On the Complexity of the Single Individual SNP Haplotyping Problem

We present several new results pertaining to haplotyping. These results concern the combinatorial problem of reconstructing haplotypes from incomplete and/or imperfectly sequenced haplotype fragments. We consider the complexity of the problems Minimum Error Correction (MEC) and Longest Haplotype Reconstruction (LHR) for different restrictions on the input data. Specifically, we look at the gapless case, where every row of the input corresponds to a gapless haplotype-fragment, and the 1-gap case, where at most one gap per fragment is allowed. We prove that MEC is APX-hard in the 1-gap case and still NP-hard in the gapless case. In addition, we question earlier claims that MEC is NP-hard even when the input matrix is restricted to being completely binary. Concerning LHR, we show that this problem is NP-hard and APX-hard in the 1-gap case (and thus also in the general case), but is polynomial time solvable in the gapless case.Comment: 26 pages. Related to the WABI2005 paper, "On the Complexity of Several Haplotyping Problems", but with more/different results. This papers has just been submitted to the IEEE/ACM Transactions on Computational Biology and Bioinformatics and we are awaiting a decision on acceptance. It differs from the mid-August version of this paper because here we prove that 1-gap LHR is APX-hard. (In the earlier version of the paper we could prove only that it was NP-hard.

On some combinatorial properties of graph states

I graph state sono particolari stati quantistici, rappresentabili tramite grafi indiretti semplici, che giocano un ruolo fondamentale in informatica quantistica, in particolare nell'ambito dei codici a correzione di errore e nel modello di computazione one-way. Lo scopo di questa tesi è studiare alcune proprietà dei graph state attraverso un approccio combinatorio. Innanzitutto si sono analizzate le proprietà di un'invariante dei grafi: il numero di sottografi indotti con numero dispari di archi. Questo numero è stato valutato per famiglie importanti di grafi ed è stato trovato un algoritmo efficiente per calcolarlo. Abbiamo inoltre caratterizzato questo numero rispetto all'azione di local complementation. Alcune proprietà delle funzioni booleane associate ai graph state sono state studiate analizzando la struttura dei grafi di Cayley delle suddette funzioni. Infine sono stati definiti, e ne è stata analizzata la struttura, grafi che permettono di tracciare le trasformazioni di local complementation e switching fra graph state. Un congettura è stata fatta sulla struttura di questi grafi. A margine del lavoro, sono state introdotte alcune estensioni alla definizione classica di graph state. In particolare sono stati definiti gli "edge graph state" e i "3-hypergraph state"

Classical computing, quantum computing, and Shor's factoring algorithm

This is an expository talk written for the Bourbaki Seminar. After a brief introduction, Section 1 discusses in the categorical language the structure of the classical deterministic computations. Basic notions of complexity icluding the P/NP problem are reviewed. Section 2 introduces the notion of quantum parallelism and explains the main issues of quantum computing. Section 3 is devoted to four quantum subroutines: initialization, quantum computing of classical Boolean functions, quantum Fourier transform, and Grover's search algorithm. The central Section 4 explains Shor's factoring algorithm. Section 5 relates Kolmogorov's complexity to the spectral properties of computable function. Appendix contributes to the prehistory of quantum computing.Comment: 27 pp., no figures, amste

Mixed radix design flow for security applications

The purpose of secure devices, such as smartcards, is to protect sensitive information against software and hardware attacks. Implementation of the appropriate protection techniques often implies non-standard methods that are not supported by the conventional design tools. In the recent decade the designers of secure devices have been working hard on customising the workflow. The presented research aims at collecting the up-to-date experiences in this area and create a generic approach to the secure design flow that can be used as guidance by engineers. Well-known countermeasures to hardware attacks imply the use of specific signal encodings. Therefore, multi-valued logic has been considered as a primary aspect of the secure design. The choice of radix is crucial for multi-valued logic synthesis. Practical examples reveal that it is not always possible to find the optimal radix when taking into account actual physical parameters of multi-valued operations. In other words, each radix has its advantages and disadvantages. Our proposal is to synthesise logic in different radices, so it could benefit from their combination. With respect to the design opportunities of the existing tools and the possibilities of developing new tools that would fill the gaps in the flow, two distinct design approaches have been formed: conversion driven design and pre-synthesis. The conversion driven design approach takes the outputs of mature and time-proven electronic design automation (EDA) synthesis tools to generate mixed radix datapath circuits in an endeavour to investigate the added relative advantages or disadvantages. An algorithm underpinning the approach is presented and formally described together with secure gate-level implementations. The obtained results are reported showing an increase in power consumption, thus giving further motivation for the second approach. The pre-synthesis approach is aimed at improving the efficiency by using multivalued logic synthesis techniques to produce an abstract component-level circuit before mapping it into technology libary. Reed-Muller expansions over Galois field arithmetic have been chosen as a theoretical foundation for this approach. In order to enable the combination of radices at the mathematical level, the multi-valued Reed-Muller expansions have been developed into mixed radix Reed-Muller expansions. The goals of the work is to estimate the potential of the new approach and to analyse its impact on circuit parameters down to the level of physical gates. The benchmark results show the approach extends the search space for optimisation and provides information on how the implemented functions are related to different radices. The theory of two-level radix models and corresponding computation methods are the primary theoretical contribution. It has been implemented in RMMixed tool and interfaced to the standard EDA tools to form a complete security-aware design flow.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo

X-Codes: Theory and Applications of Unknowable Inputs

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNSF / ACI-99-84492-CAREE

Classical simulation of measurement-based quantum computation on higher-genus surface-code states

We consider the efficiency of classically simulating measurement-based quantum computation on surface-code states. We devise a method for calculating the elements of the probability distribution for the classical output of the quantum computation. The operational cost of this method is polynomial in the size of the surface-code state, but in the worst case scales as $2^{2g}$ in the genus $g$ of the surface embedding the code. However, there are states in the code space for which the simulation becomes efficient. In general, the simulation cost is exponential in the entanglement contained in a certain effective state, capturing the encoded state, the encoding and the local post-measurement states. The same efficiencies hold, with additional assumptions on the temporal order of measurements and on the tessellations of the code surfaces, for the harder task of sampling from the distribution of the computational output.Comment: 21 pages, 13 figure

Probabilistic representation and manipulation of Boolean functions using free Boolean diagrams

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (p. 145-149).by Amelia Huimin Shen.Ph.D