The Meeting problem for @math searchers in a polygon @math (possibly with holes) consists in making the searchers move within @math , according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of @math , we minimize the number of searchers @math for which the Meeting problem is solvable. Specifically, if @math has a rotational symmetry of order @math (where @math corresponds to no rotational symmetry), we prove that @math searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with @math searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.
To the best of our knowledge, only two papers deal with this problem. @cite_6 , a definition of computation is not explicitly given, but it is said that a certain point is not computable, in the sense that its coordinates cannot be computed with basic arithmetic operations and extractions of roots of any degree. @cite_16 , computation is explicitly defined in terms of algebraic functions, although all that is actually needed is the ability to construct regular polygons, as well as all points that can be geometrically obtained with compass and straightedge.
@cite_16 @cite_6
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Publisher Summary This chapter discusses the mathematical theory of computation. Computation essentially explores how machines can be made to carry out intellectual processes. Any intellectual process that can be carried out mechanically can be performed by a general purpose digital computer. There are three established directions of mathematical research that are relevant to the science of computation—namely, numerical analysis, theory of computability, and theory of finite automata. The chapter explores what practical results can be expected from a suitable mathematical theory. Further, the chapter presents several descriptive formalisms with a few examples of their use and theories that enable to prove the equivalence of computations expressed in these formalisms. A few mathematical results about the properties of the formalisms are also presented. We design an algorithm for computing the generalized (algebraic circuits with root extracting; cf. Pippenger [J. Comput. System Sci., 22 (1981), pp. 454--470], Ja'Ja' [Proc. 22nd IEEE FOCS, 1981, pp. 95--100], Grigoriev, Singer, and Yao [ SIAM J. Comput., 24 (1995), pp. 242--246]) additive complexity of any rational function. It is the first computability result of this sort on the additive complexity of algebraic circuits. We give necessary and sufficient combinatorial conditions characterizing the computational tasks that can be solved by N asynchronous processes, up to t of which can fail by halting. The range of possible input and output values for an asynchronous task can be associated with a high-dimensional geometric structure called a simplicial complex. Our main theorem characterizes computability y in terms of the topological properties of this complex. Most notably, a given task is computable only if it can be associated with a complex that is simply connected with trivial homology groups. In other words, the complex has “no holes!” Applications of this characterization include the first impossibility results for several long-standing open problems in distributed computing, such as the “renaming” problem of Attiya et. al., the “k-set agreement” problem of Chaudhuri, and a generalization of the approximate agreement problem. The field of computational geometry is concerned with the design and analysis of geometric algorithms. For such algorithms, correctness and efficiency proofs are constructed, or problems are proven to be hard when no correct and efficient algorithms exist. In order to be able to do this, several assumptions about the input data for geometric algorithms are made. One of them is that this data is correct, with absolute certainty and infinite precision. In practical applications, this is often not the case, and as a result the value of these theoretical guarantees may be questionable. If we want to supply geometric algorithms with theoretical guarantees that are actually observed in practice, we have to loosen our assumptions about the input data to a more realistic level. Depending on the application, we may be confident that each data point, for example, is not more than some value epsilon away from its given position. We can then construct algorithms that are guaranteed to be correct and efficient as long as the input satisfies this weaker assumption. Furthermore, we can analyse how the imprecision in the input influences the accuracy of the output. In this thesis, we present new algorithms for classical geometric problems, that explicitly take data imprecision into account. These algorithms cannot produce the real answers to those problems, but instead produce information about the possible values that the answers can have. In several cases, this can be done without adding any extra cost to the asymptotic running times of the classical solutions. In some cases, though, computing this information is significantly more costly than using classical algorithms, and in some cases we prove that indeed no efficient algorithms exist.