We investigate random variables arising in occupancy problems, and show the variables to be negatively associated, that is, negatively dependent in a strong sense. Our proofs are based on the FKG correlation inequality, and they suggest a useful, general technique for proving negative dependence among random variables. We also show that in the special case of two binary random variables, the notions of negative correlation and negative association coincide.

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We review how to solve the all‐pairs shortest‐path problem in a nonnegatively weighted digraph with n vertices in expected time O(n2 log n). This bound is shown to hold with high probability for a wide class of probability distributions on nonnegatively weighted digraphs. We also prove that, for a large class of probability distributions, Ω(n log n) time is necessary with high probability to compute shortest‐path distances with respect to a single source. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 205–220 (1997)

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We review how to solve the all-pairs shortest-path problem in a nonnegatively Ž 2. weighted digraph with n vertices in expected time O n log n. This bound is shown to hold with high probability for a wide class of probability distributions on nonnegatively weighted Ž. digraphs. We also prove that, for a large class of probability distributions, ⍀ n log n time is necessary with high probability to compute shortest-path distances with respect to a single Ž .