Tight Approximation Bounds on a Simple Algorithm for Minimum Average Search Time in Trees

TL;DR

This work studies the EPT-sum, the minimum leaf-depth sum over all edge partition trees (EPTs) of a graph, which corresponds to the average search time in tree-like posets. It analyzes a simple balanced-cut algorithm that repeatedly cuts a most balanced edge to build an EPT, and proves a tight $1.5$-approximation for vertex-weighted trees, resolving a question posed by Cicalese et al. (2014). The core technique introduces an augmented tree $aug(T)$ with cost at most $1.5$ times the optimum and shows how to transform it into the balanced-cut EPT while preserving or lowering the EPT-sum, using a detailed case analysis. The results connect to clustering objectives (Dasgupta's objective) and yield a fast $O(n\log n)$ balanced-EPT construction, with implications for the practical efficiency of optimized search strategies in trees and open questions about unweighted-tree complexity.

Abstract

The graph invariant EPT-sum has cropped up in several unrelated fields in later years: As an objective function for hierarchical clustering, as a more fine-grained version of the classical edge ranking problem, and, specifically when the input is a vertex-weighted tree, as a measure of average/expected search length in a partially ordered set. The EPT-sum of a graph $G$ is defined as the minimum sum of the depth of every leaf in an edge partition tree (EPT), a rooted tree where leaves correspond to vertices in $G$ and internal nodes correspond to edges in $G$. A simple algorithm that approximates EPT-sum on trees is given by recursively choosing the most balanced edge in the input tree $G$ to build an EPT of $G$. Due to its fast runtime, this balanced cut algorithm can be used in practice, and has earlier been analysed to give a 1.62-approximation on trees. In this paper, we show that the balanced cut algorithm gives a 1.5-approximation of EPT-sum on trees, which amounts to a tight analysis and answers a question posed by Cicalese et al. in 2014.

Figures (8)

• Figure 1: An unweighted tree $G$ and an EPT $T$ of $G$. Adding up the depth of each leaf, one sees that $\mathsf{EPT\text{-}sum}(G,T) = 39$. This is not optimal for $G$; a better EPT can be made by making the edge $ef$ (which incidentally is also the most balanced edge in $G$) root.
• Figure 2: Case 1.
• Figure 3: Case 2.
• Figure 4: Case 3.
• Figure 5: Case 4.

Theorems & Definitions (17)

• Definition 1: EPT-sum
• Definition 2: EPT-sum, alternative def.
• Theorem 1
• proof
• Claim 1
• Claim 2
• Definition 3: Augmented tree
• Lemma 1
• proof
• Definition 4: Splitting