The height of Mallows trees

Abstract

Random binary search trees are obtained by recursively inserting the elements $\mathit{\sigma }(1),\mathit{\sigma }(2),\dots ,\mathit{\sigma }(\mathit{n})$ of a uniformly random permutation σ of $[\mathit{n}]=\{1,\dots ,\mathit{n}\}$ into a binary search tree data structure. Devroye (J. Assoc. Comput. Mach. 33 (1986) 489–498) proved that the height of such trees is asymptotically of order ${\mathit{c}}^{\ast }log\mathit{n}$, where ${\mathit{c}}^{\ast }=4.311\dots$ is the unique solution of $\mathit{c}log((2\mathit{e})/\mathit{c})=1$ with $\mathit{c}\ge 2$. In this paper, we study the structure of binary search trees ${\mathit{T}}_{\mathit{n},\mathit{q}}$ built from Mallows permutations. A $Mallows(\mathit{q})$ permutation is a random permutation of $[\mathit{n}]=\{1,\dots ,\mathit{n}\}$ whose probability is proportional to ${\mathit{q}}^{Inv(\mathit{\sigma })}$, where $Inv(\mathit{\sigma })=|\{\mathit{i}<\mathit{j}:\mathit{\sigma }(\mathit{i})>\mathit{\sigma }(\mathit{j})\}|$. This model generalizes random binary search trees, since $Mallows(\mathit{q})$ permutations with $\mathit{q}=1$ are uniformly distributed. The laws of ${\mathit{T}}_{\mathit{n},\mathit{q}}$ and ${\mathit{T}}_{\mathit{n},{\mathit{q}}^{-1}}$ are related by a simple symmetry (switching the roles of the left and right children), so it suffices to restrict our attention to $\mathit{q}\le 1$.

We show that, for $\mathit{q}\in [0,1]$, the height of ${\mathit{T}}_{\mathit{n},\mathit{q}}$ is asymptotically $(1+\mathit{o}(1))({\mathit{c}}^{\ast }log\mathit{n}+\mathit{n}(1-\mathit{q}))$ in probability. This yields three regimes of behaviour for the height of ${\mathit{T}}_{\mathit{n},\mathit{q}}$, depending on whether $\mathit{n}(1-\mathit{q})/log\mathit{n}$ tends to zero, tends to infinity or remains bounded away from zero and infinity. In particular, when $\mathit{n}(1-\mathit{q})/log\mathit{n}$ tends to zero, the height of ${\mathit{T}}_{\mathit{n},\mathit{q}}$ is asymptotically of order ${\mathit{c}}^{\ast }log\mathit{n}$, like it is for random binary search trees. Finally, when $\mathit{n}(1-\mathit{q})/log\mathit{n}$ tends to infinity, we prove stronger tail bounds and distributional limit theorems for the height of ${\mathit{T}}_{\mathit{n},\mathit{q}}$.