Algorithms for deflating regular and singular spectral characteristics of m χ η polynomial matrix pencils of degree s ^ 1 are treated. To deflate regular spectral characteristics, algorithms extending known deflation techniques for matrices are proposed (deflation is applied to general type pencils, new types of spectral transformations are introduced, and block deflation to convert part of the pencil spectrum into an arbitrary point of the complex plane is suggested). To deflate singular spectral characteristics (polynomial solutions), we present algorithms utilizing along with unitary matrices some simple unimodular and regular matrices differing from unitary ones in one or in several columns.Deflation algorithms make it possible to update known spectral characteristics so as to provide more stable and fast numerical computation of spectral characteristics not computed as yet.

This is the sequel to the papers by V. B. Khazanov and V. N. Kublanovskaya 'Spectral problems for matrix pencils. Methods and algorithms. I and IT.A review of methods and algorithms for the solution of spectral problems for singular matrix pencils with linear and nonlinear (polynomial) dependence on the spectral parameter is presented. The methods are constructed using new principles which differ from those presented in Parts I and II of the paper. Here, we apply, as a rule, equivalent transformations with very simple unimodular matrices constructed using elementary plane rotation and reflection matrices. This property will be exploited when selecting a permutation matrix in the ZW decomposition algorithm which ensures the validity of inequalities (1.3).

No abstract

No abstract

The Congested Clique is a distributed-computing model for single-hop networks with restricted bandwidth that has been very intensively studied recently. It models a network by an n-vertex graph in which any pair of vertices can communicate one with another by transmitting O(log n) bits in each round. Various problems have been studied in this setting, but for some of them the best-known results are those for general networks. For other problems, the results for Congested Cliques are better than on general networks, but still incure significant dependency on the number of vertices n. Hence the performance of these algorithms may become poor on large cliques, even though their diameter is just 1. In this paper we devise significantly improved algorithms for various symmetry-breaking problems, such as forests-decompositions, vertex-colorings, and maximal independent set.We analyze the running time of our algorithms as a function of the arboricity a of a clique subgraph that is given as input. The arboricity is always smaller than the number of vertices n in the subgraph, and for many families of graphs it is significantly smaller. In particular, trees, planar graphs, graphs with constant genus, and many other graphs have bounded arboricity, but unbounded size. We obtain O(a)-forest-decomposition algorithm with O(log a) time that improves the previously-known O(log n) time, O(a 2+ǫ )-coloring in O(log * n) time that improves upon an O(log n)-time algorithm, O(a)-coloring in O(a ǫ )-time that improves upon several previous algorithms, and a maximal independent set algorithm with O( √ a) time that improves at least quadratically upon the state-of-the-art for small and moderate values of a.Those results are achieved using several techniques. First, we produce a forest decomposition with a helpful structure called H-partition within O(log a) rounds. In general graphs this structure requires Θ(log n) time, but in Congested Cliques we are able to compute it faster. We employ this structure in conjunction with partitioning techniques that allow us to solve various symmetry-breaking problems efficiently. . This research has been supported by ISF grant 724/15 and Open University of Israel research fund.networks, a more realistic model has been studied. This is the CONGEST model that is similar to the LOCAL model, except that each edge is only allowed to transmit O(log n) bits per round. An important type of CONGEST networks that has been intensively studied recently is the Congested Clique model. It represents single-hop networks with limited bandwidth. Although the diameter of such networks is 1, which would make any problem on such graphs trivial in the LOCAL model, in the Congested Cliques various tasks become very challenging. Note that the Congested Clique is equivalent to a general n-vertex graph in which any pair of vertices (not necessarily neighbors) can exchange messages of size O(log n) in each round. Such a general graph corresponds to a subgraph of an n-clique. The subgraph constitutes the input, while the clique...