Pekka Orponen

DNA rendering of polyhedral meshes at the nanoscale

It was suggested more than thirty years ago that Watson–Crick base pairing might be used for the rational design of nanometre-scale structures from nucleic acids. Since then, and especially since the introduction of the origami technique, DNA nanotechnology has enabled increasingly more complex structures. But although general approaches for creating DNA origami polygonal meshes and design software are available, there are still important constraints arising from DNA geometry and sense/antisense pairing, necessitating some manual adjustment during the design process. Here we present a general method of folding arbitrary polygonal digital meshes in DNA that readily produces structures that would be very difficult to realize using previous approaches. The design process is highly automated, using a routeing algorithm based on graph theory and a relaxation simulation that traces scaffold strands through the target structures. Moreover, unlike conventional origami designs built from close-packed helices, our structures have a more open conformation with one helix per edge and are therefore stable under the ionic conditions usually used in biological assays.

General-purpose computation with neural networks: a survey of complexity theoretic results

We survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics. The criteria of classification include the architecture of the network (feedforward versus recurrent), time model (discrete versus continuous), state type (binary versus analog), weight constraints (symmetric versus asymmetric), network size (finite nets versus infinite families), and computation type (deterministic versus probabilistic), among others. The underlying results concerning the computational power and complexity issues of perceptron, radial basis function, winner-take-all, and spiking neural networks are briefly surveyed, with pointers to the relevant literature. In our survey, we focus mainly on the digital computation whose inputs and outputs are binary in nature, although their values are quite often encoded as analog neuron states. We omit the important learning issues.

Dempster's rule of combination is # P -complete (research note)

Abstract We consider the complexity of combining bodies of evidence according to the rules of the Dempster-Shafer theory of evidence. We prove that, given as input a set of tables representing basic probability assignments m 1 , …, m n over a frame of discernment Θ, and a set A ⊆ Θ , the problem of computing the combined basic probability value (m 1 ⊕ … ⊕ m n )(A) is # P -complete. As a corollary, we obtain that while the simple belief, plausibility, and commonality values Bel (A), Pl (A), and Q(A) can be computed in polynomial time, the problems of computing the combinations (Bel 1 ⊕ … ⊕ Bel n (A), (Pl 1 ⊕ … ⊕ Pl n )(A), and (Q 1 ⊕ … ⊕ Q n )(A) are # P -complete.

On the Effect of Analog Noise in Discrete-Time Analog Computations

We introduce a model for analog computation with discrete time in the presence of analog noise that is flexible enough to cover the most important concrete cases, such as noisy analog neural nets and networks of spiking neurons. This model subsumes the classical model for digital computation in the presence of noise. We show that the presence of arbitrarily small amounts of analog noise reduces the power of analog computational models to that of finite automata, and we also prove a new type of upper bound for the VC-dimension of computational models with analog noise.

A Survey of Continuous-Time Computation Theory

Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuous-time computation. However, while special-case algorithms and devices are being developed, relatively little work exists on the general theory of continuous- time models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions.

Instance complexity

We introduce a measure for the computational complexity of mdiwdual instances of a decision problem and study some of Its properties. The instance complexity of a string ~ with respect to a set A and time bound t, ict(x : A). is defined as the size of the smallest special-case program for A that run> m time t,decides x correctly, and makes no mistakes on other strings (“don’t know” answers are permitted). We prove that a set A is m P if and only if there exist a polynomial t and a constant c such that ic’(x : A) < c for all X; on the other hand, If A ]s NP-hard and P # NP, then for all polynomials t and constants c. lc’(~ : A) > c log I ~ I for ]nfimtely many x. Obserwng that Kf(x), the t-bounded Kolmogorov complexity of x, N roughly an upper bound on ]Ct(.t : A), we proceed to investigate the existence of mdiwdually hard problem Instances. ].e , strings whose instance complexity E close to their Kolmogorov complexity. We prove that if t(n)z n is a time-constructible function and A 1s a recurswe set not in DTIME(t), there then exist a constant c and mfimtely many I such that ic’(x : ,4) z K’ (x) – c. for some Prehmmary versions of parts of this work have appeared under the titles “What 1sa hard instance of a computational problem?” m Proceedings of tize Conference on Structare m Cornplexm Theory (Berkeley, Calif., June i 986), and “On the instance complexity of NP-hard problems” in Procecduzgs of the 5tk .4nrrual Conference on StntctLwe m Cowrpkwty Theory (Barcelona, Spain, July 1990). These Proceedings have been published by Springer-Verlag, Berlin, and IEEE, New York, respectively. The research of P. Orponen was supported by the Academy of Finland, and the research of K. Ko in part by National Science Foundation (NSF) grant CCR 8S-01575. Authors’ current addresses: P. Orponen, Department of Computer Science, Unnerslty of Helsinkl, FIN-0001 4 Helsinki, Finland; K. Ko, Department of Computer Science, State Unwersity of New York at Stony Brook, Stony Brook, NY 11794; U. Schomng, Abteiltrng Theoretische Informatik, Umversltat Ulm, D-89069 Ulm, Germany; O. Watanabe, Department of Computer Science, Tohyo Institute of Technology, Tokyo 152, Japan. Permission to copy without fee all or part of this material IS granted provided that the copies are not made or distributed for duect commercial advantage, the ACM copyright notice and the title of the pubhcdtion and Its date appear, and notice K given that copying 1s by permission of the Association for Computing Machinery. To copy otherwse, or to repubhsh, requmes a fee and/or specific permission. 01994 ACM 0004-5411/94/’0100-0096

The density and complexity of polynomial cores for intractable sets

03.50 Journal of the AwocI.tIon for Compuh.g Md.hlncry, Vii 41 No 1, January 1YY4 pp Y6-121 Instance Complexity 97 time bound t‘(n)dependent on the complexity of recognizing A. Under the stronger assumptions that the set A is NP-hard and DEXT # NEXT, we prove that for any polynomia~ t there exist a polynomial f‘ and a constant c such that for infinitely many x, ict(x : A) z K“(x) – c. If A is DEXT-hard, then the same result holds unconditionally. We also prove that there is a set A E DEXT such that for some constant c and all x, ic’xp(x : A) s K’xp (x) – 2 log ZCexPr(x)– C, where exp(n) = 2“ and exp’(n) = cn2zn + c.

Balanced Data Gathering in Energy-Constrained Sensor Networks

Let A be a recursive problem not in P. Lynch has shown that A then has an infinite recursive polynomial complexity core. This is a collection C of instances of A such that every algorithm deciding A needs more than polynomial time almost everywhere on C. We investigate the complexity of recognizing the instances in such a core, and show that every recursive problem A not in P has an infinite core recognizable in subexponential time. We further study how dense the core sets for A can be, under various assumptions about the structure of A. Our main results in this direction are that if P ≠ NP, then NP-complete problems have polynomially nonsparse cores recognizable in subexponential time, and that EXPTIME-complete problems have cores of exponential density recognizable in exponential time.

Online bin packing with delay and holding costs

We consider the problem of gathering data from a wireless multi-hop network of energy-constrained sensor nodes to a common base station. Specifically, we aim to balance the total amount of data received from the sensor network during its lifetime against a requirement of sufficient coverage for all the sensor locations surveyed. Our main contribution lies in formulating this balanced data gathering task and in studying the effects of balancing. We give an LP network flow formulation and present experimental results on optimal data routing designs also with impenetrable obstacles between the nodes. We then proceed to consider the effect of augmenting the basic sensor network with a small number of auxiliary relay nodes with less stringent energy constraints. We present an algorithm for finding approximately optimal placements for the relay nodes, given a system of basic sensor locations, and compare it with a straightforward grid arrangement of the relays.

Computing with truly asynchronous threshold logic networks

We consider online bin packing where in addition to the opening cost of each bin, the arriving items collect delay costs until their assigned bins are released (closed), and the open bins themselves collect holding costs. Besides being of practical interest, this problem generalises several previously unrelated online optimisation problems. We provide a general online algorithm for this problem with competitive ratio 7, with improvements for the special cases of zero delay or holding costs and size-proportional item delay costs.