branches and each branch is represented by a B-spline curve resulting in a C/sup 1/ global parameterization for the curve branch. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The final approximation based on quadratic spline curves keeps many important geometric features and gives a refined topological structure of the original curve.
13826196
Hourglasses and Hamilton cycles in 4-connected claw-free graphs
We show that if G is a 4-connected claw-free graph in which every induced hourglass subgraph S contains two non-adjacent vertices with a common neighbor outside S, then G is hamiltonian. This extends the fact that 4-connected claw-free, hourglass-free graphs are hamiltonian, thus proving a broader special case of a conjecture by Matthews and Sumner. 2005 Wiley Periodicals, Inc. J Graph Theory 48: 267–276, 2005