A Lower Bound on Cycle-Finding in Sparse Digraphs
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We consider the problem of finding a cycle in a sparse directed graph G that is promised to be far from acyclic, meaning that the smallest feedback arc set in G is large. We prove an information-theoretic lower bound, showing that for N-vertex graphs with constant outdegree any algorithm for this problem must make Ω̃(N^5/9) queries to an adjacency list representation of G. In the language of property testing, our result is an Ω̃(N^5/9) lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the Ω(√(N)) lower bound, implicit in Bender and Ron, which follows from a simple birthday paradox argument.
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