With random competition we propose a method for parallelizing backtracking search. We can prove high efficiency of random competition on highly parallel architectures with thousands of processors. This method is suited for all kinds of distributed memory architectures, particularly for large networks of high performance workstations since no communication between the processors is necessary dur...

Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotica...

We discuss search strategies for finding sources of particles transported in a random environment and detected by the searcher(s). The mixing of the particles in the environment is supposed to be strong, so that strategies based on concentration-gradient ascent are not viable. These dilute conditions are common in natural environments typical of searches performed by insects and birds. The spar...

A local search algorithm solving an NP-complete optimisation problem can be viewed as a stochastic process moving in an 'energy landscape' towards eventually finding an optimal solution. For the random 3-satisfiability problem, the heuristic of focusing the local moves on the presently unsatisfied clauses is known to be very effective: the time to solution has been observed to grow only linearl...

Shenvi, Kempe and Whaley’s quantum random-walk search (SKW) algorithm [2003 Phys. Rev. A 67 052307] is known to require O( √ N) number of oracle queries to find the marked element, where N is the size of the search space. This scaling is thought to be the best achievable on a quantum computer. We prove that the final state of the quantum walk in the SKW algorithm yields the nearest neighbours o...

Random forests have emerged as one of the most commonly used nonparametric statistical methods in many scientific areas, particularly in analysis of high throughput genomic data. A general practice in using random forests is to generate a sufficiently large number of trees, although it is subjective as to how large is sufficient. Furthermore, random forests are viewed as "black-box" because of ...

This entry contains proofs for the textbook results about the distributions of the height and internal path length of random binary search trees (BSTs), i. e. BSTs that are formed by taking an empty BST and inserting elements from a fixed set in random order. In particular, we prove a logarithmic upper bound on the expected height and the Θ(n log n) closed-form solution for the expected interna...