From the early days in his career, Sid Yakowitz showed interest in noisy function optimization. He realized the universality of random search as an optimization paradigm, and was particularly interested in the minimization of functions Q without making assumptions on the form of Q. Especially the noisy optimization problem appealed to him, as exact computations of Q come often at a tremendous c...

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. Notoriously hard to compute, the exact fractions ck had been determined for k ≤ 3 only. We computed c4 a...

In a randomly grown binary search tree (BST) of size n, any xed pattern occurs with a frequency that is on average proportional to n. Deviations from the average case are highly unlikely and well quantiied by a Gaussian law. Trees with forbidden patterns occur with an exponentially small probability that is characterized in terms of Bessel functions. The results obtained extend to BSTs a type o...

(a) For the cases n = 1, 2, 3 we observe that ∆n = 0, 1, 2, respectively, irrespective of the structure of the random search tree. We conclude that δ1 = 0, δ2 = 1, δ3 = 2. For the case n = 4 we observe that either ∆4 = 2 or ∆4 = 3, depending on the structure of the random search tree (∆4 > 2 because we already have ∆3 > 2 and the longest distance cannot decrease by adding more nodes, ∆4 6 3 bec...

A decentralized search algorithm is a method of routing on a random graph that uses only limited, local, information about the realization of the graph. In some random graph models it is possible to define such algorithms which produce short paths when routing from any vertex to any other, while for others it is not. We consider random graphs with random costs assigned to the edges. In this sit...