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Heights of butterfly trees

John Peca-Medlin, Chenyang Zhong

Abstract

Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST -- a structure motivated by parallel data architectures -- corresponding to composite permutations formed via Kronecker or wreath products. Extending Devroye's result that the height $h_n$ of a random BST satisfies $h_n / \log n \to c^* \approx 4.311$, we show that block BSTs with $nm$ nodes and fixed external size $m$ satisfy $h_{n,m} / \log n \to c^* + h_m$ in distribution. We then study butterfly trees: BSTs with $N = 2^n$ nodes generated from permutations built using iterated Kronecker or wreath products. For simple butterfly trees (from iterated Kronecker products of $S_2$), we give a full distributional description showing polynomial height growth: $\mathbb{E} h_n^{\operatorname{B}} = Θ(N^α)$ with $α= \log_2(3/2) \approx 0.58496$. For nonsimple butterfly trees (from wreath products), we prove power-law bounds: $cN^α\cdot (1 + o(1)) \le \mathbb{E} h_n^{\operatorname{B}} \le dN^β\cdot (1 + o(1))$, with $β\approx 0.913189$.

Heights of butterfly trees

Abstract

Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST -- a structure motivated by parallel data architectures -- corresponding to composite permutations formed via Kronecker or wreath products. Extending Devroye's result that the height of a random BST satisfies , we show that block BSTs with nodes and fixed external size satisfy in distribution. We then study butterfly trees: BSTs with nodes generated from permutations built using iterated Kronecker or wreath products. For simple butterfly trees (from iterated Kronecker products of ), we give a full distributional description showing polynomial height growth: with . For nonsimple butterfly trees (from wreath products), we prove power-law bounds: , with .

Paper Structure

This paper contains 11 sections, 16 theorems, 72 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Future directions and extensions.
  3. Outline.
  4. Preliminaries and notation
  5. Random BSTs
  6. Butterfly permutations
  7. Heights of block BSTs
  8. Heights of butterfly trees
  9. Simple butterfly trees: TnB
  10. Nonsimple butterfly trees: TnB
  11. Discussion.

Key Result

Theorem 1

If $\pi \sim \operatorname{Unif}(S_n)$ and $h_n = h(\mathcal{T}(\pi))$, then $h_n/\log n$ converges in probability and in $L^p$ for $p \ge 1$ to $c^* \approx 4.311$ as $n \to \infty$, where $c^*$ denotes the unique solution to $x \log(2e/x) = 1$ for $x \ge 2$.

Figures (9)

  • Figure 1: Step-by-step construction of $\mathcal{T}(352416)$ next to $\mathcal{T}({\mathbf x})$
  • Figure 2: $\mathcal{T}(\pi)$ for $\pi \in S_{30}$, with $h(\mathcal{T}(\pi)) = 6$ and particular LIS (from nodes 12 to 30) and LDS (from nodes 25 to 3) dotted paths are shown over the BST in red and green, respectively.
  • Figure 3: $\mathcal{T}(216534)$ formed using $\mathcal{T}(132)$ with nodes replaced by copies of $\mathcal{T}(12)$ and $\mathcal{T}(21)$.
  • Figure 4: $\mathcal{T}(12 \otimes 2143)$ and $\mathcal{T}(21 \otimes 2143)$ formed from gluing two copies of $\mathcal{T}(2143)$ together with an extra edge (in blue) connecting the top-left edges or top-right paths of each parent BST to the root of the child BST.
  • Figure 5: Simple butterfly trees of minimal height with $N = 2^{10} = 1{,}024$ nodes
  • ...and 4 more figures

Theorems & Definitions (29)

  • Theorem 1: devroye1986note
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: Slutsky's theorem, slutsky1925stochastische
  • Theorem 6: Subgroup algorithm, DiSh87
  • Corollary 1
  • proof : Proof of \ref{['thm:1wr']}
  • Lemma 1
  • Corollary 2
  • ...and 19 more