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In this article, we introduce Mallows processes, defined to be continuous-time càdlàg processes with Mallows distributed marginals. We show that such processes exist and that they can be restricted to have certain natural properties. In particular, we prove that there exists regular Mallows processes, defined to have their inversions numbers Inv j (σ) = |{i ∈ [j − 1] : σ(i) > σ(j)}| be independent increasing stochastic processes with jumps of size 1. We further show that there exists a unique Markov process which is a regular Mallows process. Finally, we study properties of regular Mallows processes and show various results on the structure of these objects. Among others, we prove that the graph structure related to regular Mallows processes looks like an expanded hypercube where we stacked k hypercubes on the dimension k ∈ [n]; we also prove that the first jumping times of regular Mallows processes converge to a Poisson point process.

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Given a permutation σ$$ \sigma $$, its corresponding binary search tree is obtained by recursively inserting the values σfalse(1false),…,σfalse(nfalse)$$ \sigma (1),\dots, \sigma (n) $$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In this article, we study the height of binary search trees drawn from the record‐biased model of permutations whose probability measure on the set of permutations is proportional to θrecordfalse(σfalse)$$ {\theta}^{\mathrm{record}\left(\sigma \right)} $$, where recordfalse(σfalse)=false|false{i∈false[nfalse]:∀jσfalse(jfalse)false}false|$$ \mathrm{record}\left(\sigma \right)=\mid \left\{i\in \left[n\right]:\forall j\sigma (j)\right\}\mid $$. We show that the height of a binary search tree built from a record‐biased permutation of size n$$ n $$ with parameter θ$$ \theta $$ is of order false(1+oℙfalse(1false)false)maxfalse{c∗logn,0.3emθlogfalse(1+nfalse/θfalse)false}$$ \left(1+{o}_{\mathbb{P}}(1)\right)\max \left\{{c}^{\ast}\log n,\kern0.3em \theta \log \left(1+n/\theta \right)\right\} $$, hence extending previous results of Devroye on the height or random binary search trees.

We consider a family of local search algorithms for the minimum-weight spanning tree, indexed by a parameter ρ. One step of the local search corresponds to replacing a connected induced subgraph of the current candidate graph whose total weight is at most ρ by the minimum spanning tree (MST) on the same vertex set. Fix a non-negative random variable X, and consider this local search problem on the complete graph Kn with independent X-distributed edge weights. Under rather weak conditions on the distribution of X, we determine a threshold value ρ * such that the following holds. If the starting graph (the "initial candidate MST") is independent of the edge weights, then if ρ > ρ * local search can construct the MST with high probability (tending to 1 as n → ∞), whereas if ρ < ρ * it cannot with high probability.