(ℓ,k)(\ell,k)-Routing on Plane Grids

The packet routing problem plays an essential role in communication networks. It involves how to transfer data from some origins to some destinations within a reasonable amount of time. In the $(\ell,k)$-routing problem, each node can send at most $\ell$ packets and receive at most $k$ packets. Permutation routing is the particular case $\ell=k=1$. In the $r$-central routing problem, all nodes at distance at most $r$ from a fixed node $v$ want to send a packet to $v$. In this article we study the permutation routing, the $r$-central routing and the general $(\ell,k)$-routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We use the \emph{store-and-forward} $\Delta$-port model, and we consider both full and half-duplex networks. The main contributions are the following: \begin{itemize} \item[1.] Tight permutation routing algorithms on full-duplex hexagonal grids, and half duplex triangular and hexagonal grids. \item[2.] Tight $r$-central routing algorithms on triangular and hexagonal grids. \item[3.] Tight $(k,k)$-routing algorithms on square, triangular and hexagonal grids. \item[4.] Good approximation algorithms (in terms of running time) for $(\ell,k)$-routing on square, triangular and hexagonal grids, together with new lower bounds on the running time of any algorithm using shortest path routing. \end{itemize} \noindent All these algorithms are completely distributed, i.e. can be implemented independently at each node. Finally, we also formulate the $(\ell,k)$-routing problem as a \textsc{Weighted Edge Coloring} problem on bipartite graphs

A Distributed Algorithm for Bandwidth Allocation in Stable Ad Hoc Networks

We propose a distributed algorithm for allocating bandwidth in stable ad hoc networks. After having discussed the problem of bandwidth allocation in such networks, we define a sequence of feasible solutions to this problem. This sequence has the property to be an increasing sequence in terms of overall used bandwidth. After a theoretical analysis of the sequence, we design a distributed algorithm based on this sequence. We test our algorithm by simulations on different topologies like chains, rings, meshes and geometric random graphs.We compare our solutions with the optimal solution in terms of global bandwidth allocation that presents the smallest standard deviation and with the the fairest solution regarding to max-min fairness. The simulations show that the global used bandwidth is less than $25\%$ from optimality in the worst case and the standard deviation is the smallest of the three tested solutions

Pojednostavljeni račun sparivanja u poligrafovima

Matching polynomial and perfect matchings for fasciagraphs, rotagraphs and twisted rotagraphs are treated in the paper. Classical transfer matrix approach makes it possible to get recursions for matching polynomial and perfect matchings, but the order of the matrix grows exponentially in the number of the linking edges between monographs. Novel transfer matrices are introduced whose order is much lower than that in classical transfer matrices. The virtue of the method introduced is especially pronounced when two or more linking edges end in the same terminal vertex of a monograph. An example of a polyacene polygraph with extended pairings is given where a novel matrix has only 16 entries as compared to 65536 entries in the classical transfer matrix. However, all pairings are treated here on equal footing, but the method introduced can be applied to selected types of pairings of interest in chemistry.U radu se razmatraju polinomi sparivanja i savršena sparivanja u fascia- i rotagrafovima te izvijenim rotagrafovima. Iako klasični postupak transfer matrice omogućava izvođenje rekurzija za polinom sparivanja i savršena sparivanja, red ove matrice eksponencijalno raste s brojem veza me|u monografovima. Ovdje su uvedene nove transfer matrice čiji je red mnogo ni`i od onoga za klasične transfer matrice, i to posebice kada jedna ili više veza me|u monografovima završava u jednom te istom čvoru. Postupak je ilustriran na primjeru poliacenskih poligrafova gdje ovdje uvedena matrica ima samo 16 elemenata u usporedbi s 65536 elemenata klasične transfer matrice. Iako se ovdje uvedeni postupak primjenjuje istovremeno na sva moguća sparivanja u poligrafovima, on je otvoren za primjenu na odabrana sparivanja od posebnoga kemijskoga interesa

Analysis of the Mean Field Annealing Algorithm for Graph Colouring

We introduce the Multi State Bitstream Neuron. By replacing the stochastic activation function with stochastic weights the MSBSN is shown to approximate a Generalised Boltzmann Machine. Benchmarks show the algorithm performs as well as the Boltzmann algorithm whilst the MSBSN lends itself to a very compact and fast hardware implementation