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Date and Time: Wednesday, April 20, 2005, 12:15 pm Duration: This information is not available in the database Location: This information is not available in the database Speaker: Nick Wormald (Univ. of Waterloo) Birth control for giantsWe consider a problem of Dimitris Achlioptas: initially a graph G
has no edges.
In each round two edges of Kn are generated independently and uniformly at
random. We must select one of those edges and add it to G. Our object is
to avoid creating a giant component, for as long
as possible. This motivates the following problem. Fix any algorithm
that determines which edge we select. Let Gm denote the
graph after m rounds. Then G0,G1,... forms a random graph
process, that evolves from the empty graph to a graph with a giant
component and, of course, beyond. For the standard random graphs, the giant
component appears after about n/2 edges. For a general class of algorithms we
analyse this process and show the existence of a phase transition in
the size of
the largest component. For the original Achlioptas problem, we also
obtain lower
bounds on how long we can avoid creating a giant (asymptotically
almost surely),
of more than 0.8n edges. For the converse problem of trying to
create a giant, we
have upper bounds of less than 0.35n.
(Joint work with Joel Spencer)
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