Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quasilinear-time eccentricities computation, and more, on median graphs

Pierre Bergé, Guillaume Ducoffe, Michel Habib

TL;DR

A combinatorial algorithm which computes for any median graph G, in quasilinear time O(n log^4 n), vertex-labels of size O(log^3 n) such that any distance of G can be retrieved in time O(log^4 n) thanks to these labels.

Abstract

Computing the diameter, and more generally, all eccentricities of an undirected graph is an important problem in algorithmic graph theory and the challenge is to identify graph classes for which their computation can be achieved in subquadratic time. Using a new recursive scheme based on the structural properties of median graphs, we provide a quasilinear-time algorithm to determine all eccentricities for this well-known family of graphs. Our recursive technique manages specifically balanced and unbalanced parts of the $Θ$-class decomposition of median graphs. The exact running time of our algorithm is O(n log^4 n). This outcome not only answers a question asked by B{é}n{é}teau et al. (2020) but also greatly improves a recent result which presents a combinatorial algorithm running in time O(n^1.6408 log^{O(1)} n) for the same problem.Furthermore we also propose a distance oracle for median graphs with both polylogarithmic size and query time. Speaking formally, we provide a combinatorial algorithm which computes for any median graph G, in quasilinear time O(n log^4 n), vertex-labels of size O(log^3 n) such that any distance of G can be retrieved in time O(log^4 n) thanks to these labels.

Quasilinear-time eccentricities computation, and more, on median graphs

TL;DR

A combinatorial algorithm which computes for any median graph G, in quasilinear time O(n log^4 n), vertex-labels of size O(log^3 n) such that any distance of G can be retrieved in time O(log^4 n) thanks to these labels.

Abstract

Computing the diameter, and more generally, all eccentricities of an undirected graph is an important problem in algorithmic graph theory and the challenge is to identify graph classes for which their computation can be achieved in subquadratic time. Using a new recursive scheme based on the structural properties of median graphs, we provide a quasilinear-time algorithm to determine all eccentricities for this well-known family of graphs. Our recursive technique manages specifically balanced and unbalanced parts of the -class decomposition of median graphs. The exact running time of our algorithm is O(n log^4 n). This outcome not only answers a question asked by B{é}n{é}teau et al. (2020) but also greatly improves a recent result which presents a combinatorial algorithm running in time O(n^1.6408 log^{O(1)} n) for the same problem.Furthermore we also propose a distance oracle for median graphs with both polylogarithmic size and query time. Speaking formally, we provide a combinatorial algorithm which computes for any median graph G, in quasilinear time O(n log^4 n), vertex-labels of size O(log^3 n) such that any distance of G can be retrieved in time O(log^4 n) thanks to these labels.

Paper Structure

This paper contains 16 sections, 37 theorems, 22 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Presentation of the contributions: ideas and impact
  2. Preliminaries
  3. Definitions and properties from metric graph theory
  4. Median graphs and $\Theta$-classes
  5. Orthogonal $\Theta$-classes, POFs and ladder sets
  6. Median set and majority rule
  7. Exploiting halfspaces for the computation of eccentricities
  8. Balanced $\Theta$-classes
  9. Retrieving all eccentricities thanks to $\Theta$-classes
  10. Computation of eccentricities on median graphs without balanced $\Theta$-classes
  11. Slice decomposition
  12. Peeling the slices
  13. Summary and analysis for the eccentricities computation
  14. Distance oracle
  15. Recursive scheme exploiting gated halfspaces
  16. ...and 1 more sections

Key Result

Theorem 1

There exists a combinatorial algorithm which computes all weighted eccentricities of a weighted median graph $(G,\omega)$ in quasilinear time $O(n\log^4(n))$.

Figures (8)

  • Figure 1: Example of median graph $G$ with $d=3$. Two $\Theta$-classes are highlighted.
  • Figure 2: The $v_0$-orientation of some median graph $G$ and some of its $\Theta$-classes. For example, $\mathcal{E}^-(u) = \{ E_1,E_2,E_3 \}$.
  • Figure 3: The largest weighted distance from $u" \in H_i"$ to some vertex $x \in H_i'$ is given by the (unweighted) distance from $u"$ to its gate $v = g_{H_i'}(u")$ in addition with the weighted eccentricity of $v$: $\hbox{ecc}\left(v ~\vert ~ (H_i',\omega)\right) = d(v,x)$.
  • Figure 4: Illustration of the slice decomposition of some median graph $G$ with $L(G) = \{E_1,E_2\}$. For example, $V(G_1) = V(S_1) \cup V(G_2)$.
  • Figure 5: Retrieving the eccentricities of $x \in V(S_i)$ in graph $G_i$: example with $\ell = 2$ and $i = 1$. Either the largest weighted distance from $x$ is with a vertex of $S_i$, or with one in the other side $V(G_{i+1})$.
  • ...and 3 more figures

Theorems & Definitions (75)

  • Theorem 1
  • Theorem 2
  • Definition 1: Convex and gated sets
  • Lemma 1: Intersection of gated subgraphs BaCh08
  • Definition 2: Fibers BaCh08
  • Definition 3: Open fibers
  • Lemma 2: ChLaRa18ChLaRa19
  • Definition 4: Median graph
  • Definition 5: Dimension $d$
  • Lemma 3: BaCh08BeChChVa20
  • ...and 65 more