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Title:
Random k-out subgraphs
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Abstract:
Random sampling a subgraph of a graph is an important algorithmic technique. Solving some problems on the (smaller) subgraph is naturally faster, and can give either a useful approximate answer, or sometimes even give a result that can be quickly modified to an exact solution, on the original graph.
We study the following simple sampling model and question about it:
Each vertex of an arbitrary simple graph on n vertices chooses k random incident edges.
What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph?
We prove that the answer is O(n/k), when k ≥ c log n, for some large enough c. We conjecture that the same holds for smaller values of k, possibly for any k ≥ 2. Such a result is best possible for any k ≥ 2.
As an application, we use this sampling result to obtain a one-way communication protocol with private randomness for finding a spanning forest of a graph in which each vertex sends only O( \sqrt(n) log n) bits to a referee.
Based on joint work with Jacob Holm, Valeria King, Mikkel Thorup and Uri Zwick.
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