The complexity of finding certain trees in tournaments
A tournament Tn is an orientation of the complete graph on n vertices. We continue the algorithmic study initiated by Hell and Rosenfeld of recognizing various directed trees in tournaments. Hell and Rosenfeld considered orientations of paths, and showed the existence of oriented paths on n vertices finding which in Tn requires Θ(n lgα n) “edge probes” where α ≤ 1 is any fixed non-negative constant. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree). In particular, we show, by proving upper and lower bounds, that the complexity of finding a vertex of outdegree k(n−1)/2) in Tn is Θ(nk). We also establish an Ω(n2) lower bound for finding a vertex of maximum outdegree in Tn. These bounds are in sharp contrast to the O(n) bounds for selection in the case of transitive tournaments.
Unable to display preview. Download preview PDF.