Infinite Networks, Halting and Local Algorithms

The immediate past has witnessed an increased amount of interest in local algorithms, i.e., constant time distributed algorithms. In a recent survey of the topic (Suomela, ACM Computing Surveys, 2013), it is argued that local algorithms provide a natural framework that could be used in order to theoretically control infinite networks in finite time. We study a comprehensive collection of distributed computing models and prove that if infinite networks are included in the class of structures investigated, then every universally halting distributed algorithm is in fact a local algorithm. To contrast this result, we show that if only finite networks are allowed, then even very weak distributed computing models can define nonlocal algorithms that halt everywhere. The investigations in this article continue the studies in the intersection of logic and distributed computing initiated in (Hella et al., PODC 2012) and (Kuusisto, CSL 2013).Comment: In Proceedings GandALF 2014, arXiv:1408.556

Almost Settling the Hardness of Noncommutative Determinant

In this paper, we study the complexity of computing the determinant of a matrix over a non-commutative algebra. In particular, we ask the question, "over which algebras, is the determinant easier to compute than the permanent?" Towards resolving this question, we show the following hardness and easiness of noncommutative determinant computation. * [Hardness] Computing the determinant of an n \times n matrix whose entries are themselves 2 \times 2 matrices over a field is as hard as computing the permanent over the field. This extends the recent result of Arvind and Srinivasan, who proved a similar result which however required the entries to be of linear dimension. * [Easiness] Determinant of an n \times n matrix whose entries are themselves d \times d upper triangular matrices can be computed in poly(n^d) time. Combining the above with the decomposition theorem of finite dimensional algebras (in particular exploiting the simple structure of 2 \times 2 matrix algebras), we can extend the above hardness and easiness statements to more general algebras as follows. Let A be a finite dimensional algebra over a finite field with radical R(A). * [Hardness] If the quotient A/R(A) is non-commutative, then computing the determinant over the algebra A is as hard as computing the permanent. * [Easiness] If the quotient A/R(A) is commutative and furthermore, R(A) has nilpotency index d (i.e., the smallest d such that R(A)d = 0), then there exists a poly(n^d)-time algorithm that computes determinants over the algebra A. In particular, for any constant dimensional algebra A over a finite field, since the nilpotency index of R(A) is at most a constant, we have the following dichotomy theorem: if A/R(A) is commutative, then efficient determinant computation is feasible and otherwise determinant is as hard as permanent.Comment: 20 pages, 3 figure

On Spectral Properties of Finite Population Processor Shared Queues

We consider sojourn or response times in processor-shared queues that have a finite population of potential users. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite population models where the total population is $N\gg 1$. Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Hermite equation. The dominant eigenvalue leads to the tail of a customer's sojourn time distribution.Comment: 28 pages, 7 figures and 5 table

Computing Hilbert Class Polynomials

We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing $H_D$, and we show that all methods have comparable run times