86,059 research outputs found
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
Quasirandom Rumor Spreading: An Experimental Analysis
We empirically analyze two versions of the well-known "randomized rumor
spreading" protocol to disseminate a piece of information in networks. In the
classical model, in each round each informed node informs a random neighbor. In
the recently proposed quasirandom variant, each node has a (cyclic) list of its
neighbors. Once informed, it starts at a random position of the list, but from
then on informs its neighbors in the order of the list. While for sparse random
graphs a better performance of the quasirandom model could be proven, all other
results show that, independent of the structure of the lists, the same
asymptotic performance guarantees hold as for the classical model. In this
work, we compare the two models experimentally. This not only shows that the
quasirandom model generally is faster, but also that the runtime is more
concentrated around the mean. This is surprising given that much fewer random
bits are used in the quasirandom process. These advantages are also observed in
a lossy communication model, where each transmission does not reach its target
with a certain probability, and in an asynchronous model, where nodes send at
random times drawn from an exponential distribution. We also show that
typically the particular structure of the lists has little influence on the
efficiency.Comment: 14 pages, appeared in ALENEX'0
- …