Tight bounds for finding degrees from the adjacency matrix
to find a vertex of outdegree k in a simple n vertex directed graph, Θ(n(n−k)) probes to its adjacency matrix are necessary and sufficient in the worst case, and
to find a vertex of degree k in an n vertex undirected graph, Θ(n2) probes to its adjacency matrix are necessary and sufficient in the worst case.
to test whether T n has a vertex of outdegree k≤(n−1)/2, (2k+2)n+O(k log k) probes to the adjacency matrix are necessary in the worst case, and
to test whether T n has a vertex of outdegree k, for k=0 and 1, (2k+2)n+o(n) probes to the adjacency matrix are sufficient, by exhibiting two different algorithms, one for k=0 and k=1.
We conjecture that this lower bound for tournaments is optimal (up to lower order terms) for most k≤(n−1)/2.
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