Sampled longest common prefix array

When augmented with the longest common prefix (LCP) array and some other structures, the suffix array can solve many string processing problems in optimal time and space. A compressed representation of the LCP array is also one of the main building blocks in many compressed suffix tree proposals. In this paper, we describe a new compressed LCP representation: the sampled LCP array. We show that when used with a compressed suffix array (CSA), the sampled LCP array often offers better time/space trade-offs than the existing alternatives. We also show how to construct the compressed representations of the LCP array directly from a CSA

The Problem of Implementation and its Relation to the Philosophy of Cognitive Science

According to certain arguments, computation is observer-relative either in the sense that many physical systems implement many computations (Hilary Putnam), or in the sense that almost all physical systems implement all computations (John Searle). If sound, these arguments have a potentially devastating consequence for the computational theory of mind: if arbitrary physical systems can be seen to implement arbitrary computations, the notion of computation seems to lose all explanatory power as far as brains and minds are concerned. David Chalmers and B. Jack Copeland have attempted to counter these relativist arguments by placing certain constraints on the definition of implementation. In this thesis, I examine their proposals and find both wanting in some respects. During the course of this examination, I give a formal definition of the class of combinatorial-state automata , upon which Chalmers s account of implementation is based. I show that this definition implies two theorems (one an observation due to Curtis Brown) concerning the computational power of combinatorial-state automata, theorems which speak against founding the theory of implementation upon this formalism. Toward the end of the thesis, I sketch a definition of the implementation of Turing machines in dynamical systems, and offer this as an alternative to Chalmers s and Copeland s accounts of implementation. I demonstrate that the definition does not imply Searle s claim for the universal implementation of computations. However, the definition may support claims that are weaker than Searle s, yet still troubling to the computationalist. There remains a kernel of relativity in implementation at any rate, since the interpretation of physical systems seems itself to be an observer-relative matter, to some degree at least. This observation helps clarify the role the notion of computation can play in cognitive science. Specifically, I will argue that the notion should be conceived as an instrumental rather than as a fundamental or foundational one.

Genetic algorithm for satellite customer assignment

The problem of assigning customers to satellite channels is considered. Finding an optimal allocation of customers to satellite channels is a difficult combinatorial optimization problem and is shown to be NP-complete in an earlier study. We propose a genetic algorithm (GA) approach to search for the best/optimal assignment of customers to satellite channels. Various issues related to genetic algorithms such as solution representation, selection methods, genetic operators and repair of invalid solutions are presented. A comparison of this approach with the standard optimization method is presented to show the advantages of this approach in terms of computation time

A Reactive Tabu Search Based Equalizer for Severely Delay-Spread UWB MIMO-ISI Channels

We propose a novel equalizer for ultrawideband (UWB) multiple-input multiple-output (MIMO) channels characterized by severe delay spreads. The proposed equalizer is based on reactive tabu search (RTS), which is a heuristic originally designed to obtain approximate solutions to combinatorial optimization problems. The proposed RTS equalizer is shown to perform increasingly better for increasing number of multipath components (MPC), and achieve near maximum likelihood (ML) performance for large number of MPCs at a much less complexity than that of the ML detector. The proposed RTS equalizer is shown to perform close to within 0.4 dB of single-input multiple-output AWGN performance at 10(-3) uncoded BER on a severely delay-spread UWB MIMO channel with 48 equal-energy MPCs.

On The Ground State Properties Of The +/-J Ising Model

In the combinatorial method or Grassmann algebra formalism the ground state properties of the f J Ising model can be expressed in terms of the behaviour of the eigenvectors of a matrix. It is shown that a transition from localized to extended eigenvectors signals the breakdown of ferromagnetic rigidity.

Patterns in permuted binary matrices

Reorganizing a dataset so that its hidden structure can be observed is useful in any data analysis task. For example, detecting a regularity in a dataset helps us to interpret the data, compress the data, and explain the processes behind the data. We study datasets that come in the form of binary matrices (tables with 0s and 1s). Our goal is to develop automatic methods that bring out certain patterns by permuting the rows and columns. We concentrate on the following patterns in binary matrices: consecutive-ones (C1P), simultaneous consecutive-ones (SC1P), nestedness, k-nestedness, and bandedness. These patterns reflect specific types of interplay and variation between the rows and columns, such as continuity and hierarchies. Furthermore, their combinatorial properties are interlinked, which helps us to develop the theory of binary matrices and efficient algorithms. Indeed, we can detect all these patterns in a binary matrix efficiently, that is, in polynomial time in the size of the matrix. Since real-world datasets often contain noise and errors, we rarely witness perfect patterns. Therefore we also need to assess how far an input matrix is from a pattern: we count the number of flips (from 0s to 1s or vice versa) needed to bring out the perfect pattern in the matrix. Unfortunately, for most patterns it is an NP-complete problem to find the minimum distance to a matrix that has the perfect pattern, which means that the existence of a polynomial-time algorithm is unlikely. To find patterns in datasets with noise, we need methods that are noise-tolerant and work in practical time with large datasets. The theory of binary matrices gives rise to robust heuristics that have good performance with synthetic data and discover easily interpretable structures in real-world datasets: dialectical variation in the spoken Finnish language, division of European locations by the hierarchies found in mammal occurrences, and co-occuring groups in network data. In addition to determining the distance from a dataset to a pattern, we need to determine whether the pattern is significant or a mere occurrence of a random chance. To this end, we use significance testing: we deem a dataset significant if it appears exceptional when compared to datasets generated from a certain null hypothesis. After detecting a significant pattern in a dataset, it is up to domain experts to interpret the results in the terms of the application.

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game

Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.

A Deterministic Optimization Approach to Protein Sequence Design Using Continuous Models

Determining the sequence of amino acid residues in a heteropolymer chain of a protein with a given conformation is a discrete combinatorial problem that is not generally amenable for gradient-based continuous optimization algorithms. In this paper we present a new approach to this problem using continuous models. In this modeling, continuous "state functions" are proposed to designate the type of each residue in the chain. Such a continuous model helps define a continuous sequence space in which a chosen criterion is optimized to find the most appropriate sequence. Searching a continuous sequence space using a deterministic optimization algorithm makes it possible to find the optimal sequences with much less computation than many other approaches. The computational efficiency of this method is further improved by combining it with a graph spectral method, which explicitly takes into account the topology of the desired conformation and also helps make the combined method more robust. The continuous modeling used here appears to have additional advantages in mimicking the folding pathways and in creating the energy landscapes that help find sequences with high stability and kinetic accessibility. To illustrate the new approach, a widely used simplifying assumption is made by considering only two types of residues: hydrophobic (H) and polar (P). Self-avoiding compact lattice models are used to validate the method with known results in the literature and data that can be practically obtained by exhaustive enumeration on a desktop computer. We also present examples of sequence design for the HP models of some real proteins, which are solved in less than five minutes on a single-processor desktop computer Some open issues and future extensions are noted.

Information and Decision Technologies

Production scheduling in a flexible manufacturing system (FMS) is a real-time combinatorial optimization problem that has been proved to be NP-complete. Solving this problem needs on-line monitoring of plan execution and requires real-time decision-making in selecting alternative routings, assigning required resources, and rescheduling when failures occur in the system. Expert systems provide a natural framework for solving this kind of NP-complete problems.In this paper an expert system with a novel parallel heuristic approach is implemented for automatic short-term dynamic scheduling of FMS. The principal features of the expert system presented in this paper include easy rescheduling, on-line plan execution, load balancing, an on-line garbage collection process, and the use of advanced knowledge representational schemes. Its effectiveness is demonstrated with two examples.

On the Nash-Williams′ Lemma in Graph Reconstruction Theory

A generalization of Nash-Williams′ lemma is proved for the Structure of m-uniform null (m − k)-designs. It is then applied to various graph reconstruction problems. A short combinatorial proof of the edge reconstructibility of digraphs having regular underlying undirected graphs (e.g., tournaments) is given. A type of Nash-Williams′ lemma is conjectured for the vertex reconstruction problem.

On kuhnels 9-vertex complex projective plane

We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C P2. The original proof of existence, due to Kuhnel, as well as the original proof of uniqueness, due to Kuhnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kuhnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.

Multiple UAV Coalitions for a Search and Prosecute Mission

Unmanned aerial vehicles (UAVs) have the potential to carry resources in support of search and prosecute operations. Often to completely prosecute a target, UAVs may have to simultaneously attack the target with various resources with different capacities. However, the UAVs are capable of carrying only limited resources in small quantities, hence, a group of UAVs (coalition) needs to be assigned that satisfies the target resource requirement. The assigned coalition must be such that it minimizes the target prosecution delay and the size of the coalition. The problem of forming coalitions is computationally intensive due to the combinatorial nature of the problem, but for real-time applications computationally cheap solutions are required. In this paper, we propose decentralized sub-optimal (polynomial time) and decentralized optimal coalition formation algorithms that generate coalitions for a single target with low computational complexity. We compare the performance of the proposed algorithms to that of a global optimal solution for which we need to solve a centralized combinatorial optimization problem. This problem is computationally intensive because the solution has to (a) provide a coalition for each target, (b) design a sequence in which targets need to be prosecuted, and (c) take into account reduction of UAV resources with usage. To solve this problem we use the Particle Swarm Optimization (PSO) technique. Through simulations, we study the performance of the proposed algorithms in terms of mission performance, complexity of the algorithms and the time taken to form the coalition. The simulation results show that the solution provided by the proposed algorithms is close to the global optimal solution and requires far less computational resources.

Proof of a conjecture of Frankl and Furedi

We give a simple linear algebraic proof of the following conjecture of Frankl and Furedi [7, 9, 13]. (Frankl-Furedi Conjecture) if F is a hypergraph on X = {1, 2, 3,..., n} such that 1 less than or equal to /E boolean AND F/ less than or equal to k For All E, F is an element of F, E not equal F, then /F/ less than or equal to (i=0)Sigma(k) ((i) (n-1)). We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace of R-/F/. Finally, the desired bound on /F/ is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray-Chaudhuri-Wilson theorem. (C) 1997 Academic Press.