Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

Mittagsseminar Talk Information

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 giants

We 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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