[Paper Review] Isometric embeddings in trees and their use in the diameter problem

Guillaume Ducoffe|arXiv (Cornell University)|Oct 29, 2020

Complexity and Algorithms in Graphs82 references4 citations

TL;DR

This paper presents a near-optimal algorithm for computing the diameter of graphs and finite metric spaces that admit a quasi-isometric embedding in a system of $k$ trees, running in $\mathcal{O}(2^{\mathcal{O}(k\log k)}(N+n)^{1+o(1)})$ time, where $N$ is the total size of the trees. It further provides a faster $\mathcal{O}(N + kn)$-time algorithm for embeddings in the strong product of $k$ trees, and introduces a novel data structure for efficient eccentricity queries on trees.

ABSTRACT

We prove that given a discrete space with $n$ points which is either embedded in a system of $k$ trees, or the Cartesian product of $k$ trees, we can compute all eccentricities in ${\cal O}(2^{{\cal O}(k\log{k})}(N+n)^{1+o(1)})$ time, where $N$ is the cumulative total order over all these $k$ trees. This is near optimal under the Strong Exponential-Time Hypothesis, even in the very special case of an $n$-vertex graph embedded in a system of $ω(\log{n})$ spanning trees. However, given such an embedding in the strong product of $k$ trees, there is a much faster ${\cal O}(N + kn)$-time algorithm for this problem. All our positive results can be turned into approximation algorithms for the graphs and finite spaces with a quasi isometric embedding in trees, if such embedding is given as input, where the approximation factor (resp., the approximation constant) depends on the distortion of the embedding (resp., of its stretch). The existence of embeddings in the Cartesian product of finitely many trees has been thoroughly investigated for cube-free median graphs. We give the first-known quasi linear-time algorithm for computing the diameter within this graph class. It does not require an embedding in a product of trees to be given as part of the input. On our way, being given an $n$-node tree $T$, we propose a data structure with ${\cal O}(n\log{n})$ pre-processing time in order to compute in ${\cal O}(k\log^2{n})$ time the eccentricity of any subset of $k$ nodes. We combine the latter technical contribution, of independent interest, with a recent distance-labeling scheme that was designed for cube-free median graphs.

Motivation & Objective

• To develop fast algorithms for computing the diameter in graphs and finite metric spaces that can be quasi-isometrically embedded in a system of trees.
• To design a data structure enabling efficient eccentricity queries for subsets of nodes in a tree, with $\mathcal{O}(n\log n)$ preprocessing and $\mathcal{O}(k\log^2 n)$ query time for $k$ nodes.
• To provide the first quasi-linear time algorithm for computing the diameter in cube-free median graphs without requiring an embedding as input.
• To establish tight conditional lower bounds under the Strong Exponential-Time Hypothesis (SETH), showing the near-optimality of the proposed algorithms.
• To extend results to general metric spaces with quasi-isometric embeddings in trees, providing approximation algorithms with distortion-dependent approximation factors.

Proposed method

• Leverages isometric and quasi-isometric embeddings of finite metric spaces into systems of $k$ trees or products of $k$ trees to reduce the diameter computation problem to tree-based computations.
• Introduces a dynamic programming approach on trees that computes eccentricities by propagating values upward and downward in the tree, maintaining four possible states per node to capture path extremities.
• Uses a recursive divide-and-conquer strategy on gated subgraphs (fibers) induced by the star of a centroid in median graphs, exploiting the convexity and gatedness of fibers.
• Applies a distance-labeling scheme for median graphs to enable efficient computation of distances and vertex weights in subgraphs, reducing the overall running time.
• Combines the above techniques with a centroid decomposition strategy in median graphs, ensuring that each recursive level processes at most half the vertices, leading to $\mathcal{O}(\log n)$ levels.
• Employs a modified BFS to compute gates and distances from vertices to fibers, updating vertex weights to reflect maximum eccentricity contributions from outside the fiber.

Experimental results

Research questions

• RQ1 Can the diameter of a graph or finite metric space with a quasi-isometric embedding in $k$ trees be computed faster than $\mathcal{O}(n^2)$ time?
• RQ2 What is the optimal time complexity for computing all eccentricities in a space embedded in a system of $k$ trees, and how does it depend on $k$?
• RQ3 Is there a near-optimal algorithm for diameter computation in cube-free median graphs without requiring an embedding as input?
• RQ4 How does the Strong Exponential-Time Hypothesis (SETH) constrain the time complexity of diameter computation in such tree-like spaces?
• RQ5 Can approximation algorithms be derived for spaces with quasi-isometric embeddings in trees, with approximation factors depending on the embedding's distortion or stretch?

Key findings

• The diameter of a finite metric space with $n$ points that is quasi-isometrically embedded in a system of $k$ trees can be computed in $\mathcal{O}(2^{\mathcal{O}(k\log k)}(N+n)^{1+o(1)})$ time, where $N$ is the total size of the trees.
• For embeddings in the strong product of $k$ trees, the diameter can be computed in $\mathcal{O}(N + kn)$ time, which is significantly faster than the general case.
• The paper presents the first quasi-linear time algorithm for computing the diameter in cube-free median graphs, running in $\mathcal{O}(n^{1+o(1)})$ time under mild conditions on the graph's dimension and degree.
• A data structure is proposed with $\mathcal{O}(n\log n)$ preprocessing time that supports $\mathcal{O}(k\log^2 n)$ time queries for the eccentricity of any $k$-node subset in a tree.
• The algorithm's time complexity is shown to be near-optimal under the Strong Exponential-Time Hypothesis (SETH), even for embeddings in $\omega(\log n)$ spanning trees.
• For graphs with a quasi-isometric embedding in trees, the paper provides approximation algorithms where the approximation factor depends on the embedding's distortion, and the approximation constant on its stretch.